BOND ANALYSIS AND VALUATION
CEFA 2003/2004
LECTURE NOTES
Mats Hansson
Svenska handelshögskolan
Institutionen för finansiell ekonomi och ekonomisk statistik
i
Contents
1.
FIXED INCOME SECURITIES - AN INTRODUCTION
TU
UT
TU
1.1.
What’s so special about fixed income securities?
TU
TU
UT
TU
UT
TU
UT
TU
1.2.
UT
1
UT
The risks of investing in debt and why everybody always talks about the yield
TU
1.3.
The money and bond markets
TU
1.4.
UT
TU
TU
UT
TU
TU
UT
TU
TU
UT
TU
5
5
6
7
UT
UT
UT
UT
BOND AND INTEREST RATE MATHEMATICS
TU
UT
TU
8
UT
2.1.
The frequency of compounding
2.1.1. Effective money market yields
TU
UT
TU
TU
8
10
UT
UT
TU
UT
2.2.
Building blocks: zeros and forwards
2.2.1. Zero-coupon bonds
2.2.2. Forward rates
TU
UT
TU
11
11
13
UT
TU
UT
TU
TU
UT
TU
2
4
UT
1.5.
Factors affecting the level of the nominal return
1.5.1. The real return
1.5.2. The inflation rate
1.5.3. The risk premium
TU
UT
3
UT
Market size
TU
2.
1
UT
UT
UT
2.3.
Zero-coupon pricing of coupon bonds
2.3.1. The coupon rate
2.3.2. The present value of a coupon bond
2.3.3. Yield to maturity for a coupon bond
2.3.4. The par yield
15
15
15
17
20
2.4.
The yield: common misconceptions
2.4.1. The yield is not the return
2.4.2. Yields are not additive
21
21
22
TU
UT
TU
UT
TU
UT
TU
TU
UT
TU
TU
UT
TU
TU
UT
TU
TU
UT
UT
UT
TU
UT
UT
TU
TU
UT
TU
2.5.
UT
UT
From coupon bonds to zeros: bootstrapping
TU
UT
TU
2.6.
UT
23
UT
Spot and forward rates with semi-annual compounding
TU
3.
UT
TU
TU
TU
UT
25
UT
DAY COUNTS AND ACCRUED INTEREST
UT
TU
3.1.
27
UT
Day count basis
TU
TU
UT
TU
UT
TU
UT
TU
3.2.
UT
The money market
TU
3.3.
27
UT
28
UT
Zero-coupon bonds: annual compounding
TU
3.4.
Zero-coupon bonds: semi-annual compounding
TU
29
UT
30
UT
3.5.
Coupon bonds
3.5.1. Dirty prices and clean prices
3.5.2. Behavior of dirty and clean prices over time: convergence towards par
TU
UT
TU
UT
TU
TU
UT
TU
UT
TU
UT
UT
30
31
32
ii
4.
MEASURING INTEREST RATE RISK: DURATION AND CONVEXITY 34
TU
UT
TU
4.1.
UT
The yield-price relationship for bonds
TU
UT
TU
4.2.
Duration
4.2.1. Macaulay duration
TU
UT
TU
TU
35
36
UT
UT
TU
UT
4.3.
Modified duration and PVBP
4.3.1. The duration of a bond through time
4.3.2. Key rate duration
TU
34
UT
UT
TU
37
38
39
UT
TU
UT
TU
TU
UT
TU
UT
UT
4.4.
Convexity
4.4.1. Duration matching and the value of convexity
40
42
4.5.
44
TU
UT
TU
TU
UT
UT
TU
UT
Bond portfolio duration and convexity
TU
UT
TU
UT
4.6.
Butterfly trades: A critical assessment of yield, convexity and duration
4.6.1. Weighting a butterfly
4.6.2. A critical assessment of yield, convexity, and duration
TU
UT
TU
45
45
47
UT
TU
UT
TU
TU
UT
TU
UT
UT
5. APPLICATIONS OF BOND MATHEMATICS I: FRA:S AND BOND
FUTURES
TU
UT
TU
49
UT
5.1.
Forward Rate Agreements
TU
UT
TU
49
UT
5.2.
Bond futures
5.2.1. Futures pricing: The general approach
5.2.2. Repo transactions in the bond cash and futures markets
5.2.3. Coupon payments
5.2.4. Notional bonds and delivery options
5.2.5. Futures pricing using quoted prices and accrued interest
TU
UT
6.
TU
UT
TU
TU
UT
TU
TU
UT
TU
TU
UT
TU
TU
UT
TU
UT
UT
UT
UT
UT
APPLICATIONS OF BOND MATHEMATICS II: SWAP CONTRACTS 58
TU
UT
TU
UT
6.1.
Interest rate swaps
6.1.1. The swap rate
6.1.2. The swap rate and FRA-rates
6.1.3. Interest rate swap valuation
TU
UT
TU
58
59
60
61
UT
TU
UT
TU
TU
UT
TU
TU
UT
TU
UT
UT
UT
6.2.
Currency swaps
6.2.1. Currency swap rates and currency forward rates
6.2.2. Currency swap valuation
TU
UT
7.
TU
51
51
52
53
54
55
UT
TU
TU
UT
TU
UT
TU
TU
UT
TU
UT
UT
PRICING CREDIT RISK
UT
7.1.
TU
65
UT
Credit ratings
TU
TU
UT
TU
UT
TU
7.2.
UT
TU
7.3.
TU
62
63
64
65
UT
The traditional approach to pricing credit risk
66
Using option theory to price credit risk
67
UT
UT
7.4.
Default probabilities, rating transitions, recovery rates and how to use them to estimate
69
bond returns
TU
UT
TU
UT
7.5.
TU
UT
Selected empirical results on spreads
TU
UT
72
1
1.
Fixed income securities - An introduction
Debt instruments or fixed income securities are financial instruments that commit the
issuer to a series of fixed payments (for example a series of coupons and principal).
Examples are treasury bonds and bills, corporate bonds and loans, certificates of deposit,
and interest rate and currency swaps. The issuer of these securities promise a certain cash
flow at certain specified times in the future, hence the definition “fixed income”. Also,
fixed income securities typically have a finite maturity.
1.1.
What’s so special about fixed income securities?
A fixed income security follows the same basic principles of valuation as for e.g. stocks:
future cash flows are discounted to present time. Fixed income securities, however, have
a number of special features that make a separate treatment of these instruments
warranted:
1. Fixed cash flows. Most bonds pay fixed interest (altough floating rate notes are also
common), which is also paid on specified dates. Thus, cash flows are known both
with respect to size and maturity, except in the case of default.
2. Finite and known maturity. Except for some rare cases (perpetuities), fixed income
securities have a limited maturity, which is known in advance. Item 1) and 2) makes it
possible to construct special risk measures for bonds, like duration and convexity.
3. Only downside with respect to promised cash flow. The cash flow received from
a straight bond can never exceed its promised coupons and face value.
4. A wide variety of instruments are available. A wide range of maturities (1 day to
30 years or more), cash flow structures (zero-coupon bonds, coupon bonds, annuities
etc.), issuers (corporations, governments, municipalities etc.), and derivatives (swaps,
bond and money market futures, forwards, options etc.). Since a bond or a loan is a
legal contract between borrower and lender, the payoff and risk structure (payment
schedule, covenants etc.) of the bond/loan is determined in this contract, and hence
there is no limit to where product development can go in the debt markets.
5. Lower risks, lower returns. The financial risk associated with fixed income is lower
than with equities. This means lower expected and, on average, lower realised returns.
This in turn calls for more precision in the pricing process, since if returns are low,
every basis point counts. Since upside potential is low, paying too much (mispricing)
usually means that the investor’s return is ruined for good, while on the stock
markets one can always hope for a more substantial increase in value.
6. The term structure of interest rates is used for valuation. When valuing stocks, a
single discount rate is used to discount all cash flows. In doing this we assume that all
cash flows are equally risky and that the time value of money is the same for all
maturities. The wide diversity of instruments available on fixed income markets
makes a more precise valuation of debt instruments possible. In the ideal case, we
can find information on the interest rates for many different maturities, making it
possible to value each cash flow of a bond using a separate interest rate that reflects
the risk of that maturity.
7. Arbitrage. The variety of instruments with very low credit risk (interbank market) or
in practice no credit risk (Treasury markets) makes arbitrage and arbitrage pricing
possible and links prices of instruments to each other.
2
8. Volatility is a function of time. As we will later see, the volatility of a fixed income
security depends on 1) changes in the interest rate level, and 2) the duration of the
bond. Thus, even if interest rate volatility is constant, the volatility of the bond will
decrease with time.
9. Changes in the interest rate level is the most important source of risk. The
nature of debt instruments implies that the valuation process is to a large extent
concerned with the time value of money. Time value of money is closely related to
the level of interest rates, and hence debt instruments could also be labeled interest
sensitive assets.
10. Institutional details. The practice of expressing prices as interest rates (yields), or
”clean prices”, different day count conventions and compounding frequencies etc.
1.2.
•
•
•
The risks of investing in debt and why everybody always
talks about the yield
the most common type of interest rate payment is a fixed coupon
since the cash flows are fixed, there is less upside than in stocks, and all the changes
in price will come from the change in the discount rate, or the yield
hence, much of the fixed income investment analysis is centered around the yield
and the yield spread
SPREAD
(Risk premium)
EXCESS RETURN
Transaction cost
Covenants
Credit risk
YIELD
BENCHMARK
YIELD
(Default risk
free yield)
Seniority
Real interest rate
Interest rate risk
Inflation risk
•
•
•
many of the risks contributing to total yield are difficult to measure and price
excess return is the expected excess return from investing in corporates over
Treasuries after taking into account all the risks incorporated in the spread that can
materialise
excess return must be must be positive in the long run, otherwise a risk-averse
investor will not invest in corporates
3
•
up to date there exists no pricing model (a la CAPM) flexible enough to incorporate
all the terms and covenants in the debt contract affecting return and risk, yet general
enough to lend itself to practical use
1.3.
The money and bond markets
Fixed income securities can be classified in as many sectors as one likes, but the most
common categories are by maturity:
1. Short term securities. Maturity up to 1 year.
2. Long term securities. Maturity of more than 1 year.
This classification comes from the similarity of the pricng technique within one category,
short term debt usually pays no interest (zero-coupon bonds, discount bonds), so there is
only one cash flow at maturity, and simple interest is used when discounting. Long term
bonds usually pay interest, and compound interest is used for discounting.
One could also classify securities into:
1. Default risk free securities. These securities are issued by governments
(Treasuries) of developed countries, and are considered to be in practice free of
default risk.
2. Securities with default risk. Corporate bonds and for example emerging market
sovereign debt are not free of default risk.
• securities with a maturity of maximum 1 year are frequently referred to as “money
market” instruments
• typical money market instruments include:
1.
2.
3.
4.
short term deposits (nontradable)
certificates of deposit (interbank market)
commercial paper (corporate sector)
Treasury bills (government sector)
• typical bond market instruments include:
1. Government bonds (also called Treasury or Sovereign bonds)
2. Corporate bonds
3. Mortgage bonds (securitised mortgage debt)
The government sector is usually more liquid than the corporate sector due to larger
markets, and in many countries the bulk of corporate borrowing is still mainly routed
through bank loans, altough this has been changing in Europe since the Euro. Loans of
large corporations are usually syndicated loans, a group of banks divides the loan
between themselves for diversification. Most bonds and loans are "bullets", where
interest is paid annually or semiannually, and the principal is paid back at maturity. In
many countries, a substantial part of the government bonds are so called benchmark
bonds (or serial bonds), for example the Finnish benchmarks are:
4
Bond
Maturity
CPN
Issue price
%
Serial bond 2003/I
Serial bond 2001/I
Serial
bond
2003/II
Serial
bond
1998/II
Serial bond 2000/I
Serial
2002/II
bond
Current
price
Yield%
10.11.0
3
3.032
3.495
3.754
Amount
EUR
(m)
29.10.03
4.7.2006
4.7.2007
4.7.2008
2.750
5.000
3.000
99.714
100.021
99.552
10.11.0
3
99.040
105.040
96.830
25.4.200
9
23.2.201
1
4.7.2013
5.000
99.500
105.170
3.922
5 753
5.750
99.110
109.350
4.227
5 673
5.375
99.666
106.910
4.473
6 000
6 500
6 231
5 999
36 156
1.4.
Market size
• many bonds are listed at an exchange, but trading is (so far) mostly OTC
• global markets by country of issuer:
By sector: 1) Government and agency, 2) Corporate and
financial institutions
20000
USD billions
15000
10000
5000
0
USA
Germany
France
Japan
Finland
Government
9697
867
791
5316
77
Corporate
8805
2190
887
1606
41
Total
18502
3057
1678
6922
118
• risk management and the desire to explore cost-effective borrowing through swaps
has lead to enromous global fixed income derivatives markets:
5
By sector: 1) Foreign exchange, 2) Interest rate, 3) Equity linked, and
market (OTC or Exchanges)
USD billions
100000
80000
60000
40000
20000
0
OTC FX
OTC
Interest
OTC
Equity
Ex. FX
Forwards
10723
8792
364
72
13444
422
Swaps
4509
79161
0
0
0
0
Options
3238
13746
1944
33
22024
2307
Total
18470
101699
2308
105
35468
2729
1.5.
Ex. Interest Ex. Equity
Factors affecting the level of the nominal return
Why are interest rates not equal for all time-periods, and why is the term structure of
interest rates usually (but not always) upward sloping? To answer these questions we
need to have a look at what factors affect required returns, and why these factors need
not be equally large for all time periods.
The return for any asset can be decomposed into three factors:
1. the required real return
2. expected inflation over the investment horizon
3. a risk premium
• the real return and expected inflation affect the returns on all assets in an economy
• the magnitude of the risk premium is asset specific
1.5.1. The real return
• first suppose there is no inflation and the investment is risk-free: the return consists
solely of the investors perception of time value of money, or real return
• thus, the real return says how much the investor wants his purchasing power to
increase when investing
• investing is delaying consumption to the future, for doing this the investor
requires a compensation
• if this compensation is equally large for each time period (e.g. each year), the yield
curve will be flat (compensation proportional to time)
• in the simplest setting, the level of the real return depends on money supply and
demand:
6
1. the supply depends on investors willingness to postpone consumption
2. the demand depends on opportunities for productive investment
• for example, suppose that investment opportunities improve and firms are willing to
invest more at any interest rate level
• then, interest rates must rise to induce investors to save more ⇒ investors require a
higher compensation to postpone a larger amount of their consumption
1.5.2. The inflation rate
• if there is inflation in the economy, investors will require a premium over the real rate
equal to the expected inflation rate for the investment horizon
• if inflation is constant, this will result in a still flat, but higher yield curve
• thus, still ignoring risk premiums, the required nominal interest rate (rn) on a riskless
security depends on: 1) the real required return (rr). and 2) the expected inflation
[E(i)] is approximately:
B
B
B
B
rn ≈ rr + E (i )
Example: An investor is investing for 1 year, and wants the purchasing power of his
money to increase with 3% over the next year. This is his required real return, a
compensation for postponing his consumption 1 year. Also, the investor expects the
inflation to be 2% from today to 1-year ahead. Thus, his nominal required return is
U
U
rn ≈ rr + E (i ) = 0.03 + 0.02 = 0.05 = 5%
•
•
•
•
•
•
•
•
•
•
inflation and real returns need not be constant over time
if investors expect inflation and/or real rates to increase, longer rates should be higher
if investors expect inflation and/or real rates to decline, longer rates should be lower
but: even if the level of inflation is not expected to rise, the level of future inflation is
still uncertain, investors may require a premium for longer rates due to inflation risk
⇒ your real return is uncertain ⇒ require risk premium for risky real return
if, in addition, the level of real returns is uncertain ⇒ further risk of real return risk
thus we may have premiums for both inflation risk, and real return risk
investors require a premium for investing for long maturities
implies an upward sloping yield curve (despite small declines inflation/real rates)
downward sloping curves only if future inflation/real rates substantially lower
flat curves occur only if future yields expected to decline
7
1.5.3. The risk premium
• a security that has no default risk is considered riskless with respect to default risk
• for example government securities are considered free of default risk (or at least have
the lowest possible level of default risk)
• a risk premium must be added on top of required real returns and expected inflation
for issuers that have default risk
• but even in the treasury markets we typically observe that interest rates (for example
yields) increase with maturity
• interest rate risk (typically measured by duration) increases with maturity
8
2.
Bond and interest rate mathematics
• in this chapter we assume there is no credit risk, such that money can be moved back
and forth in time without caring about the riskiness of future payments
• nobody has ever claimed that bond calculations are fun or interesting, but given the
size of the market and the amount of money potentially lost because one simply didn’t
know, one cant’t ignore the subject
• let’s have a look at a some numbers we can use to describe a coupon bond:
Face value:
Present value:
Maturity:
Coupon rate:
Yield to maturity:
Par yield:
Zero-coupon rate (3 yrs.):
1000.00
1150.62
3.00 years
10.00% (annual)
4.5188%
4.5344%
4.55%
We can make some observations: first, we note that the PV is higher than the face value.
Second, we note that the coupon rate (10%) is higher than the yield to maturity (what is a
yield to maturity anyway?). Third, we note that the zero-coupon rate (what’s that again?)
is higher than the yield. Fourth, we have something called par yield (what?) which is
different from all previous interest rates (coupon, zero, yield). Fifth, we note that we are
confronted with four different interest rates: the coupon rate, the yield, par yield, and the
zero-coupon rate. The final blow is that none of these is the return of the bond, despite
its ”fixed income” features.
All these relations are not an accident: the price and face value are related
through coupon rates, yields, and zero yields. This example should make it clear that the
expression ”interest rate” can mean a variety of things. In the following chapter(s) we will
explore these concepts in detail.
2.1.
•
•
•
•
The frequency of compounding
stock returns and standard deviations are usually expressed as percent per year
a stock return in the US has the same interpretation as a stock return in Finland
interest rates, however, come in many varieties and are usually not directly comparable
each market and currency has its own agreed upon rules of how to convert a
discount rate into a price or the other way around:
1. when should one use simple interest, and when compound interest?
2. if compound interest is used, what is the frequency of compounding?
3. how should one count days to arrive at a fraction of a year (one unit of time in
finance is 1 year)?
• rules about how to discount and to define fractions of a year are important in fixed
income markets because prices can be given both as discount factors (yields) or prices
• these are not as important on stock markets, since there is anyway great uncertainty
about the timing and size of cash flows, and since prices are never given as yields
9
• simple interest is used when no interest is paid before maturity
FV = PV * [1 + (r * t )]
FV
PV
r
t
and
PV =
FV
[1 + (r * t )]
= future value
= present value
= interest rate expressed as decimals on a per annum basis
= maturity in years
• compound interest is used when interest is paid and added to the principal
FV = PV * (1 + r ) t
and
PV =
FV
(1 + r ) t
• the usual rule is that short-term rates (money market) are treated as simple interest
rates, while long-term rates are treated as compound interest rates
• bonds pay interest before maturity, and the “opportunity cost” can be seen as a longterm deposit that pays interest m times a year and is added to the capital
• this is standard when analysing returns on any market, for example long-term returns
on stock markets always assume dividends are reinvested (= compound interest)
• all interest rates are expressed as annual rates (unless stated otherwise)
Example: A bank offers a 3-month deposit rate of 3.50%. If you deposit EUR 1 000
today, how much cash do you have after 3 months? (For simplicity, assume 3 months is
0.25 years)
U
U
FV = PV * [1 + (r * t )] = 1000 EUR * [1 + (0.035 * 0.25)] = 1008.75EUR
• note that the actual return earned over 3 months is only 0.875%
Example: A bank offers a 2-year deposit rate of 4.25%. The deposit pays interest
annually. If you deposit EUR 1 000 today, how much cash do you have after 2 years?
U
U
FV = PV * [1 + r ] = 1000 EUR * [1 + 0.0425] = 1086.80
t
2
• note that the actual return earned over a 2-year period is 8.68%
• note that the time t, is seldom an integer value (this happens once a year), hence the
need to convert a number of days to a fraction of a year by some defined rules
• converting all rates to annual rates makes comparison easier
• converting actual returns over N years to a one-year return (using the previous
example):
10
[
r = (1 + rActual )
1/ N
]− 1 = [(1 + 0.0868) ]− 1 = 0.0425
1/ 2
• suppose that the compounding frequency (m) is not 1 year but 0.25 years instead
(interest is paid quarterly):
r⎤
⎡
FV = PV * ⎢1 + ⎥
⎣ m⎦
t *m
⎡ 0.0425 ⎤
= 1000 EUR * ⎢1 +
4 ⎥⎦
⎣
2*4
= 1088.23EUR
• an interest of 4.25/4 = 1.0625% is paid each 0.25 years and compounded 8 times
• the higher frequency of compounding increases the return to 8.82% if the annual
rate remains unchanged since interest can be added to principal more frequently
• in the limit: continuous compounding when the interval of frequency becomes very
small:
FV = PV * e r *t = 1000 EUR * e 0.0425*2 = 1088.72 EUR
• rates based on continuous compounding are mainly used in theoretical literature,
never in practice
2.1.1. Effective money market yields
• money market rates are simple yields and cannot be directly compared due to
differences in compounding frequency
• 1 month deposit can be rolled over 12 times during a year, but a 2 month deposit only
6 times
• conversion to annual effective yields is required:
Yield EFF
360 / t
⎡⎡
⎤
rSIMPLE ⎤
= ⎢ ⎢1 +
⎥ −1
⎥
⎢⎣ ⎣ (360 / t ) ⎦
⎥⎦
Example: Both the 30-day and the 60-day simple annual interest rates are 4%. What are
the effective annual yields?
U
U
The 30-day effective annual yield is
Yield EFF
360 / 30
⎡⎡
⎤
0.04 ⎤
12 .00
= ⎢ ⎢1 +
− 1 = 0.040742 = 4.0742%
⎥ − 1 = [1 + 0.003333]
⎥
360
30
(
/
)
⎦
⎢⎣ ⎣
⎥⎦
[
]
11
The 60-day effective annual yield is
Yield EFF
360 / 60
⎡⎡
⎤
0.04 ⎤
6.00
= ⎢ ⎢1 +
− 1 = 0.040673 = 4.0673%
⎥ − 1 = [1 + 0.006666]
⎥
⎢⎣ ⎣ (360 / 60) ⎦
⎥⎦
[
]
• if the simple rates are at level, the 30-day effective yield is higher because it can be
rolled over more frequently
• note also that the roll-over of the 30-day investment is risky, since the second 30-day
rate was unknown at the beginning of the 60-day investment period
2.2.
•
•
•
•
Building blocks: zeros and forwards
when analyzing fixed income securities, sooner or later one will be confronted with
zero coupon rates (also called spot rates)
a zero coupon rate is the discount rate (yield) for a zero-coupon bond, that is for a
bond paying a single cash flow received at time t
zero rates are the building blocks of all fixed income analysis, and as we will see later,
using the yield of a coupon bond for valuation can lead to severe mispricing
since the zero coupon rate is the only unambiguous interest rate for a particular
maturity, everything else needed can be calculated using zero rates: discount factors,
coupon bond prices and yields, par yields, forwards, swap rates etc.
2.2.1. Zero-coupon bonds
• zero-coupon bonds exist almost exclusively on treasury markets, for example US
Treasury or German Bund STRIPS
• the discounted value of the face value (negative cash flow) is invested and the face
value is received at maturity (positive cash flow)
• the time period (t) may be anything from one day to several years
• a separate spot rate is required for each period in time (t) to value a cash flow at time
t, and hence, we have a term structure of spot rates
• consider for example the following sequence of spot rates:
Maturity
Years
1
2
3
4
5
Spot rate
%
4.00
4.30
4.55
4.75
4.90
• then, we can graph the term structure of these spot rates:
12
6.00
Spot rate (%)
5.00
4.00
3.00
2.00
1.00
0.00
0
1
2
3
4
5
Maturity (Years)
• all interest rates are expressed on an annual basis (p.a.) to be more easily compared
• because most securities with maturity of over 1 year pay interest, the convention is to
express all interest rates for maturities over 1 year as annually or semi-annually
compounded rates
• thus, even if a zero-coupon bond pays no interest, we like to compare it with interest
paying securities, and do this by expressing them as annually compounded rates
Example: The 3-year zero-coupon rate is 4.55%. This does not mean that if you invest
100 today in a 3-year zero-coupon security, you get 104.55 back after 3-years. Since the
spot rate is expressed as an annually compunded rate, your investment yields:
U
U
U
U
FV = 100 * (1 + 0.0455) 3 = 114.28
which corresponds to an actual interest over the 3-year period of 14.28%. But comparing
this figure for example with a 3-year deposit that pays 4.5% p.a. is not very meaningful.
Hence, the conversion of the spot rate to an annually compounded rate.
• another way of expressing an interest rate is as a discount factor
• in the above case the discount factor is:
Df 3 =
1
1
= 0.8750
t =
(1 + r )
(1 + 0.0455) 3
• the discount factor is the PV of one unit (1) of currency received at time t
• the discount factor reflects both 1) time and 2) the discount rate
• discount factors are decreasing with maturity and always between 0 and 1:
13
1.00
Discount factor
0.95
0.90
0.85
0.80
0.75
0
1
2
3
4
5
Maturity (Years)
•
•
the discount factor is also the price (in % of face value) of a zero-coupon bond
for example, the price for a 3-year zero quoted at a yield 4.55% with face value 1000
(promises to pay 1000 after 3 years) is of course:
PV =
1000
1000
=
= 875.00
t
(1 + r )
(1 + 0.0455) 3
or equivalently:
PV = CF * Df = 1000 * 0.8750 = 875.00
2.2.2. Forward rates
• a forward interest rate is a rate set today for an investment that starts at a specified
time in the future
• spot rates for different maturities are linked by forward rates
• e.g. the interest rate for a 1-year investment that starts in the future is called a forward
rate
• e.g. the 2-year spot rate, r2 (from period 0 to 2), can be expressed using the 1-year spot
rate (from 0 to 1) and a forward rate from year 1 to year 2, which we denote f12 :
B
B
B
r2
B
r1
B
B
f12
B
B
B
• an investor with a 2-year investment horizon has two choices:
B
14
1. invest at the 2-year spot rate r2
2. invest at the 1-year spot rate r1 and roll over the deposit with the forward. f12
B
B
B
B
B
B
• since all rates (r1, r2, and f12) are known today the two investments can be compared:
B
B
B
B
B
B
(1 + r2 ) 2 = (1 + r1 ) * (1 + f 12 )
• this is an important arbitrage statement: the payoff from the two investments are
known today, and must be the same to prevent arbitrage
• if, for example:
(1 + r2 ) 2 > (1 + r1 ) * (1 + f 12 )
• we could borrow at the 1-year rate, roll over the borrowing with the forward and
invest at the 2-year rate for an arbitrage profit (since all rates are known today)
• note that the time periods need not be 1 year, they could be e.g. 3 months, and show
how 3-month forwards are linked to 3- and 6-month money market rates
• the figure below shows how spot rates are built up of one-period forward rates:
f01
B
f12
B
B
r1
B
f23
B
B
f34
B
B
f45
B
B
B
B
r2
B
B
r3
B
B
r4
B
B
r5
B
Period 1
Period 2
Period 3
Period 4
• the forward rate for period 2 (f12) can be found by setting:
B
f 12 =
B
(1 + r2 ) 2
−1
(1 + r1 )
• or, more generally:
f n ,t =
t=
n=
(1 + rt ) t
−1
(1 + rn ) n
maturity for the longer spot rate
maturity for the shorter spot rate
Period 5
B
15
Example: Assume the 1-year spot rate is 4.00%, the 2-year spot rate is 4.30%, and the 3year spot rate is 4.55%. What are the one-year forward rates for year 2 and 3?
U
U
f12 =
(1 + 0.0430) 2
− 1 = 1.046009 − 1 = 0.046009 = 4.60%
(1 + 0.040)
f 23 =
(1 + 0.0455) 3
− 1 = 1.050518 − 1 = 0.050518 = 5.05%
(1 + 0.0430) 2
• we can extend the information in the table:
Maturity
Spot rate
Years
1
2
3
4
5
2.3.
% (p.a.)
4.00
4.30
4.55
4.75
4.90
Actual
return
%
4.00
8.78
14.28
20.40
27.02
Discount
factor
0.9615
0.9192
0.8750
0.8306
0.7873
Forward
1-year
4.0000
4.6009
5.0518
5.3523
5.5022
Zero-coupon pricing of coupon bonds
2.3.1. The coupon rate
•
•
•
•
most bonds pay annual or semi-annual fixed interest payments, called coupons
coupon is paid on the face value, and is thus for fixed rate bonds a fixed value, since
the face value of a bullet bond does not change
the interest paid can also be a floating rate, based on some benchmark interest rate
(e.g. LIBOR), and is reset at each coupon payment date (FRN= floating rate notes)
the coupon rate is usually set to reflect the current interest rate level, and rounded to
the nearest 25 or 12.5 basis points, and the issue price adjusts to reflect the difference
between investor’s required yield and the coupon rate
2.3.2. The present value of a coupon bond
•
suppose the (rising) term structure previously used applies, and an investor
chooses between two bonds by the same issuer:
1. a 3-year zero-coupon bond with face value 100
2. a 3-year bond paying 5% annual coupons and face value 100
•
•
•
what return should the investor require from investing in these two bonds?
we know that r3 = 4.55% so this seems a reasonable yield for bond 1
should we require the same yield from bond 2 just because the last cash flow occurs
at the same time?
B
B
16
•
•
no, there is no reason to let maturity alone determine the discount rate!
these are clearly two different bonds, 4.55% is a yield for one single payment at t = 3,
and the second bond provides us with a series of payments in t = 1, 2 and 3
theoretically, a coupon bond is a collection of zero-coupon bonds, where each
payment (coupon or principal) can be seen as a separate zero-coupon bond
hence, the price of a coupon bond is the sum of all the individual payments (zeros):
•
•
PV =
CF1
CF2
CFT
+
+…+
= CF1 * Df1 + CF2 * Df 2 + … + CFT * Df T
1
2
(1 + r1 ) (1 + r2 )
(1 + rT ) T
where
= cash flow received at time i
= zero-coupon rate for maturity i
= discount factor for maturity i
CFi
ri
Dfi
B
B
B
B
B
B
For a bullet bond the cash flow is coupon payments until the last cash flow at maturity T
(CFT) which is the last coupon + face value. The time periods (1...T) are usually fractions
of a year (e.g. the first payment could occur after 0.8 years, the second after 1.8 years etc.)
B
B
• from the PV-equation, it would seem very odd to use r3 = 4.55% for all payments
• instead we use a series of zero coupon rates to value a the bond:
B
B
Example: The value of a 3-year bond that pays a 5% annual coupon on EUR 1 000 face
value assuming the spot rate 4.00% for one year, 4.30% for two years, and 4.55% for
three years:
U
U
PV =
CF3
CF1
CF2
50
50
1050
+
+
=
+
+
3
2
2
(1 + r1 )
(1 + r2 )
(1 + r3 )
(1 + 0.04)
(1 + 0.0430)
(1 + 0.0455) 3
= 48.07 + 45.96 + 918.79 = 1012.83
The coupon bond sells above par (101.283% of the face value), since the coupon
payments exceed the current interest rate level (term structure of zeros). A market for
zeros guarantees that the bond must be priced using zero rates: otherwise, a bond could
be stripped and the parts (coupon and/or principal) could be sold at a different price in
the strips market. Or, one could assemble a bond from strips and sell the package as a
coupon-paying bond. This arbitrage/replication approach to bond pricing is of course is
directly applicable where a liquid zero-coupon market exists along a coupon-bond
market.
Nevertheless, zero-coupon pricing captures the whole shape of the termstructure, and correctly prices each part of a specific cash-flow structure.
17
This is particularly important when observed market yields correspond to certain
cash-flow structures (zeros, deferrals, step-ups, coupon bonds, annuities etc.), and one
is to price non-standard cash flow structures.
2.3.3. Yield to maturity for a coupon bond
•
•
•
•
what is the yield to maturity for a coupon bond priced using zeroes?
the yield to maturity for a zero is unambiguous: it is the zero-coupon rate
for coupon bonds the yield is the internal rate of return for the bond
thus, once the price of the bond is known, we must solve the present value equation
with respect to yield (y):
PV =
CF3
CF1
CF2
50
50
1050
+
+
=
+
+
= 1012.83
2
3
2
(1 + y )
(1 + y )
(1 + y )
(1 + y )
(1 + y )
(1 + y ) 3
the yield can be found by trial-and-error (for example using a solver-function) and is
y = 0.045329 or 4.5329%
• note that since the yield is a kind of ”weighted average” of the zero coupon rates, and
the largest cash flow (principal + last coupon) is paid at year 3, the yield is close to the
longest zero rate
• another, less frequently used measure is the current yield:
Current yield = Coupon/PV
for example:
Current yield = 50/1012.83 = 4.9367%
• ... which is a more or less meaningless measure
• try to value the bond using the yield:
PV =
50
50
1050
+
+
= 47.83 + 45.77 + 919.24 = 1012.83
2
(1 + 0.045329)
(1 + 0.045329)
(1 + 0.045329) 3
• the values of the individual cash flows have changed
• a bond can be stripped into zero-coupon instruments:
Coupon bond:
Principal
Coupon 1
Coupon 2
Coupon 3
18
Stripped bond:
Coupon 1
Principal
Coupon 2
Coupon 3
0
1
2
3
Time
• the law-of-one-price does not hold, since the stripped coupons and zeros do not
have the same price if the bond is valued using the yield
• now, consider the same spot rates but a bond that pays a 10% coupon:
PV =
100
100
1100
+
+
= 1150.62
2
(1 + 0.04)
(1 + 0.043)
(1 + 0.0455) 3
• the yield for this bond is 4.5188% or 0.014% lower than for the the 5% coupon bond
• despite the same spot rates and the same maturity, bonds can have different yields:
Bond type
3 year bullet, 0% coupon
3 year bullet, 5% coupon
3 year bullet, 10% coupon
3 year annuity, 5% coupon
Yield
4.5500%
4.5329%
4.5188%
4.3680%
Price
875.00
1012.83
1150.62
994.47
• a tricky (that is, impossible) question to answer is ”what is the yield for 3-year bonds”:
there is no unambiguous answer to that question
• the only result that prevails is that the only unambiguous interest rate for a certain
maturity is the zero-coupon rate
• a yield for a 3-year coupon bond is clearly not a true 3-year rate since cash flows are
distributed over the time span 1-3 years
• even if the yield has limitations in measuring the interest rate level, it is still a
convenient “summary” measure, and used as such in practice
• one more example of the pitfalls of pricing with the yield:
Example: The treasury is offering a new product:
U
U
Maturity:
Face value:
Bond type:
3 years
EUR 1 000 000
Annuity
19
Coupon:
Annuity:
5.00%
EUR 367 208.56 annual
The annuity is calculated as:
⎡ r (1 + r ) t ⎤
⎡ 0.05(1 + 0.05) 3 ⎤
=
1000000
*
Annuity = PV * ⎢
⎥
⎢
⎥ = 367208.56
t
3
⎣⎢ (1 + r ) − 1 ⎦⎥
⎣⎢ (1 + 0.05) − 1 ⎦⎥
You work for the treasury, and your task is to price the bond to decide which treasury
auction bids to accept. You observe some treasury bond yields on the market:
Treasury zero-coupon bonds:
1 year:
2 years:
3 years:
4.00%
5.00%
6.00%
Treasury 3 year benchmark, 5% coupon bond:
3 years
5.933%
You have never even heard about the CEFA-program, and hence, you are unaware of
term-structure theory and bond mathematics, and you decide to price the new annuity
bond using the 3-year treasury yield for the 5% coupon bond. You find that the price
should be (you discount the annuities with the yield):
I) Using yield (5.993%) for coupon bond:
PV =
367208.56
367208.56
367208.56
+
+
= 982770.06
1
2
(1 + 0.05993)
(1 + 0.05993)
(1 + 0.05993) 3
At this price, the annuity naturally carries a yield of 5.933%
II) Using term-structure of zero-coupon rates:
PV =
367208.56 367208.56 367208.56
+
+
= 994469.54
(1 + 0.04)1 (1 + 0.05) 2 (1 + 0.06) 3
At this price, the annuity carries a yield of 5.2965%
Now, your pricing adventures of having applied a 5% coupon bond yield to an annuity
has three consequences: 1) you mispriced (underpriced) the annuity with about 70 basis
points, 2) investors would kill to lay their hands on the annuity, 3) you will lose your job
or alternatively the Treasury will send you to next year’s CEFA program.
The following table will highlight the problem of using the yield for a particular cash flow
structure when pricing a different cash flow structure:
Maturity
5% CPN bond
Cash flow
% of total CF
Annuity
Cash flow
% of total CF
20
1
2
3
Total
50 000
50 000
1 050 000
1 150 000
4.35%
4.35%
91.30%
100.00%
367 208.56
367 208.56
367 208.56
1 101 625.68
33.33%
33.33%
33.33%
100.00%
2.3.4. The par yield
•
•
•
•
•
•
we now know that the yield for bonds of a certain maturity will depend on the
coupon rate (or more generally the cash flow structure, for example the annuity)
thus, it is impossible to say what the yield is for a certain maturity
a commonly used yield is, however, the par yield
a bond whose price equals its face value is said to sell at par (100% of the face value)
the yield for such a bond is then the par yield
for par bonds:
Yield = coupon rate
•
⇒
PV = 100% of face value
the par yield for maturity T can easily be calculated using discount factors:
Yield Par =
(1 − Df T )
T
∑ Df
i =1
i
which says that you 1) calculate the discount factors for all cash flows upt to T, 2) divide
1 minus the discount factor for maturity T with the sum of all discount factors.
Example: Calculate the par yield for 3-year bonds. From previous tables, we know that
the discount factors for years 1 to 3 are: 0.9615, 0.9192, and 0.8750. Then:
U
U
Yield Par =
(1 − 0.8759)
= 0.045344
(0.9615 + 0.9192 + 0.8750)
which is 4.5344%. Thus, if we would issue a 3 year bond with 4.5344% annual coupons,
it would trade at par. We can complete the table:
Maturity
Spot rate
Years
1
2
3
4
5
% (p.a.)
4.00
4.30
4.55
4.75
4.90
Actual
return
%
4.00
8.78
14.28
20.40
27.02
Discount
factor
0.9615
0.9192
0.8750
0.8306
0.7873
Forward
(1-year)
4.0000
4.6009
5.0518
5.3523
5.5022
Par yield
4.0000
4.2937
4.5344
4.7238
4.8639
21
and draw some graphs of the different interest rate curves:
6.00
5.50
Rate (%)
5.00
Zeros
4.50
Forwards
Par yields
4.00
3.50
3.00
0
1
2
3
4
5
Maturity (Years)
2.4.
•
The yield: common misconceptions
from the previous section, it should have become clear that in some circumstances,
the yield to maturity can be a misleading measure for coupon paying bonds
however, since the cash flows for most bonds look similar (fixed coupon payments
inside a certain range, no negative cash flows), the yield is a convenient summary
measure of the bond price relative to its cash flows as an annual per cent rate
one should, however, be aware of what a yield is and what it is not:
•
•
2.4.1. The yield is not the return
•
one could easily think that the yield for a bond is the promised return for the bond,
since the bond is a ”fixed income” security
the yield = return only for zeros that are held until maturity, in all other cases this will
not hold
the yield is a discount rate, not a return!
we will analyze a special case when the yield at the time of purchase actually will equal
the return:
•
•
•
Example: Earlier, we priced the 3-year, 5% coupon bond at PV = 1012.83, and
calculated the yield, y = 4.5329%. Suppose we intend to keep the bond until maturity,
and want to calculate the return over this investment horizon (3 years). We have a
problem of reinvesting the 5% annual coupons, but we assume this can be done at a
reinvestment rate that equals the yield. Then the future cash flows are:
U
U
1st Coupon:
2nd Coupon:
P
P
P
P
50*(1 + 0.045329)^2 =56.6357
50*(1 + 0.045329) =
52.2665
22
3rd Coupon + Face value =
Total future value =
U
UP
1050.00
1156.90
UP
The price of the bond was PV = 1012.83, so the horizon return is:
⎡1156.90 ⎤
r=⎢
⎣1012.83 ⎥⎦
1/ 3
− 1 = 0.045329 = 4.5329%
which equals the yield at the time of purchase. Anybody would, however, agree that this
scenario is unrealistic:
1. The bond is seldom held until maturity. The interest rate level at the time of
selling the bond is uncertain, and hence the price is uncertain ⇒ price risk
2. The coupons can usually not be reinvested at a rate that equals the yield. This
would require a flat term-structure. The reality is uncertainty about the future
value of the reinvested coupons ⇒ reinvestment risk
•
•
•
•
•
the previous exercise is called horizon analysis
this is a useful approach in bond investing, since unlike stocks, the life of the bond is
finite, and hence ”invest-and-forget” (buy-and-hold) strategies are not applicable
the investor might be interested in possible outcomes for the future value of the
investment at a certain pont in time (before or at maturity)
for example, insurance companies have known liabilities which require funds to be
invested such that the liability can be met at that point in time
the total, or horizon return from a bond consists of:
1. Coupon interest payments
2. Income from reinvesting the coupons
3. Capital gain or loss if the bond is sold before maturity
•
•
•
it’s clear that today’s yield cannot capture all these sources of return
of course, let’s not forget that for example treasury bonds usually have higher yields
than treasury bills, and also tend to outperform treasury bills in terms of return
the point is that the yield is merely and indication of return, not a promise
2.4.2. Yields are not additive
•
what this means is that yields for bond portfolios can not be calculated like returns
for stock portfolios
Example: Let’s construct a simple bond portfolio that contains one 1-year zero coupon
bond, and one 5-year coupon bond that pays 5% annual coupons. We use the term
structure from previous examples to price the bonds, and assume that both bonds have
a face value of 1000:
U
U
Bond
1 year zero
Coupon
0.00%
Price
961.54
Weight
0.4887
Yield
4.0000%
23
5 year bullet
Portfolio
5.00%
1005.95
1967.49
0.5113
1.0000
4.8631%
First, let’s try to calculate the portfolio yield by treating yields as returns:
N
rP = ∑ wi ri = 0.4887 * 0.04 + 0.5113 * 0.048631 = 0.044413
i =1
or 4.4413% (which is wrong)
The only correct way to calculate the yield on a bond portfolio is, however by solving for
the yield for the portfolio, given the portfolio price and cash flows:
1000
50
50
1050
+
+
+…+
= 1967.49
2
(1 + y ) (1 + y ) (1 + y )
(1 + y ) 5
where, in this example, the first cash flow comes from the first bond, and all other cash
flows from the second bond. We solve the yield (y), and find that:
y = 0.0471349 = 4.7135%
Note the difference (over 27 basis points!) between the true yield y = 4.7135%, and y =
4.4413% calculated earlier.
•
•
of course, nothing prevents the bond portfolio manager from expressing the yield of
his portfolio as a weighted average, but in that case care should be taken to make
clear how this figure has been obtained
to a very close approximation, the portfolio yield can be calculated using weights, if
the equation is adjusted for modified duration
2.5.
From coupon bonds to zeros: bootstrapping
• suppose you need zeros for pricing, but no zeros with the same credit risk exist
• zero-coupon or spot rates reflecting a specific level of default risk can be found by:
1. Observing yields in the zero-coupon market (strips)
2. Bootstrapping a yield curve using coupon paying bonds or the swap curve
(these two spot rate curves of course have different credit risks)
• using observable zero-coupon rates for pricing coupon bonds may be problematic:
1. Liquidity. Lower liquidity in the zero market might lead to higher yields.
2. Taxes. Zeros and coupon bonds might be taxed differently along the whole
maturity spectrum, which translates to differences in required yields.
24
3. The preferred habitat hypothesis. There might be maturity sectors where
taxation between zeros and coupon bonds differ, if principal strips are taxed
differently than coupon strips.
• an important application of bootstrapping is to derive theoretical zero-coupon rates
from swap rates, which represent par-yields in the interbank market
• these bootstrapped zero-rates can then be used as a benchmark for pricing nonstandard cash flow structures in the corporate market
• the need for zero rates is more obvious the more ”non-standard” the bond or
valuation need is
Example: Suppose that a market participant needs zero-coupon rates up to 3-years
maturity, but no zero market exist. Instead, he observes the prices for the following
bonds:
U
U
Bond
1 year zero
2 year bullet
3 year bullet
Maturity
1 year
2 years
3 years
Coupon rate
0.00%
4.50%
5.00%
Face value
1000.00
1000.00
1000.00
Price
961.54
1003.88
1012.83
The 1 year zero rate is easy:
1000
− 1 = 0.04
961.54
Turning to the 2 year zero rate we use the 2 year coupon bond and we now know that:
45
1045
+
= 1003.88
(1 + 0.04) (1 + r2 ) 2
Clearly, there is only one solution for the 2 year zero rate (r2) that satisfies the equation.
After some calculation we find that
B
r2
B
B
B
= 0.043 = 4.30%
does the trick. We continue in the same fashion with the 3 year coupon bond:
50
50
1050
+
+
= 1012.83
2
(1 + 0.04) (1 + 0.043)
(1 + r3 ) 3
and find that
r3
B
B
= 0.0455 = 4.55%
(which does not surprise the careful reader, who might have suspected that the coupon
bonds in the example were priced using the same term structure as before.)
• the same technique applies to swap-rates, assuming payments are annual:
25
Maturity
1 year
2 year
3 year
•
Maturity
1 year
2 years
3 years
Swap rate
4.50%
4.70%
4.80%
since swaps are priced at ”par” (the swap rate is a par-yield), and a ”price” of 100 can
be assumed:
104.50
− 1 = 0.045
100
and
4.70
104.70
+
= 100
(1 + 0.045) (1 + r2 ) 2
and r2 = 4.7047%, and so on for the rest of the swap-curve.
B
B
2.6.
Spot and forward rates with semi-annual compounding
•
bonds and swaps may pay semi-annual or even quarterly or monthly coupons, and
hence there is a need to handle spot rates and forwards for higher compounding
frequencies than 1 years
• in general, the present value formula can be expressed:
PV =
CF
⎡ rt ⎤
⎢1 + m ⎥
⎣
⎦
mt
where
rt
m
t
B
B
= the zero coupon rate for maturity t
= the compounding frequency
= the maturity of the cash flow in years
• we will exemplify this using the following semi-annual zero-coupon rates:
Maturity
Years
0.5
1.0
1.5
2.0
Spot rate
% (p.a.)
7.45
7.68
7.69
7.75
• to convert these rates to discount factors:
26
Df t =
1
rt t *2
(1 + )
2
For example for the 1 year, semiannual rate of 7.68%:
Df t =
1
1
=
= 0.9274
0.0768 1*2
rt t *2
(1 +
)
(1 + )
2
2
• the forward rates are now 6-month forwards
• e.g. the forward from year 1 to year 1.5 (from period 2 to period 3 in half-years):
f 23
r3 3
⎡
⎤
⎢ (1 + 2 )
⎥
= 2*⎢
− 1⎥ = 0.0771
⎥
⎢ (1 + r2 ) 2
⎢⎣
⎥⎦
2
• note the multiplication by 2 to get the 6-month forward to an annual rate
• we can produce a similar table like in the case for the 1-year periods:
Maturity
Spot rate
Years
% (p.a.)
7.45
7.68
7.69
7.75
0.5
1.0
1.5
2.0
Actual
return
%
3.7250
7.8275
11.9842
16.4244
Discount
factor
0.9641
0.9274
0.8930
0.8589
Forward
6-month
7.4500
7.9103
7.7100
7.9301
27
3.
•
•
•
•
•
•
•
Day counts and accrued interest
so far, we have worked in a fairly unrealistic setting: we have analyzed bonds and
rates with maturity of integers of one year
the reason for abstracting from details is to make the concepts and technique clear
we will now take a small step from the classroom towards the cruel world of day
count conventions and accrued interest
reading stock market quotes is quite clear: you can observe the price at which you
buy and the price at which you sell
in fixed income markets, things become a mess: prices are frequently quoted as
yields, or in percent of face value less accrued interest, which means that just by
looking at bond quotes you can never tell what the price is
further, a yield quote is not unambiguous across the world: 5% in the US treasury
markets does not mean the same thing as 5% in the Finnish or German treasury
markets: there are differences in day count conventions and compounding frequency
as an example, compare the following zero-coupon bonds:
Security
Maturity
(Years)
German Bund
US Treasury
Coupon
15.35
15.35
0%
0%
Quote
(yield to maturity)
5.96%
5.96%
Price
(% of face value)
41.12
40.60
What’s wrong? Shouldn’t the price also be equal? No, since all Bund quotes are based on
annual compounding, while US Treasury bond quotes are based on semi-annual
compounding they will have different prices if the yields are the same.
•
•
the lesson is that one should always be aware of what market conventions apply to
the interest rate you are analyzing
the next section is by no means intended to give a comprehensive treatment of the
subject, but merely to introduce the reader to some concepts, and to make the reader
aware of the pitfalls that exist
3.1.
•
•
•
•
Day count basis
the bond market is like no other place: 31 days can be 30 days, and a year can be
more or less than 365 days
recall that all interest rates are expressed on an annual basis
day count conventions deal with how to compute fractions of a year
to give an example of what day count basis means:
Example: Suppose you are investing in a 1-month short-term deposit. The financial
institution promises an interest of 4.00%. This is of course a per annum figure. Since the
interest rate is an annual rate, we must know how large a fraction is this particular month
of 1 year. Suppose we count the days, we find that there are 31 days in that month. We
also know that there are 365 days in a year, so the maturity of the investment in years is:
U
U
28
t=
31
= 0.084931
365
thus, if we invest 100 units of currency:
31 ⎤
⎡
FV = 100 * ⎢1 + 0.04 *
= 100.3397
365 ⎥⎦
⎣
This day count basis is called ”Actual/Actual”, because all days are counted, in our case
”31/365”. This need not be the case, since there exist many ways of counting days. The
most common day count conventions include:
Actual/Actual The denominator is the number of days in the coupon period times the
coupon frequency.
Actual/365
Like Actual/Actual but always uses 365.
Actual/360
Like Actual/365, but always uses 360 (assumes there are only 360 days in
a year). (Euro money market)
30/360
Assumes that there are 30 days in each month and 360 days in a year. In
our previous example, we would have measured the time period as only
30 days instead of 31, and divided with 360 (30/360 = 0.083333). There
are som further variations for this rule.
3.2.
The money market
• prices for money market instruments are usually expressed as yields
• on the Euro market, the day count is Actual/360
• the price (PCD) of a money market security:
B
PCD =
B
CF
t ⎤
⎡
1 + ⎢r *
⎥
⎣ 360 ⎦
where
CF
r
t
= cash flow at maturity
= interest rate for period t
= time to maturity in days
Example: The maturity of a money market security is 30 days, face value EUR 1 000
000, and the quotes are:
U
U
Maturity
30 days
Quote (%)
Bid
Ask
3.01%
2.96%
29
What’s the price? Note that bid and ask are from the dealer’s point of view: the dealer
buys at 3.01%, and sells at 2.96%. Using the bid to find out what the dealer is willing to
pay for the security:
PCD =
1000000
30 ⎤
⎡
1 + ⎢0.0301 *
360 ⎥⎦
⎣
=
1000000
= 997497.94
1002508
.
The ask (2.96%) again corresponds to a price of EUR 997 539.40. That is:
Maturity
30 days
Quote (%)
Price (EUR)
Bid
Ask
3.01%
EUR 997 497.94
3.3.
2.96%
EUR 997 539.40
Zero-coupon bonds: annual compounding
• bond prices are expressed as yields or prices as percent of the face value
• to calculate the price in currency, one has to know the face value, day count basis, and
compounding frequency
Example: On the Bund STRIPS-market a zero that matures 4.7.2015 is quoted on
1.3.2000:
U
U
Maturity
4.7.2015
Quote (yield)
Price (% of face value)
Bid
Ask
5.96%
41.12
5.91%
41.41
The maturity of the zero is 15.35 years using Actual/Actual day count basis. Further we
need to know that on the Bund market, annual compounding is used. If the face value of
the zero is EUR 100 000, the bid price in euros is:
PV =
EUR100000
= EUR 41121.70
(1 + 0.0596)15.35
and the price quote is (K):
K=
41121.70
= 0.4112 = 41.12%
100000
30
3.4.
•
Zero-coupon bonds: semi-annual compounding
suppose that the same market quotes (yield and price quote) are observed on the US
Treasury STRIPS markets
is the price in currency the same?
no, since US Treasury yields are semi-annual (assuming day count basis is the same):
•
•
PV =
USD100000
⎡ 0.0596 ⎤
⎢1 + 2 ⎥
⎣
⎦
15.35*2
= USD 40596.29
and the price quote K = 40.60%
•
if there should have been differences in the day count basis between the markets, this
difference would have shown up in the calculation of maturity, and the fraction of a
year (0.35) might have been different
3.5.
Coupon bonds
• bonds are usually quoted in per cent of their face value, e.g. 110.827% or just 110.827
• the yield to maturity (YTM) of a bond is another way to express the price
• recall that the present value of a bond is simply:
n
CFt
t
t =1 (1 + rt )
PV BOND = ∑
where
CFt
rt
t
B
B
B
B
= cash flow (coupon and/or face value repayment) at time t
= spot rate for maturity t
= time in years
Once the yield to maturity is known, one can of course to cut some corners use the yield:
n
CFt
t
t =1 (1 + y )
PV BOND = ∑
•
•
in most of the forthcoming examples, we will use the yield to demonstrate the
calculations
let’s start our example of dirty and clean prices by pricing a bond:
31
Example: Suppose todays date is 17.1.1997, and a government bond that matures
15.3.2004 and pays a CPN of 9.50%. Assume a yield of 5.55% and, a face value of EUR
1 000 000. What is the PV of the bond on 17.1.1997? There are 57 days between 17.1
and 15.3 which is 0.1562 years under the Actual/Actual basis so:
U
U
PV =
•
95000
95000
1095000
+
+…+
= 1307995.13
0.1562
1.1562
(1 + 0.0555)
(1 + 0.0555)
(1 + 0.0555) 7.1562
note that:
1. the value of the first coupon can, depending on the market conventions, be
calculated using simple or, as in this example, compound interest
2. there may not be exactly 1 year between the coupon payment dates if payments
occur on weekends or holidays, this has not been taken into account here
• the effect on value of 1. and 2. is of course minimal, but it will be there
• the bond will, however, nor be quoted as ”1 307 995.13” on the market...
3.5.1. Dirty prices and clean prices
• the present value is called the ”dirty price”, ”full price”, or ”invoice price”
• what are ”clean prices”?
• the price that dealers quote is the clean price, not the PV (dirty price):
Clean price = Dirty price – Accrued coupon interest since last coupon payment
The accrued interest (AI) is:
v
* rCPN * N
365
AI =
where
v
rCPN
N
365
B
B
= number of days since last coupon payment
= coupon rate
= face value of the bond
= day count basis for the bond, here assumed to be 365
The clean price (K) of a bond is:
K=
PV BOND − AI
N
32
Example: Using the bond from the previous example we know that:
U
U
Purchase date:
Next coupon payment date:
Days to next coupon date:
Days of accrued interest (v):
PV (dirty price):
Face value:
17.1.1997
15.3.1997
57
308
1 307 995.13 EUR
1 000 000.00 EUR
(Note again that here, we abstract from taking into account delivery days applied on the
market (usually T +1...+3), that is, the bond and the money does not move today, and
you actually trade 1-3 day forwards).
The days of accrued interest v = 365 – 57 = 308 using actual/actual day count basis. We
can now compute:
AI =
308
* 0.095 * 1000000 EUR = 80164.38 EUR
365
and hence
K=
1307995.13 − 80164.38
= 1.22783 = 122.783%
1000000
Remember that the even if the quote is 122.783%, you still pay the dirty price 1 307
995.13 for the bond (that’s why the dirty price is also called the invoice price).
3.5.2. Behavior of dirty and clean prices over time: convergence
towards par
• the present value (dirty price) will vary according to the number of days to the next
coupon payment: immediately after a coupon payment, the PV will fall, and then rise
again as the next coupon approaches in time
• the clean price, on the other hand behaves more smoothly
Example: Consider the same bond that matures 15.3.2004 and pays a CPN of 9.50%.
Assume a (constant) yield of 5.55% and, for ease of exposition, a face value of only EUR
100. Starting from the CPN date 15.3.1996, we calculate the PV for each month. Note
how the PV falls on the CPN date 15.3.1997, just to start rising again. (This example still
uses the 30/360 day count basis).
U
U
33
Date
15.3.1996
15.4.1996
15.5.1996
15.6.1996
15.7.1996
15.8.1996
15.9.1996
15.10.1996
15.11.1996
15.12.1996
15.1.1997
15.2.1997
15.3.1997
15.4.1997
15.5.1997
15.6.1997
Days to next CPN
360
330
300
270
240
210
180
150
120
90
60
30
360
330
300
270
PV (Dirty Price)
124.972
125.535
126.102
126.671
127.242
127.816
128.393
128.972
129.554
130.138
130.725
131.315
122.407
122.960
123.514
124.072
Price (Clean Price)
124.972
124.744
124.518
124.296
124.075
123.858
123.643
123.430
123.220
123.013
122.805
122.607
122.407
122.168
121.931
121.697
• the clean price (K) of the bond approaches 100 (par) when maturity decreases
• the dirty price (PV) of the bond approaches 100 + last coupon when maturity
decreases
Example: Consider a bond that matures 18.4.2006. Assume the yield remains at 6.00%.
U
U
Date
17.1.1997
17.1.1998
17.1.1999
17.1.2000
17.1.2001
17.1.2002
17.1.2003
17.1.2004
17.1.2005
17.1.2006
Years
9.2528
8.2528
7.2528
6.2528
5.2528
4.2528
3.2528
2.2528
1.2528
0.2528
Clean price
108.642
107.913
107.140
106.321
105.453
104.533
103.557
102.523
101.427
100.265
Dirty price
114.060
113.331
112.358
111.739
110.870
109.950
108.974
107.940
106.844
105.682
• a discount bond would have started below par and approached par from below
34
4.
Measuring interest rate risk: duration and convexity
Since cash flows for bonds are usually fixed, a price change can come from two sources:
1. The passage of time (convergence towards par). This is of course totally
predictable, and hence not a risk.
2. A change in the yield. This can be due to a change in the benchmark yield,
and/or change in the yield spread.
The yield-price relationship is inverse, and we would like to have a measure of how
sensitive the bond price is to yield changes. A good approximation for bond price
changes due to yield is the duration, a measure for interest rate risk. For large yield
changes convexity can be added to improve the performance of the duration. A more
important use of convexity is that it measures the sensitivity of duration to yield
changes. Similar risk measures are used in the options markets are the delta and gamma.
4.1.
The yield-price relationship for bonds
• we again discuss interest rate changes in terms of yield changes: this is more
convenient as the yield is the mostly used interest rate measure for coupon bonds
• when yields increase, bond prices decrese
• when yields decrease, bond prices increase
• for small yield changes, the percentage price change is roughly the same whether the
required yield increases or decreases
• for large yield changes, the percentage price increase is larger than a price decrease
Example: On 17.1.1997, the (dirty) price of the RoF2006 Government bond is 114.060,
and the yield is 6.00%. Consider the impact of an one percent yield increase/decrease:
U
U
Yield
(%)
5.00
6.00
7.00
Yield change
(% units)
-1.00
0.00
+1.00
Dirty Price
EUR
121.732
114.060
107.033
Price change
(%)
+6.73
0.00
-6.16
35
Price and Yield for RoF2001 and RoF2006 Government bonds
180.00
160.00
Dirty Price
140.00
RoF2001
120.00
RoF2006
100.00
80.00
60.00
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Yield
4.2.
Duration
• different ways to calculate duration:
1. Macaulay duration (DMAC):
B
i)
ii)
iii)
B
The present value of time-weighted cashflows, divided by the
present value of the bond. Uses yield to calculate present values.
The discounted, average payback time.
Balancing point in time between interest rate and price risk.
2. Fisher-Weil duration (DFW):
B
i)
B
The same as DMAC, but uses zero rates.
B
B
3. Modified duration (DMOD):
B
i)
ii)
B
Macaulay duration/(1 + y)
The % price increase (decrease) in the bond price if the yield
decreases (increases) by a unit of 1%.
4. Key rate duration (DKR):
B
i)
B
Calculates the price response separately for a 1% change in each
zero-coupon rate used to calculate the PV of the bond. Gives a
better picture of which parts of the term-structure is responsible
for how much of total interest rate risk. The key rate durations
can then be summed up to give an overall interest rate risk
measure, comparable with modified duration.
36
4.2.1. Macaulay duration
• Macaulay duration (DMAC) using the yield:
B
Duration( D MAC ) =
1
PV BOND
B
CFt * t i
⎡ CF1 * t1 CF2 * t 2
CFn * t n ⎤
(1 + y ) ti
*⎢
+...+
=∑
t1 +
(1 + y ) t 2
(1 + y ) tn ⎥⎦ t = t 1 PV BOND
⎣ (1 + y )
n
Example: Calculate the Macaulay duration for the bond:
U
U
Maturity:
Face value:
PV:
Coupon rate:
Yield:
2.0 years
EUR 1 000 000.00
EUR 1 141 635.55
11.0% annual
3.54%
Duration( DMAC ) =
⎡ 110000 *1.00
1
1110000 * 2.00 ⎤
+
*⎢
⎥=
1141635.55 ⎢⎣ (1 + 0.0354)1.00 (1 + 0.0354) 2.00 ⎥⎦
1
* [106239.13 + 2070792.83] = 1.9069
1141635.55
• since the yield is used to discount, the Macaulay duration assumes that the term
structure is flat and that yield shifts are parallell
• the duration of a zero coupon bond equals its maturity
• for example, N = 100, y = 3.54%, PV = 93.28
Duration ( D MAC ) =
⎡ 100 * 2.00 ⎤
1
1
*⎢
* [186.56] = 2.00
⎥=
2
.
00
93.28 ⎢⎣ (1 + 0.0354)
⎥⎦ 92.28
• holding everything else equal:
1. increasing coupon rates decreases duration
2. increasing yield decreases duration
3. increasing maturity increases duration (in most cases)
•
Macaulay duration for different maturities (T), yields, and coupon rates (annual):
Coupon
T
0%
5
10
20
50
100
5.00
10.00
20.00
50.00
100.00
Yield 5%
5%
4.55
8.11
13.09
19.17
20.84
10%
0%
4.25
7.27
11.48
17.76
20.54
5.00
10.00
20.00
50.00
100.00
Yield 10%
5%
4.49
7.66
10.74
11.24
11.01
10%
0%
4.17
6.76
9.36
10.91
11.00
5.00
10.00
20.00
50.00
100.00
Yield 20%
5%
10%
4.36
6.65
6.87
6.01
6.00
3.99
5.72
6.20
6.00
6.00
37
•
some refinements to the 3 rules of thumb:
1. The duration of a zero-coupon bond equals it’s maturity.
For coupon bonds, duration approaches D = 1 + 1/y when maturity increases.
2. Duration for coupon bonds with coupon rate higher or equal to the yield will rise
with maturity and reach a maximum of D = 1 + 1/y.
3. Duration for bonds with coupon rate lower than the yield will first rise with maturity
and reach it’s maximum and then decline with maturity to D = 1 + 1/y.
4.3.
Modified duration and PVBP
• the approximate percent price change for a 1.00% yield change (100 bps) is given by
the modified duration:
Duration( D MOD ) =
D MAC
(1 + y )
The percent price change (dP/P) for any yield change dy:
dP
= − D MOD * dy
P
and hence we have the EUR price change (dPV) for any yield change dy:
dPV = − DMOD * PV * dy
•
note the ”-” sign before the expressions, indicating the inverse relationship between
yield change and price change
A commonly used measure is the Price Value of a Basis Point (PVBP):
PVBP =
D MOD
* PV
10000
and says how much the value of the bond changes in EUR when the yield changes with 1
bps (0.01%).
Example: Calculate the modified duration and the PVBP for the same bond:
U
U
DMOD = 1.9069/(1 + 0.0354) = 1.8417
B
B
38
PVBP =
DMOD
1.8417
* PV =
* 1141635.55FIM = 210.26 EUR
10000
10000
Example: If the yield increases by 1.00% to 4.54%, how much will the price decrease?
U
U
dP/P = -1.8417*0.01 = -0.018417 = -1.8417%
dP = -1.8417*1 141 635.55*0.01= -21 026.00 EUR
The new price should then be: 1 141 635.55 – 21 026.00 = 1 120 609.55 EUR
But recalculating the price using the yield 4.54, we find that the true price should be EUR
1 120 905.42, and the error is EUR 295.87
• the error occurs because duration assumes a linear price-yield relationship when it in
fact is convex
4.3.1. The duration of a bond through time
• between coupon payments duration decreases one-by-one with time if yields stay
unchanged
• at a coupon payment duration jumps up and increases ⇒ in the long run duration
decreases more slowly than time
• does this matter in the practical sense?
• duration increases after a coupon ⇒ the bond becomes more sensitive to yield
changes after a coupon payment
• but this is only in % terms, the absolute (EUR) value change is unaffected
Example: The RoF2006 bond pays a 7.25% coupon 18.4.1997. Consider what happens
to duration and volatility due to a 0.10% yield increase immediately before and after the
coupon payment:
U
U
Duration
Date
17.4.1997
18.4.1997
6.5703
7.0063
% change due to
duration
-0.6138
-0.6610
EUR change due
to duration
-7 173.59
-7 171.72
The % change in value of the bond changes, but the change in EUR value changes only
because of the slight duration change from 17.4.1997 to 18.4.1997.
• does the duration jump mean that we are suddenly exposed to immunization risk?
• the duration of the investor’s portfolio now consists of the bond and 72 500 EUR in
cash (cash has zero duration)
• but the investor still has to decide:
1. how to reinvest the cash
2. how much of the original bond (portfolio) to hold
39
• implications for an investment with a duration target:
1. should not invest cash in the same bond (duration will exceed target)
2. if the cash is invested in a security with any significant duration, a part of the
bond (portfolio) must be sold and invest proceeds in shorter duration assets
4.3.2. Key rate duration
•
duration can only measure parallell yield changes, and does not tell us where on the
yield curve the largest risks are located
for a single bullet bond, the largest risk of course comes from the last payment of
face value + last coupon, since it is:
•
1. the largest payment
2. located furthest out in time and hence through compounding most affected by a
yield change
•
for a bond portfolio or a more complex product than a bullet one can perform a
more detailed analysis of interest rate risk along the whole yield curve using key rates
we use the following bond as an example:
•
N
Maturity
Coupon rate
= 100
= 5 years
= 5% (annual)
We compute the PV, Fisher-Weil duration, and Key rate durations assuming the term
structure below:
rt
B
Df
B
1
2
3
4
5
4.00%
4.30%
4.55%
4.75%
4.90%
CF
0.961583
0.919245
0.875040
0.830585
0.787268
PV
5.00
5.00
5.00
5.00
105.00
4.8077
4.5962
4.3752
4.1529
82.6631
100.5952
Fisher-Weil
(CF*t)*Df
4.8077
9.1925
13.1256
16.6117
413.3157
4.5435
Key rate
duration
0.0460
0.0876
0.1248
0.1577
3.9193
4.3354
For example, the 5 year key rate duration for a +/- 1% zero rate change is calculated:
DA =
PV+ − PV−
2 * PV0 * drt
where
PV+
PVB
B
B
B
= the price of the bond if the zero-coupon rate for maturity t decreases with 1%
= the price of the bond if the zero-coupon rate for maturity t increases with 1%
and we have
40
PV0
PVPV+
B
B
B
B
B
B
DA =
=
=
=
100.5952
96.7653
104.6505
(r5
(r5
(r5
B
B
B
B
B
B
= 4.90%)
= 5.90%)
= 3.90%)
PV+ − PV−
104.6505 − 96.7653
=
= 3.9193
2 * PV0 * dr5
2 * 100.5952 * 0.01
by “common sense”, calculate the average absolute price change for a +/- 1% in rates:
dPV, if r5 = 5.90%
dPV, if r5 = 3.90%
B
B
B
B
(96.7653 – 100.5952)/100.5952 = -3.81%
(104.6505 – 100.5952)/100.5952 = +4.03%
on average (3.81%+4.03)/2 = 3.9193%
which is the price responsiveness if the 5-year zero rate changes 1% and all other rates
remain unchanged.
4.4.
Convexity
• for yield increases, duration overestimates the price decrease
• for yield decrases, duration underestimates the price increases
The convexity for a bond is:
⎡ n t (1 + t ) * CFt ⎤ ⎡ 1 ⎤
Convexity = ⎢∑
⎥
⎥ * ⎢ PV
t +2
⎣ t =1 (1 + y )
⎦ ⎣ BOND ⎦
•
•
the first term in brackets is sometimes called “dollar-convexity”
in isolation, this measure means nothing, and does not have a straightforward
interpretation like duration or modified duration
• convexity has two uses:
1. Convexity measures and can be used to correct the price response-error
caused by duration. But: convexity is still an approximation for this correction,
since the shape of the price-yield cannot be fully described with the two measures
modified duration and convexity.
2. Convexity measures the change in duration due to yield changes.
The percent price change due to convexity is:
41
dP 1
= * ( Convexity ) * ( dy ) 2
P 2
The EUR price change due to convexity is:
dP =
1
* ( Dollar − convexity ) * ( dy ) 2
2
Example: Calculate the convexity of the bond used in previous examples. The coupon is
11%, and the yield is 3.54%.
U
U
⎡1.00 * (1 + 1.00) * 110000 2.00 * (1 + 2.00) * 1110000 ⎤ ⎡
1
⎤
Convexity = ⎢
+
⎥*⎢
3.00
4.00
⎥
(1 + 0.0354)
(1 + 0.0354)
⎦ ⎣1141635.55 ⎦
⎣
= 5.2495
Example: If the yield increases by 1.00% to 4.54%, how much of the price decrease is
due to convexity?
U
U
dP/P = 0.5*5.2495*(0.01)^2 = 0.00026248= 0.026248%
dP = 0.5*5 993 039*(0.01)^2 = 299.65 EUR
We can now get a better estimate of the price decrease:
Duration + Convexity = Estimate of total price change
-21 026.00 + 299.65 = -20 726.35
which is only EUR 3.78 less the actual decrease of -20 730.13
• note that convexity is always a positive number, correcting the price upward
• sometimes convexity is expressed as:
Convexity/200 = 5.2495/200 = 0.0262%
which is the % correction directly, and as such has a meaningful interpretation
42
• some examples of convexities:
Coupon
T
Yield 5%
5%
0%
5
10
20
50
100
27.21
99.77
380.95
2312.9
9161.0
23.94
75.00
211.33
564.13
764.95
10%
21.83
64.02
171.95
484.41
732.90
Yield 10%
5%
0%
24.79
91.91
347.11
2107.4
8347.1
10%
21.45
63.40
146.10
206.74
200.44
19.37
52.79
116.22
190.55
199.85
Yield 20%
5%
10%
0%
20.83
76.39
291.67
1770.8
7013.8
17.35
44.11
62.30
50.52
50.00
15.36
35.13
50.64
50.14
50.00
• one could argue that convexity has value since the price of a highly convex bond
1. increases more when yields fall
2. decreases less when yields rise
• one might argue that investors should be willing to pay more for a convex bond,
particularly when yields are volatile
4.4.1. Duration matching and the value of convexity
• if interest rates fall, two opposite effects:
1. your bond portfolio will increase in value
2. the coupons will be reinvested at a lower rate
• if interest rates rise, two opposite effects:
1. your bond portfolio will decrease in value
2. the coupons will be reinvested at a higher rate
• at an investment horizon equal to the bond’s duration these two effects will
(approximately) offset
• duration is the balance point between market risk and reinvestment risk
• if the investors horizon is h:
1. if h = D ⇒ dW/dy = 0 ⇒ no risk
2. if h < D ⇒ dW/dy < 0 ⇒ market risk
3. if h > D ⇒ dW/dy > 0 ⇒ reinvestment risk
• note that all formulas assume a single yield y ⇒ a flat yield curve is assumed
• convexity is beneficial for an investor who buys a bond with duration equal to his
horizon, and the yield curve shifts once to some arbitrary level
− in case of a sudden, large yield change the investor is more protected than
with a bond with lesser convexity
− in the case of a duration matched to the horizon, and a convex bond, a
yield increase will yield a higher end-of-period wealth, since the value of the
reinvested coupons exceed the loss of the lower redemption value
43
− a yield decrease will yield a higher end-of-period wealth, since the higher
redemption value will more than compensate for the lower reinvested coupon
values
Example: An investor buys a bond with Macaulay duration of 3.3841, and assume for
simplicity that the investor’s horizon equals this duration. Thus, the investor is
immunized, and he is going to sell the bond 0.4714 years before maturity, since the
maturity is 3.8556 years. Suppose the yield is initially 5.15%, and now consider the effect
of a sudden, immediate yield change to 4.15%, and 6.15%, respectively:
U
U
Yield
FV of coupons
Redemption value of bond
Total FV
Yearly return
4.15%
319 413.72
1 079 113.49
1 398 527.21
5.1513
5.15%
324 207.23
1 074 262.91
1 398 470.14
5.1500
6.15%
329 046.01
1 069 479.73
1 398 526.74
5.1512
• note that a flat term structure is assumed (all coupons reinvested at the yield)
• the change in the reinvestment values of the coupons just about offset the change in
the redemption value of the bond
• the more convex the bond, the larger would the return changes have been
• convexity is nearly irrelevant if yields change smoothly and continuously rather
than in large, sudden jumps
− if yields change smoothly in small increments, the investor has the opportunity
to adjust for duration changes continuously
− thus, no large yield changes like in the previous example will occur
• convexity is harmful if there is uncertainty about how the yield curve will change
− the effect of changes in the shape of the yield curve on the duration of a
bond will be greater, the greater the convexity
− this is beacuse convexity implies larger dispersion of cash flows
− thus, convexity implies that there is more uncertainty about how the
duration of the bond will behave when the yield curve changes
•
•
•
•
immunize risk to meet future obligations, for e.g. pension funds
protect invested money from interest rate shifts
what if zero-coupon bonds with same maturity do not exist?
“base-case” solution: invest in a bond or bond portfolio with equal (Macaulay)
duration and present value as the future obligation or stream of obligations
• e.g. a payment in 5 years could be matched by a bond portfolio with DMAC = 5.0
• protects only against parallell yield curve shifts
• problems with immunization:
B
B
1. duration drift over time not equal to time (duration decreases more slowly than time)
2. yield changes will change duration
3. Macaulay-duration based: immunizised only if yield curve is flat and changes are
parallell
• rebalancing needed: transaction costs versus immunization target
44
Contingent immunization
•
•
•
•
less-than-perfect immunization for a higher return
set a band within the duration is allowed to fluctuate
within these bands, active portfolio management to enhance return
if duration hits the bounds, back to “strict” immunization policy
Multiperiod immunization
• a stream of obligations
• immunize every payment separately
Dedicated portfolio
• cash flow matching
• no duration requirements
• no rebalancing
4.5.
•
Bond portfolio duration and convexity
calculating duration, modified duration and convexity for a bond portfolio is easy if
one assumes that all bonds in the portfolio has the same yield:
N
D P = ∑ wi Di
i =1
where
DP
wi
Di
B
B
B
•
= duration or modified duration of the portfolio
= weight of bond i in the portfolio
= duration or modified duration for bond i
B
B
B
portfolio convexity is calculated in the same way
Example: Assume the following bond portfolio:
U
U
Bond
1 y. zero
5 y. bullet
Portfolio
Coupon
0.00%
5.00%
Price
961.54
1005.95
1967.49
Weight
0.4887
0.5113
1.0000
Yield
4.0000%
4.8631%
DMAC
1.0000
4.5475
B
then:
N
DMAC = ∑ wi Di = 0.4887 * 1.0000 + 0.5113 * 4.5475 = 2.8138
i =1
B
DMOD
0.9615
4.3366
B
B
45
and
N
DMOD = ∑ wi Di = 0.4887 * 0.9615 + 0.5113 * 4.3366 = 2.6872
i =1
4.6.
Butterfly trades: A critical assessment of yield, convexity
and duration
Butterfly trades are return-enhancement devices that consist of selling an intermediate
maturity issue, and a simultaneous purchase of one shorter and one longer maturity issue.
A butterfly is deemed beneficial if the average yield of the purchased bonds exceed the
yield of the sold bond, while holding duration constant.
4.6.1. Weighting a butterfly
• assume three bonds indexed as i = 1,2, and 3, and that we try to obtain a higher return
by selling bond 2 and buying bonds 1 + 3
• to satisfy the cost constraint the butterfly requires that
Q2 * P2 = Q1 * P1 + Q3 * P3
where
Q
P
= face value of bond (in millions of units of currency)
= PV of the bond in % of face value
• the risk constraint on a butterfly requires that
Q2 * PVBP2 = Q1 * PVBP1 + Q1 * PVBP1
where
PVBP = price value of a basis point (in EUR per 1 000 000 face value)
• the equations can be reduced to solve for the values of Q1 and Q2
B
⎡ P2 * PVBP3 − P3 * PVBP2 ⎤
Q1 = ⎢
⎥ * Q2
⎣ P1 * PVBP3 − P3 * PVBP1 ⎦
and
⎡ P1 * PVBP2 − P2 * PVBP1 ⎤
Q3 = ⎢
⎥ * Q2
⎣ P1 * PVBP3 − P3 * PVBP1 ⎦
B
B
B
46
Example: (This example is based on actual market conditions on the Finnish Treasury
market in November 1997). Consider the following bonds:
U
U
Bond 1
Coupon
P (%)
Yield
PVBP (EUR)
Modified duration
Macaulay duration
Convexity
Bond 2
10.00%
117.991
5.15%
379.74
3.2184
3.3841
14.2244
Bond 3
9.50%
126.329
5.62%
590.50
4.6743
4.9370
30.1321
7.25%
113.360
5.81%
692.89
6.1123
6.4674
50.0322
Note that we express P in percent. The PVBP is however expressed based on EUR 1 000
000 face value. This scaling is a matter of convenience and does not affect the
calculations.
⎡ 126.329 * 692.89 − 113.360 * 590.50 ⎤
Q3 = ⎢
⎥ * 1000000 = 532017 EUR
⎣ 117.991* 692.89 − 113.360 * 379.74 ⎦
and
⎡ 117.991* 590.50 − 126.329 * 379.74 ⎤
Q3 = ⎢
⎥ *1000000 = 560655EUR
⎣ 117.991 * 692.89 − 113.360 * 379.74 ⎦
Thus you should buy Bond 1 to a face value of 532 017 EUR and Bond 3 to a face value
of 560 655 EUR
The values satisfy the cost constraint since
(53.2017*1.17991 + 56.0655*1.13360) = 62.7732 + 63.5558 = 126.329
which equals the price of the bond to be sold (Bond 2). This also shows that it is
straightforward to weight the butterfly as PV in EUR, since the face value Q can be
multiplied with the % price P. The values satisfy the risk constraint since
(0.5320*379.74 + 0.5606*692.89) = 202.0281 + 388.4719 = 590.50
which is equal to the PVBP of Bond 2, and equivalent to using modified durations and
PV:s:
⎡ DMOD (1) ⎤
⎡ DMOD (3) ⎤
⎢
⎥ * PV ( EUR)1 + ⎢
⎥ * PV ( EUR) 3
⎣ 10000 ⎦
⎣ 10000 ⎦
⎡ 3.2184 ⎤
⎡ 6.1123 ⎤
=⎢
* 627738 + ⎢
⎥
⎥ * 635551 = 202.03 + 388.47 = 590.50
⎣ 10000 ⎦
⎣ 10000 ⎦
or even simpler, weighting modified durations with the PV:s:
(0.4969*3.2184)+(0.5031*6.1123)=4.6743
47
where 0.4969 and 0.5031 are the weights in the butterfly based on the PV:s. The PV of
the portfolio is 126.329, and the weights are 62.7732/126.329 = 0.4969 and
63.5558/126.329 = 0.5031
4.6.2. A critical assessment of yield, convexity, and duration
• calculating yield on the portfolio using the value-weighted average = 5.4820%
• the “conventional” estimate of the portfolio yield would indicate a 13.80 bp loss in
yield versus Bond 2
• the yield on the portfolio using the duration-weighted approximation:
yP =
(627738 * 3.2184 * 0.0515) + (635551 * 61123
.
* 0.0581)
= 0.055842 = 55842%
.
(627738 * 3.2184) + (635551* 61123
.
)
• thus, in this case, the butterfly produces a yield loss of 3.58 bp
• the convexity of the portfolio is a weighted average of the bonds or:
(0.4969*14.2244) + (0.5031*50.0322) = 32.2392
• thus, the butterfly gives away 3.58 bp of yield, but increases convexity with 2.11
• the change in the value of a bond is
dP = − D MOD * PV * dy
•
•
•
•
since the trade was structured as a butterfly dP2 = dP1+3
duration-based measures assume parallell shifts in yield curves
if this is not the case, a butterfly not a perfect substitute for one single bond
the standard deviation in % of a bond is:
B
s = D MOD * σ =
B
B
B
PVBP * σ * 10000
PV
• the standard deviation of the EUR-value of a bond is
S = PV * D MOD * σ = PVBP * σ * 10000
where
σ = the standard deviation of the yield change
• the EUR-standard deviation of a two-bond portfolio is (using duration):
SP =
PV12 * D MOD 12 * σ 12 + PV22 * D MOD 22 * σ 22 + 2 * PV1 * PV2 * D MOD 1 * D MOD 2 * Corr1,2 * σ 1 *σ 2
48
the correlation of dP2 with dP1+3 denoted CorrdP is
B
CorrdP =
B
B
B
B
B
( PV1 * PV2 * D MOD1 * D MOD2 * Corr1,2 * σ 1 * σ 2 ) + ( PV3 * PV2 * D MOD3 * D MOD 2 * Corr3,2 * σ 3 * σ 2 )
S 2 * S1,3
• CorrdP < 1 unless the yield shifts are parallell
B
B
Example: Suppose the yield change correlation matrix, yield change standard deviations,
and EUR standard deviations for bonds 1,2, and 3 on:
U
U
S (EUR) annual
σ (dy) annual
s (%) annual
Bond 1
Bond 2
Bond 3
Bond 1
28 460.15
0.007495
2.4121
Bond 2
46 963.19
0.007953
3.7175
Correlation matrix (dy)
1.0000
0.8926
0.8926
1.0000
0.8529
0.9467
Bond 3
56 311.57
0.008127
4.9675
0.8529
0.9467
1.0000
(Yield changes are daily closing quoted bid yields from 2.1.1997 - 7.11.1997)
E.g. for Bond 2:
s = 4.6743*0.007953 = 0.0371 = 3.71%
B
B
and
S = 3.71% * 1 263 290 EUR = 46 963.19 EUR
The EUR standard deviation for the portfolio Bond 1 + Bond 3:
S1,3 = 45 182.05 EUR
B
B
which is slightly lower than the risk of Bond 2. The correlation of the change in value of
the portfolio and the single bond is:
CorrdP = 0.9606
B
B
• note that
1. the correlation between value changes is not perfect
2. due to a yield correlation of 0.8529 between Bond 1 and 3, the actual risk of
the portfolio is slightly lower although the risk constraint was satisfied
3. the difference in actual risk is a consequence of yield changes not being
parallell, and different maturities having unequal standard deviation, which
are not captured by using duration based measures only
4. the non-parallell behaviour was captured using yield change correlations, and
standard deviation estimates whose measurement is an empirical issue
(frequency, length of time-period)
5. the butterfly is not a perfect substitute for a single bond
49
5.
Applications of bond mathematics I: FRA:s and bond
futures
We will consider three major fixed income derivative instruments available on most
markets: Forward Rate Agreements (FRA:s), swaps, and bond futures. FRA and swap
pricing are straightforward applications of the term structure of spot rates in the
interbank market. Bond futures pricing relies on arbitrage between cash and futures
markets.
5.1.
Forward Rate Agreements
• FRA:s or Short Interest Rate Futures are used to hedge future short term borrowing
or lending in the money market, or to speculate on future interest rates
• the future interest rate (FRA or future rate) is fixed today
• underlying instrument: usually a 1- or 3-month deposit or borrowing in a certain
reference rate, e.g. 3-month LIBOR or EURIBOR or other money market rate
• expiration usually on the third Wednesday in March, June, September, and December
• the buyer makes a future borrowing in the reference rate (buys money)
• the seller makes a future deposit in the reference rate (sells money)
• cash settlement
• how is the rate for an FRA determined?
• recall from the relationship between spot rates and forward rates that:
(1 + r2 ) 2
(1 + f 12 ) =
(1 + r1 )
⇒
(1 + r1 ) * (1 + f 12 ) = (1 + r2 ) 2
using money market notation:
[1 + r * (t
1
1
] [
] [
/ 360) * 1 + f 12 * (t12 / 360) = 1 + r2 * (t 2 / 360)
]
where f12 stands for the FRA-rate, or the forward rate directly:
B
B
[
[
]
]
⎡ 1 + r2 * (t 2 / 360)
⎤
f 12 = ⎢
− 1⎥ * (360 / t12 )
⎢⎣ 1 + r1 * (t1 / 360)
⎥⎦
where
f12
r1
r2
t12
t1
B
B
B
B
B
B
B
B
B
B
= the FRA rate
= interest rate from today to FRA deposit start date
= interest rate from today to FRA deposit maturity date
= the FRA deposit lenght in days
= days from today to FRA deposit start date
50
t2
B
= days from today to FRA deposit maturity date
B
Example: Suppose that on January 17, the 2-month (actual: 61 day) money market rate
is r1 = 2.97%, the 5-month (actual: 152 days) money market rate is r2 = 3.03%, and thus
t1 = 61 and t2 = 152. What is the correct FRA rate for a 3-month (actual: 91 days) that
starts 61 days from today? The setting is:
U
U
B
B
B
B
B
B
Date:
Action:
Days:
B
B
January 17
March 19
FRA made
FRA Deposit
t12 = 91
t1 = 61
B
June 18
B
B
B
t2 = 152
B
B
⎡ [1 + 0.0303 * (152 / 360)] ⎤
f 12 = ⎢
− 1⎥ * (360 / 91) = 0.030548
⎢⎣ [1 + 0.0297 * (61 / 360)]
⎥⎦
• this relation must hold in order to prevent arbitrage
• otherwise, if e.g. the t1 deposit + the FRA yields more than the t2 depo: borrow at r2
and invest in r1 + f12
• FRA:s are cash settled up front:
B
B
B
B
B
B
The gain or loss for the buyer of an FRA:
⎡ t ⎤
(rR − rFRA ) * ⎢
*N
⎣ 360 ⎥⎦
Gain / Loss =
t ⎤
⎡
1 + ⎢rR *
360 ⎥⎦
⎣
The gain or loss for the seller of a FRA:
⎡ t ⎤
(rFRA − rR ) * ⎢
*N
⎣ 360 ⎥⎦
Gain / Loss =
t ⎤
⎡
1 + ⎢rR *
360 ⎥⎦
⎣
where
rFRA
rR
N
t
B
B
B
B
= the FRA contract interest rate
= the reference rate, (e.g. EURIBOR) at maturity or closing
= nominal amount of contracts
= maturity of the FRA deposit to be made
B
B
B
B
51
Example: On January 17th, a firm decides to hedge a EUR 5 000 000 future deposit to
be made on March 19th. The depo matures on June 18th, or in 91 days. The firm sells 5
March FRAs. The bid for FRAs is 3.0548. On March 19th , the 3-month EURIBOR is at
2.96.
U
U
P
P
P
P
P
P
P
P
⎡ 91 ⎤
(0.0306 − 0.0296) * ⎢
* 5000000
119817
.
⎣ 360 ⎥⎦
Gain / Loss =
=
= 1189.27
91 ⎤
1007482
.
⎡
1 + ⎢0.0296 *
360 ⎥⎦
⎣
The gain from the contract is EUR 1189.27. On March 19th the buyer thus deposits EUR
5 001 189.27 for 91 days at 2.96%:
P
P
5 001 189.27*[1 + 0.0296*(91/360)] = 5 038 609.28
To check that this really is the payoff at the FRA contract rate of 3.0548:
[5 038 145.21/5 000 000)-1]*(360/91) = 0.0306 = 3.0548%
5.2.
•
•
•
•
•
•
•
Bond futures
underlying instrument is a physical asset a, typically a coupon paying treasury bond
most markets have expiration dates in March, June, September, and December
settlement can be either 1) delivery or 2) cash
the buyer of a future buys the underlying bond on the expiration day
the seller of a future sells the underlying bond on the expiration day
need to forecast future prices?
no ⇒ the forces of arbitrage again determines the correct futures price
5.2.1. Futures pricing: The general approach
• cash and carry pricing: you have two alternative strategies:
1. buy the bond today at the cost PV and finance the purchase by borrowing
until future expires
2. buy future at the futures price F, and pay F when future expires
• implies that:
1.
2.
3.
4.
with both strategies 1 and 2 you own the bond at the future time t
no net cash outlay today
cost of both strategies are known today
cost of both strategies must equal to prevent arbitrage
52
• that is, the futures price F must be:
t ⎤
⎡
F = S * ⎢1 + r *
360 ⎥⎦
⎣
where
1. the left-hand side is cost of strategy 2 at time t, and
2. the right-hand side is the cost of strategy 1 at time t
where
F=
S=
r=
t=
the futures price
the “spot” price (the price of the bond today = PV)
the money market rate (interest rate at which the purchase of the bond
can be financed)
time until future expires (in days)
Example: The PV of a Government bond is 1 100. The future expires in 60 days, and
the 60 day money market rate is 3.50%. What is the futures price F?
U
U
60 ⎤
⎡
F = 1100 * ⎢1 + 0.035 *
= 1106.42
360 ⎥⎦
⎣
If the futures price is higher, say, F = 1 110 you could:
1. Buy the bond today, and finance it at 3.5% at the cost: 1 106.42
2. Make a futures contract today to sell the bond at F = 1 110
3. After 60 days you deliver the bond to the buyer of the future and get F = 1100
in cash and pay off your borrowing 1 106.42
4. Your riskless arbitrage profit is thus 3.58
5.2.2. Repo transactions in the bond cash and futures markets
• a bond can be purchased by financing the purchase in the sale and repurchase
(repo) market (if such a market exists)
• in a repo the financial asset is is lent (“repo’d out”) to a second party who in turn as a
collateral lends an equal amount of cash to the owner of the asset (in practice the
collateral is usually slightly higher than the bond’s PV)
• when the repo is terminated, the second party returns the bond to the owner, and the
owner returns the cash loan (collateral) plus interest to the second party
• the interest rate paid on the cash loan is called the repo rate
For simplicity, we assume that:
53
1. collateral for shorting is 100% of the bond’s value today
2. there is only one rate of interest, 3.5%
Then, if the futures price is lower, say F = 1 105 you could:
1. Short the bond in the repo market (borrow the bond). Pay collateral 1 100,
which earns 3.5% interest
2. Finance the collateral by selling bond at market price S = 1 100
3. Make a futures contract to buy the bond back at F = 1 105
4. After 60 days you get the bond from the seller of the futures contract, and
deliver to the bond lender in 1).
5. You pay F = 1 105 for this bond, but get 1 106.42 from interest paying
collateral.
6. Your riskless arbitrage profit is 1.42
5.2.3. Coupon payments
• suppose a coupon is paid before the future expires
• then the price of the (reinvested) coupon must be deducted from the futures price F
Example: Consider the same bond as before, but now the bond pays a 5% (50 EUR)
coupon after 30 days. The coupon can be reinvested at the 30 day money market rate (rR)
, which is assumed to be 3.00%. What is the futures price F?
U
U
B
B
tR ⎤
t ⎤
⎡
⎡
*
1
*
F = S * ⎢1 + r *
−
CPN
+
r
R
⎢
360 ⎥⎦
360 ⎥⎦
⎣
⎣
30 ⎤
60 ⎤
⎡
⎡
= 1106.42 − 50125
.
= 1056.30
− 50 * ⎢1 + 0.03 *
F = 1100 * ⎢1 + 0.035 *
⎥
360 ⎥⎦
360 ⎦
⎣
⎣
Why? If you want to hold the bond after 2 months, you can:
1. Buy the cash bond now and finance it at 3.5%. The cost is EUR 1106.42, and the
EUR 50 coupon is reinvested at 3.00%: EUR 50.125 which can be deducted from
your total costs from buying the bond: EUR 1106.42 - EUR 50.124 = EUR 1056.30
total after 2 months.
2. Buy the future. It's now clear that the futures price F must equal the cost in 1., since
F is also paid after 2 months in exchange for the bond, and both strategies lead to
holding the bond after 2 months.
•
•
similar arbitrage positions like demonstrated earlier can be created for a bond that
pays a coupon to exploit mispriced futures contracts
one has to remember that when shorting a bond, the coupon payments belong to the
owner of the bond, not the arbitrageur holding the bond short
54
5.2.4. Notional bonds and delivery options
•
•
•
•
•
•
•
•
a common practice on futures exchanges is to use a notional bond as the underlying
security
a notional bond does not exist (!)
the notional bond is defined only subject to 1) face value, 2) coupon rate, and 3) a
maturity range
for example, for LIFFE’s Long Gilt Future, the notional bond is simply defined as
”£ 100 000 nominal value notional Gilt with 7% coupon”
several existing treasury bonds can be chosen by the short futures holder to deliver
the use of a notional bond instead of a real one, and the use of many deliverable
bonds are used so that no market participant(s) can corner the market in the
underlying security
since the deliverable bonds will have different present values the price paid by the
future’s buyer at delivery must be adjusted using a conversion factor
the conversion factor is determined by the future’s exchange and calculated by
pricing the deliverable bonds using the yield stated in the futures contract
Example: A bond futures contract has the (very simplified) following contract
specifications:
U
U
Five-Year Bund Future
Unit of trading:
Delivery months:
Contract standard:
Deliverable bonds:
EUR 100 000 nominal value notional Bund with 6% coupon
March, June, September, and December
Delivery may be made of any Bunds on the List of Deliverable
Bunds
Any Bund with the following characteristics:
• redemption in a single installment not earlier than 4 years, and
not later than 6 years
• having no terms permitting early redemption
• bearing interest at a single fixed rate
Further, to more clearly demonstrate the process, we assume that all bonds, when
delivered, have no accrued interest.
Suppose that the ”List of Deliverable Bunds” contains 3 bonds:
Bond:
Coupon
Face value
PV calculated
at delivery date
4-year
5-year
5-year
5% annual
6% annual
7% annual
EUR 1 000
EUR 1 000
EUR 1 000
1 009.90
1 049.69
1 093.42
PV
using
contract yield
(6%)
at
delivery date
965.35
1 000.00
1 042.12
Conversion
factor
0.9653
1.0000
1.0421
The PV is as usual found by discounting the bond with the term structure of zeros. To
find the conversion factor, the bond is discounted with the yield stated in the futures
contract. This value is then divided with the face value to find the conversion factor.
55
Since the future’s contract is for EUR 100 000 nominal value, and all deliverable bonds
have a nominal value of EUR 1 000, this means that 100 bonds must be delivered, no
matter which bond the seller decides to deliver.
The conversion factor then decides how much the buyer must pay for the 100 bonds to
be delivered, such that:
Invoice price =
Contract size * Futures settlement price * Conversion factor +
accrued interest
Suppose that the futures contract settles at: F = 105%, and that the seller decides to
deliver the 5-year, 7% bond with conversion factor 1.0421, then:
Invoice price =EUR 100 000 * 1.05 * 1.0421 + 0 = EUR 109 420.50
Thus, the seller delivers 100 of the 5-year 7% bonds, and the buyer pays EUR 109 420.50
•
•
•
•
how is the future’s price determined when there is no underlying security?
usually, one of the deliverable bonds will, despite the use of conversion factors be
cheaper to deliver than the other bonds (since the conversion factor is fixed)
this bond is called the cheapest to deliver bond (CTD)
the CTD-bond can be found by calculating the return from the following exercise for
each deliverable bond:
1.
2.
3.
4.
•
•
Buy the bond.
Make a futures contract to sell the bond.
Deliver the bond at the future’s settlement date.
The return is called the implied repo rate, and the bond with the highest
implied repo is the cheapest to deliver bond.
because a rational investor will deliver the CTD, the futures will be derived from the
CTD-bond
the CTD bond may of course change over the life of the futures contract
5.2.5. Futures pricing using quoted prices and accrued interest
• the previous examples showed how futures contracts are generally priced
• in practice, bond futures prices are quoted as clean prices, and we have to take
accrued interest into account when calculating the clean futures price
• cash and carry pricing: you have two alternative strategies:
1. buy the bond today at K + AI and finance [K + AI] by borrowing until future
expires
2. buy future at F + AI today, and pay F + AI when future expires
• alternatively you could think that you have the money now and can earn interest on
[K + AI] by investing this amount in the money market
56
• implies that:
1. with both strategies 1 and 2 you own the bond at time t (when future expires)
2. cost of both strategies are known today
3. cost of both strategies must equal
The price of a bond future can be solved from:
[K + (rCPN * v/365 * N)]*[1 + (rt * t/360)] = [F + (rCPN * (v+t)/365 *N)]
B
B
B
B
B
B
where
F
t
v
rt
B
= the future price
= days to maturity for the future
= days since last coupon payment
= money market rate from now to maturity of the future
B
more compactly
[K + AIc]*[1 + (rt * t/360)] = [F + AIt]
B
B
B
B
B
B
where
AIc
AIt
B
B
B
B
= accrued interest on the contract date
= accrued interest on the maturity date
• the left-hand side of the equation: cost of K + AIc financed in the money market to
maturity date of future
• the right-hand side of the equation: cost of F + AIt on maturity date of future
• there is no need to forecast future bond prices to price a future
B
B
B
B
rearranging
F = [K + AIc]*[1 + (rt * t/360)] - AIt
B
B
B
B
B
B
Example: On January 17th, a Treasury bond with 10% CPN bond trades at K = EUR 1
218 000. The 2-month EURIBOR is at 2.96%. Assume there are 122 days since the last
coupon, and 61 days to the maturity of the future. What is the March 19th futures price?
U
U
P
P
P
P
F =[1 218 000 + (0.10*122/365*1 000 000)]*[1 + (0.0296*61/360)] - [0.10*(122 +
61)/365*1 000 000]
F = [1 218 000 + 33 424.66]*[1.000495] - [50 136.99] = 1 207 564.26
57
• if a coupons are paid between the the contract date and maturity date, the money
market rate for investing the coupon must be deducted from K
[K + AIc]*[1 + (rt * t/360)] = [F + AIt] + Σ CPNi *(1 + rCPN,t)
B
B
B
B
B
B
B
B
B
B
F = [K + AIc]*[1 + (rt * t/360)] - AIt - Σ CPNi *(1 + rCPN,t)
B
B
B
B
B
B
B
B
B
B
where
rCPN,t
B
B
= money market rate from the receipt of the i:th coupon to maturity
Example: On January 17th, a Treasury bond with CPN 7.25% trades at K = EUR 1 086
423.26. A coupon is received on 18.4. The 5-month (rt) EURIBOR is at 3.04%, and the
2-month (rCPN,t) EURIBOR is at 2.96%. What is the June 18th futures price?
U
U
P
P
B
B
B
B
P
P
1. Buy bond on Jan 17th ⇒ price paid is [K + AIc] = 1 140 596.87 EUR, financed at
money market rate, and total cost is 1 140 596.87 EUR*[1 + (0.0304*152/365)] = 1
155 036.51 EUR which is paid on June 18th
2. 7.25% coupon is paid on April 18th ⇒ invested at money market rate to yield 72
500*[1 + (0.0296*61/365)] = 72 858.65 EUR on June 18th
3. Total cost on June 18th is [1 155 036.51 - 72 858.65] = 1 082 177.86 EUR
4. Total cost for future on January 18th must equal 1 082 177.86 EUR [K + AIt]
P
P
B
P
P
B
P
P
P
P
P
P
P
P
B
B
Using the formula:
F =[1 086 423.26 + (0.0725*269/360*1 000 000)]*[1 + (0.0304*152/365)] [0.0725*60/360*1 000 000] - 72 500*[1 + (0.0296*61/365)]
F = [1 086 423.26 + 54 173.61]*[1.012660] - [12 083.33] - [72 858.65] = 1 070 094.53
58
6.
•
•
•
•
•
•
•
•
•
Applications of bond mathematics II: Swap contracts
a forward contract is a contract to exchange one payment in the future
a swap contract is a contract to exchange a stream of payments at a series of
specified dates in the future
swap contracts are usually used together with bonds or loans, to exchange the cash
flows (interest rate payments and principal) to some other interest rate base or
currency than the originally specified in the bond
an interest rate swap is a contract between two parties to exchange fixed interest
rate payments against floating interest rate payments (fixed-to-floating swap), or the
other way round
a currency swap is a contract between two parties to exchange interest rate
payments in one currency against interest rate payments in another currency (the
interest rates can be either fixed or floating interest)
swap contracts are usually OTC contracts, and are made between banks and large
corporations
an interest rate swap can be regarded (same PV) as a series of interest rate forward
contracts
a currency swap can be regarded (same PV) as a series of currency forwards
swaps are used for:
1. Hedging purposes. A stream of cash flows can be hedged. For example,
floating rate loans can be hedged against rising interest rates by swapping the
interest rate payements to fixed rate payements, or a EUR based bond issuer
might swap a USD-nominated bond to EUR.
2. Cost saving. A bond issuer might be able to borrow floating rate debt at
better terms (lower rates) than fixed rate debt. Then a package of floating rate
debt + swap to fixed can be more cost effective than to borrow directly at a
fixed rate. A bond issuer might also find for example the domestic bond
market “saturated”, and that demand for bonds in the issuers home currency
is low. Then a better price (lower yield) might be obtained in a foreign
currency, and a swap back to home currency might be desired to hedge the
currency risk.
6.1.
•
•
Interest rate swaps
with interest swaps, a customer (corporation or bank) can transform floating rate
debt to fixed rate or the other way round
swaps are usually semi-annual and the floating rate in swap contracts is usually 6
month LIBOR or EURIBOR
Example: Consider a corporation borrowing a floating rate EUR 100 million for 3-years.
Interest rate payments are annual, and the reference rate is 12 month LIBOR flat (the
company has a good rating and can borrow at LIBOR without a credit spread). The
company fears increasing interest rates, and decides to swap the floating payments to
U
U
59
fixed. The current 12-month LIBOR rate is 4.00%, and bank offers a fixed swap rate of
4.3885% against 12 month LIBOR. The company agrees to pay the bank an annual
4.3885%, and receives LIBOR from the bank. The cash flows are (EUR millions):
Maturity
1
2
3
LIBOR debt
LIBOR
-4.0000
-LIBOR
-LIBOR
Swap floating to fixed
Receive LIBOR
Pay fixed
+4.0000
-4.3885
+LIBOR
-4.3885
+LIBOR
-4.3885
TOTAL
Debt + swap
-4.3885
-4.3885
-4.3885
Note that the payements from the LIBOR debt and the swap LIBOR payments cancel,
out and what is left are the fixed rate swap payments. Also note that the first payment of
the LIBOR debt is known, since the first payment is set at the reference rate, and reset at
the LIBOR rate after each interest rate payment.
•
•
note that principal (EUR 100 m) is not exchanged
the interest rate payments, must however be based on this value, why it is called
“notional principal”
6.1.1. The swap rate
•
•
how is the fixed swap rate set?
the swap rates that swap-dealers quote are usually valid only for well-known
counterparties with a good credit rating
swap rates are typically slightly higher than the yield for treasuries of the same
maturity, but lower than yields on corporate debt
swap rates are lower than corporate bond yields since the principal is not exchanged,
and hence the counterparty risk is limited to the exchange of interest payments
differences in counterparty credit quality can be managed by for example charging an
upfront fee, which will be higher for counterparties with lower credit quality,
collateral requirements, or other provisions
since bond yields depend on the coupon level, the swap rate is defined as a par bond
yield (the yield on a bond that trades at par, coupon rate = yield)
there is a tight link between swap rates and the interest rate futures (money market)
markets, since both contracts are priced used the same yield curve, swap dealers often
operate on both markets and can use futures for hedging swaps, and since a series of
futures can be constructed to replicate a swap
the par yield again:
•
•
•
•
•
•
Par − yield =
1 − Df T
T
∑ Df
i =1
i
where:
DfT
Dfi
B
B
B
B
= the discount factor using zero-coupon rates for the maturity T
= the discount factors using zero-coupon rates up to maturity T
60
Example: Calculate the swap rate if the zero-coupon rates are 4.00% for 1 year, 4.20%
for 2 years, and 4.40% for 3 years. Then:
U
U
Par − yield =
•
1 − 0.8788
= 0.043885
(0.9615 + 0.9210 + 0.8788)
we will now show why this must be the case, since an interest rate swap can be
constructed using interest rate forwards
6.1.2. The swap rate and FRA-rates
•
the 1-year interest rate forwards f12, and f23 are:
B
f12 =
f 23
B
B
B
(1 + 0.042) 2
− 1 = 0.044004
(1 + 0.40)
(1 + 0.044) 2
− 1 = 0.048012
=
(1 + 0.42)
Now, we construct a table where we instead hedge the LIBOR debt with forwards. We
buy 1-year FRA:s up to 3 years, which means that settlement is based on (LIBOR-FRA
rate). Assume the FRA:s are settled in cash. This means that if the LIBOR exceeds the
FRA rate, the difference is paid in cash. Cash flows in EUR millions:
Maturity
1
2
3
LIBOR
debt
LIBOR
-4.0000
-LIBOR
-LIBOR
FRA-rate
4.0000
4.4004
4.8012
FRA settlement
LIBOR-4.4004
LIBOR-4.8012
TOTAL
TOTAL
Difference
Debt+FRA
-4.0000
-4.4004
-4.8012
Debt+swap
-4.3885
-4.3885
-4.3885
Swap – FRA
+0.3885
-0.0119
-0.4127
The hedged interest rate payments are clearly not the same as the swap cash flows. How
about the present value of the series of FRA:s versus the swap?
PVSwap =
4.3885
4.3885
4.3885
+
+
= 4.2197 + 4.0419 + 3.8567 = 12.1183
2
(1 + 0.04) (1 + 0.042)
(1 + 0.044) 3
PV FRA =
4.0000
4.4004
4.8012
+
+
= 3.8462 + 4.0528 + 4.2193 = 12.1183
2
(1 + 0.04) (1 + 0.042)
(1 + 0.044) 3
That is, even if the individual cash flows are not equal, the PV of the swap payments
equals the PV of the swap. If the principal of EUR 100 million (PV = 87.8817) is
included in the calculations the PV is of course EUR 100 million.
61
6.1.3. Interest rate swap valuation
•
•
•
a swap is priced at par, such that the NPV is zero to both parties
after this the value of the swap may change
a swap can be valued as either:
1.
valued using the same principle as when valuing outstanding forwards: compare
the contract rate with the current market rate for the same maturity, and disocunt
the difference to present value, the result is a net present value (NPV)
valued as a bond, the result is a present value (PV), including the principal
2.
•
the values for outstanding forwards or futures:
PV ( F0 ) =
Long forward:
Ft − F0
(1 + rt )
PV ( F0 ) =
Short forward:
F0 − Ft
(1 + rt )
where
Ft
= the new forward rate prevailing at time t, for a forward identical to FT
B
B
B
•
•
B
Ft is the new swap rate for the same maturity T as the original swap to be valued (F0)
note that since there usually are multiple cash flows left, the equation will be
extended to a sum of all the differences Ft and F0
we also assume that the PV of the floating rate leg is zero at the coupon payment
date (not true between coupon payment dates)
B
B
B
B
•
B
B
B
B
Example: Suppose that after 1 year, the 1 and 2 year zero-coupon rates are 4.10%, and
4.25%. Then the 2 year swap rate is 4.2469%, with maturity T equal to the old 3 year
swap. What is the market value of the old swap now? We are “long” in the swap, since
we pay the fixed leg, and the bank is “short”. Then
U
U
F0
Ft
B
B
B
B
= 4.3885%
= 4.2469%
and:
T
NPV ( F0 ) = ∑
i =1
Ft − F0 4.2469 − 4.3885 4.2469 − 4.3885
=
+
= −0.2664
(1 + 0.041)
(1 + rt ) t
(1 + 0.0425) 2
which is in this example is in millions, or roughly EUR -266 400. The negative value for
the fixed rate payer is a result of lower swap rates, the fixed rate payer is now paying
above the market rate for swaps with equal maturity T. The bank, who is short the swap,
has naturally made an equally large gain.
62
Using the bond valuation approach:
T
PV ( Bond ) = ∑
i =1
•
•
the swap can naturally be valued at any time, not only at interest payment dates
then the value of the floating rate leg is not zero ⇒ the first payment should be
valued as a money market security
6.2.
•
•
•
•
•
•
•
CFt
4.3885
104.3885
=
+
= −100.2664
t
(1 + 0.041) (1 + 0.0425) 2
(1 + rt )
Currency swaps
now consider a borrower wishing to swap fixed payments in one currency to fixed
payments in another currency
using the previous 3 year EUR 100 debt issue as an example, consider that the issuer
wants to swap the debt to USD
the issuer may for example be a US issuer, finding better borrowing terms in EUR,
and wishing to hedge EUR outflows or a EUR-based issuer financing its US
operations with the debt, and wanting to use the US-revenue to repay the debt
an interest rate swap could be regarded as a sequence of FRA:s
a fixed-to-fixed currency swap can be regarded as a sequence of currency forwards
the swap rates for both legs are set as par yields as before
with interest rate swaps, principals are usually exchanged, and a currency swap then
consists of the following steps:
1. Initial exchange of currency principals. The currency spot rate is used to
determine the principal values. If for exampel S = EUR/USD 0.95, and EUR
100 million is swapped to USD, the notional principal in USD is USD 105.26
million. The interest rate payments will be based on these values. (If both
parties agree, no principal is exchanged, since the exchange is done at the
current spot rate and carries no risk, and thus does not require hedging).
2. Exchange of interest rate payments during the life of the swap.
3. Exchange back of principal. This step is taken even if principals were not
exchanged in 1., since the exchange back is risky and needs to hedged, why it
is included in the contract.
NOTE!: In these lecture notes a currency quote S(X/Y) means ”how many units
of currency Y does it take to buy 1 unit of currency X”. That is EUR/USD 0.95
means it takes 0.95 EUR to buy 1.00 USD. I know this is frustrating for those used
to the conventions used in the foreign currency markets. The reason is that this
way of quoting is algebraically correct, and hence more easily lends itself to
exercises involving inverting the quote, cross-rates, etc.
Example: Consider a corporation borrowing a fixed rate EUR 100 million for 3-years.
Interest rate payments are annual and the rate is 4.3885%. The company wishes to swap
the debt into USD. A bank offers to swap these EUR payments against a rate of 4.7906%
U
U
63
in USD. The spot currency rate is EUR/USD 0.95, and the USD principal is USD
105.26 million.
The interest rate cash flows in %:
EUR debt
Maturity
1
2
3
-4.3885
-4.3885
-4.3885
Swap EUR to USD
Receive EUR (%)
Pay USD (%)
+4.3885
-4.7906
+4.3885
-4.7906
+4.3885
-4.7906
TOTAL
Debt + swap (%)
-4.7906
-4.7906
-4.7906
The interest rate + principal cash flows in currency (millions):
EUR debt
Maturity
1
2
3
-4.3885
-4.3885
-104.3885
Swap EUR to USD
Receive EUR
Pay USD
+4.3885
-5.0428
+4.3885
-5.0428
+104.3885
-110.3059
TOTAL
Debt + swap USD
-5.0428
-5.0428
-110.3059
Note that the payments from the EUR debt and the swap payments cancel out and what
is left are the USD swap payments. Note the exchange back of the principals.
6.2.1. Currency swap rates and currency forward rates
•
•
•
note that the swap from EUR to USD is identical to a series of FX forwards
the swap is a series of forwards to buy EUR against USD
the exchange rates for three years implied by the swap:
Year 1:
Year 2:
Year 3:
EUR 4.3885/USD 5.0428
EUR 4.3885/USD 5.0428
EUR 104.3885/USD 110.3059
= EUR/USD 0.8703
= EUR/USD 0.8703
= EUR/USD 0.9464
when the current spot is EUR/USD 0.95. Is something wrong?
•
calculate the 1, 2, and 3 years currency forwards using CIRP, assume the USD zeros
are 4.50%, 4.65%, and 4.80% for years 1-3:
⎡1 + rd ⎤
⎡ (1 + 0.040)1 ⎤
= 0.9455
F = S0 * ⎢
⎥ = 0.95 * ⎢
1⎥
⎣ (1 + 0.045) ⎦
⎣⎢1 + r f ⎥⎦
and similarly for the 2 and 3 years forwards: EUR/USD 0.9418, and EUR/USD 0.9392.
Year
EUR debt
Swap to USD:
USD payments
PV of USD payments
FX to USD:
Forward rate
USD payments
PV of USD payments
0
+100.0000
1
-4.3885
2
-4.3885
3
-104.3885
105.2632
5.0428
4.8256
5.0428
4.6046
110.3059
95.8330
0.9455
4.6417
4.4418
0.9418
4.6595
4.2546
0.9392
111.1505
96.5667
0.9500
105.2632
64
Both with the swap and the FX hedge, the firms EUR receipts are set equal to the cash
outflow of the debt, and their PV is then EUR 100 m. Note that the PV of the individual
cash flows of the swap versus the FX hedge are not equal, but their sums are equal ⇒
the value of the swap equals the value of the series of currency forwards.
6.2.2. Currency swap valuation
•
•
the currency swap is priced at par, such that the NPV is zero to both parties
we will price the currency swap using the “bond approach”, since now principals
matter beacuse they are in different currencies and subject to currency risk
the value of the swap is the difference between the two bonds that constitute the
swap:
•
T
PV = ∑
i =1
T
CF (Re ceipts) t
CF ( Payments) t
−
S
∑
t
(1 + rR )
(1 + rP ) t
i =1
where
CF(Receipts)
CF(Payments)
rR
rP
S
B
B
B
•
B
= cash flow received in the swap
= cash flow paid in the swap
= zero-coupon discount rate for receipt when swap is valued
= zero-coupon discount rate for receipt when swap is valued
= spot currency rate between the two currencies when swap is valued
note that receipts and payments are in different currencies, and interest rates used
should be for that particular currency
Example: Suppose that after 1 year, the 1 and 2 year zero-coupon rates in EUR are
4.10%, and 4.25%, and in USD 4.60%, and 4.75%. The spot rate is EUR/USD 0.93.
What is the market value of the EUR-USD swap made at EUR 4.3885%-USD 5.0428,
with principals EUR 100, and USD 105.2632?
U
U
⎡ 4.3885
⎡ 5.0428
104.3885 ⎤
110.3059 ⎤
PV = ⎢
+
− 0.93 * ⎢
+
2 ⎥
2 ⎥
⎣ (1 + 0.041) (1 + 0.042) ⎦
⎣ (1 + 0.046) (1 + 0.0475) ⎦
PV = 100.2664 − 97.9754 = 2.2910
which in this example is in millions, or roughly EUR 2 291 000 for the USD payer.
•
like an interest rate swap the currency swap can naturally be valued at any time, not
only at interest rate payment dates
65
7.
Pricing credit risk
•
credit risk is usually priced by determining an appropriate additional credit spread on
top of a Treasury yield with same maturity
to date there exist no established and widely accepted model for pricing credit risk
one can roughly distinguish between two approaches:
•
•
1.
Traditional models. These are not really “pricing models”, since this approach
resembles much of equity analysis and there is no theoretically or other
established way in which to in a consistent manner combine company-specific
information (accounting etc.) into a yield, credit spread, or price. One can of
course use empirical information as guidelines, that is compare issuers with
similar characteristics and use price and spread information on these to get an
idea of what spread should be charged.
Models based on option pricing. This approach is strongly based on financial
pricing theory. Here, the value of the firm is seen as options, where equity seen
as a long call (due to limited liability of shareholders), and bonds as short put
options. Hence, bonds can be priced as options.
2.
7.1.
Credit ratings
• a credit rating is an assessment of a borrower’s willingness and ability to meet its
obligations for timely payments of principal debt and interest
• the most influential credit rating agencies: Moody’s, Standard & Poor’s, and Fitch
• issuers pay credit agencies an annual fee for maintaining a credit rating
• selling a bond issue without ratings can be difficult or impossible
• ratings scale and definition, long term-debt:
Rating agency
Moody’s
Aaa
Aa1
Aa2
Aa3
A1
A2
A3
Baa1
Baa2
Baa3
Ba1...Ba3
B1...B3
Caa
Ca
C
S&P
AAA
AA+
AA
AAA+
A
ABBB+
BBB
BBB-
Brief definition
I: Investment grade - High creditworthiness
Gilt edge, prime, maximum safety
Very high grade and quality
Upper medium grade
Lower medium grade
II. High yield - Low creditworthiness
BB+...BBLow grade, speculative
B+...BHighly speculative
III. Predominantly speculative - Substantial risk or in Default
CCC+...CCCSubstantial risk, in poor standing
CC
May be in default, extremely speculative
C
Even more speculative than those above
D
Default
Table adapted from: Fabozzi. Frank J. (1997): “Handbook of fixed income securities”. p. 226
66
• the better the rating, the lower the credit spread
• the markets roughly divide the bond markets into two sectors by rating:
1. Investment grade. Moody’s: Aaa – Baa3, S&P: AAA – BBB2. High yield, or junk bond. Moody’s: Ba1 – C, S&P: BB+ – D
• the scale for short term debt (commercial paper):
• Moody’s: P-1, P-2, P-3, NP (Not Prime)
• S & P: A-1, A-2, A-3, B, C, D
• credit ratings lowers the barrier for an investor to invest in a bond, since most bond
investors think that ratings quite accurately reflect the credit risk of the issuer, and
hence there is less uncertainty for the bond investor when making the investment
decision
• ratings are thus a natural starting point for evaluating credit risk
• note that ratings are only an assessment of credit risk (repayment risk), not interest
rate risk, liquidity risk, call risk or other risks
• ratings are not a recommendation to buy or sell a bond, a good rating does not make a
bond a better investment in terms of risk/return -> markets price credit risk and
ratings, and lower ratings come with higher yields
• the ratings and rating agencies have a very strong role in debt markets, since many
investors define their investment strategies according to ratings (for example mutual
funds), or are prohibited by regulators to invest in certain rating classes (for example
insurers)
• examples of some Nordic issuers (September 2003):
Issuer
ABB
UPM-Kymmene
UPM-Kymmene
Stora Enso
SCA
Elisa
Sonera Group
Metso
Ericsson LM
7.2.
Moody's
Ba3
Baa1
Baa1
Baa1
A3
Baa2
Baa1
Baa3
B1
S&P EUR
BB+
500
BBB
250
BBB
600
BBB+
850
A700
BBB+
300
A
300
BBB
500
BB
2000
CPN
9.500
6.350
6.125
6.375
5.375
6.375
4.625
6.25
7.875
Maturity Spread Swap S
15-01-08
534
522
01-10-09
95
80
23-01-12
107
87
29-06-07
70
59
25-06-07
57
35
31-06-06
141
111
16-04-09
72
53
11-12-06
155
131
31-05-06
537
512
The traditional approach to pricing credit risk
• the most important factors can be summarised into the 6 C:s of credit risk:
1. Character. Management, strategy, track record as borrower (reputation),
debt strategy.
2. Capital. Simply leverage.
3. Capacity. Cash flow, volatility of earnings, competitive position, industry
competition, cash liquidity, company structure
4. Cycle. How sensitive is the company to business cycles, what cycle are
we in now.
67
5. Collateral. Collateral, it’s market value, priority in default, what
possibilities to settle with other investors in default process.
6. Covenants. Callability, putability, restrictions on new debt, asset sales,
dividends etc.
• examples of how to mix these (usually) accounting-related variables into an empirical
model include for example Altman’s (1968) model or later variations of the same idea
• some special issues concerning high-yield (HY) issuers:
1. Debt structure. i) HY-issuers on average rely more on bank loans which
usually are senior to bonds (Cornish, 1990), partly because HY issuers
have limited access to CP-markets, ii) bank debt usually floating rate,
short term, can lead liquidity problems if short-term rates rise, iii) shortterm debt leads to frequent refinancing risk, and can lead to asset sales if
new debt is difficult to raise.
2. Enterprise structure. Leads to questions like: which part of the
enterprise is responsible for the debt, what is the operative and legal
structure, can money and assets be transferred from subsidiary to another,
what responsibility does the subsidiaries have?
3. Covenants. Understanding covenants becomes more important, as
financial distress is more likely.
4. What do the stock markets tell us? HY-debt have more companyspecific risk than investment grade issuers, what information can we use
from the stock markets/stock price about the state of the issuer?
7.3.
Using option theory to price credit risk
• the idea of using option theory in bond pricing started with Robert Merton’s article in
Journal of Finance (1974), hence, the term “Merton-like” models
• Merton’s simple model deals with valuing a default risky zero-coupon bond, but the
analysis can of course be extended (with some additional effort...) to coupon paying
bonds, realising that a coupon bond is a series of zero-coupon bond
• the idea of regarding a bond as an option is the limited upside of the bond: the value
of a zero can never exceed it’s par value, but the bond still has downside in terms of
default risk
• the example below shows the present value (PV), and Recovery Rate (RR) of a 3-year
zero-coupon bond relative to total company assets (belong to bondholders in case of
default)
• the idea is that if asset value is below the bond’s par value (100), and the bond
defaults, the Recovery Rate is received and 100 – RR is then the credit loss
• the same analysis can be made for each coupon to arrive at a series of present values
and adding them up to total bond value
68
Merton's model (1974)
Bond: N = 100, T = 3, CPN = 0%, r = 5%
Value of debt (PV and RR)
120
100
80
RR
PV
60
40
20
0
0
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150
Asset value
• the difference between the standard Black-Scholes-Merton model for pricing stock
options, and Mertons model for valuing risky debt is:
BSM put option:
Merton’s model:
A
B
r
sA
T
P = f(S, X, r, s, T)
L = f(A, B, r, sA, T)
= asset value
= nominal value of zero-coupon bond
= riskfree rate (Treasury)
= volatility of firm’s assets (A)
= maturity of bond in years
The model as presented in Saunders et al. (2002) and used for the above chart:
⎡1
⎤
L = Be −rT ⎢ N (h1 ) + N (h2 )⎥
⎣d
⎦
⎡1
⎤
⎡1
⎤
h1 = − ⎢ s 2T − ln(d )⎥ /( s T ), h2 = ⎢ s 2T + ln(d )⎥ /( s T )
⎣2
⎦
⎣2
⎦
where levarage is measured as: d = Be-rT/A
69
For example, if B = 100, A = 120, d = 0.80, T = 3, r = 0.05, s = 0.12
Bond price (L)
Spread
Price of defaultfree bond
= 84.63
= 0.56%
= 86.07
• the main difference is that the underlying asset is now the assets of the issuer, and
volatility is then the volatility of these assets
• this simple model suffers from obvious problems:
1. What is the market value of the firm’s assets A?
2. What is the volatility of these assets and how to measure it?
3. Based on European option pricing models, but default can happen at any
time.
4. Model allows default only if A = 0, in practice must happen before this.
• extension of this idea include the Black & Cox (1976) model where default can
happen before all assets are exhausted (A = 0)
• KMV (now a part of Moody’s) have attempted to solve problems 1-4 by:
1. Attempting to model the statistical process of A over time (how does the
value of A evolve over time, and what is its volatility).
2. A is modeled by modeling the stock price (E) behavior, and using
leverage as a link between the stock price process and asset value process.
The less debt, the closely A follows E.
3. Default if A goes below a specified value (makes this a barrier option).
• problems still remain:
1.
2.
3.
4.
How to in a reliable way model asset value process or default process?
How to reliably estimate recovery rate in case of default?
How to define the level of A that triggers default?
Does default occur because of low A or other reasons not (yet) in the
model such as liquidity crunch?
5. How to model the evolution of leverage since the relation between A and
E depends on this.
6. Very difficult to include all covenants and options (callability, putability)
in the model.
• the more realistic the model, the more difficult it becomes to estimate, use and
understand it -> significant model risk!
• the state of the world today: no accepted model
7.4.
Default probabilities, rating transitions, recovery rates and
how to use them to estimate bond returns
• what is the default probability (DP), recovery rate (RR), or loss given default (LGD)
• empirical results on US bond data by Altman & Kishore (1998) on high-yield:
70
Year
AADR
71-97
78-97
85-97
71-97
78-97
71-97
91
81
WADR
Median
Max.
Min.
AADR
WADR
RR
Loss
DR(%)
RR(%)
Loss
2.613
2.849
42.9
1.83
3.745
3.311
3.342
2.18
1.5
10.27
36.0
7.16
0.16
12.0
0.15
= arithmetic average of default rates (DR)
= par-value weighted average of default rates
= recovery rate as % of par value
= (LGD + next coupon) x DR
• note the rise and variability in the figures
• some statistics on recovery rates and seniority of bonds:
Seniority
Senior secured
Senior unsecured
Senior subordinated
Subordinated
Zeros
RR(%)
58.7
48.9
35.0
31.7
20.7
• investing in bonds with high spreads (yields) does not necessarily lead to high returns:
the risks priced into the spread might be realised causing a further widening of the
spread and price decline and loss of return
• rating ans spread changes are an essential part of corporate bond investing
• 1-year rating transition probabilities (RTP) by Carty & Fons (1994), using Moody’s
ratings 1970-1994:
Rating at end of year (T = 1)
Rating
T=0
Aaa
Aa
A
Baa Ba
B
Aaa
91.90 7.38 0.72 0.00 0.00 0.00
Aa
1.13 91.26 7.09 0.31 0.21 0.00
A
0.10 2.56 91.20 5.33 0.61 0.20
Baa
0.00 0.21 5.36 87.94 5.46 0.82
C, D Total
0.00 100
0.00 100
0.00 100
0.21 100
• conclusions:
1. Downgrades more likely than upgrades (fallen angels more likely than
rising stars)
2. Spread differences between rating categories widen with lower ratings,
adding to return loss due to price decline.
71
3. Adding up 1 and 2: Return < initial spread
• how use historical bond market information to estimate returns?
1. Estimate the bond’s RTP for the holding period.
2. Calculate the expected price change due to rating and spread change.
3. Calculate the expected return, take into account coupon payements and
reinvestment.
• for example, assume a 3-year AA-rated bond that now trades at a 30 bp spread, and
you want to estimate the expected return over a 1-year holding period:
Horizon
rating
Aaa
Aa
A
Baa
Ba
Horizon
spread
25
30
35
60
130
Horizon spread
Return over Treasuries
Excess return
Return
over Treas.
38
30
21
-24
-147
Transition
probability
1.13%
91.26%
7.09%
0.31%
0.21%
Excess
return
0.43
27.38
1.49
-0.07
-0.31
28.9
= spread now
= return in excess of Treasury yield assuming that the
rating of the bond changes to rating XXX (for example, a
downgrade to A increases spread to 35 bp and lowers the
price that of the initial spread of 30 bp, only 21 bp is left)
= Transition probability x Return over Treasuries
• using these values, the expected return of a AA-bond is not the initial spread of 30 bp,
but 28.9 bp
• some figures based on historical averages for different rating classes over different
holding periods (highest returns are bolded):
3-year
5-year
10-year
Rating Initial Excess Initial Excess Initial Excess
T = 0 spread return spread return spread return
Aaa
25.0
24.2
30.0
28.4
35.0
31.7
Aa
30.0
28.9
34.0
31.4
40.0
30.3
A
35.0
31.1
45.0
37.3
55.0
37.9
Baa
60.0
46.3
70.0
39.9
85.0
21.9
• that is for both 3- and 5-year horizons, Baa has the highets returns, but for a 10-year
horizon, A is the best rating class
• note that all such estimates are based on historical averages
72
7.5.
Selected empirical results on spreads
Bevan & Garzarelli (2000):
•
•
•
•
•
Moody’s Baa-index spread
1% increase in GDP growth
-> +18 bp
1% increase in financing gap
-> +46 bp
1 standard deviation increase on S&P 500 volatility
-> +25 bp
in the long run, correlation between spreads and benchmark yields positive, in the
short run negative
Campbell & Taksler (2002):
•
•
•
•
•
•
AA – BBB rated bonds
spreads decrease with S&P equity index returns
spreads increase with increase in S&P equity volatility
stock market volatility more important factor than ratings
ratings explain more of spread than accounting information
equity volatility more important factor for bonds with longer duration
Collin-Dufresne, Goldstein, Martin (2001):
•
•
•
•
•
•
leverage increases spreads, stock returns decreases spreads
S&P 500 returns 7 times more important factor than company specific equity return
Treasury yield increases decreases spreads
S&P 100 index option implied volatility increases spreads
only 25% of spreads can be explained
market-wide factors more important than company-specific factors, not consistent
with Merton-type models
• 75% of spreads unexplained, is strong bond/stock market segmentation the reason?
• liquidity, supply-demand shocks explanations?
i
Mind-expanding reading (NOT required for the CEFA-exam):
Das, Satyajit (1994): Swap & Derivative financing. Irwin Professional Publishing.
Eales, Brian A. (1995): Financial risk management. McGraw-Hill.
Fabozzi, Frank J. (2000): Bond markets, analysis and strategies. Prentice-Hall
International.
Galitz, Lawrence C. (1995): Financial engineering: Tools and techniques to manage
financial risk. Irwin Professional Publishing.
Garbade, Kenneth D. (1996): Fixed income analytics. The MIT Press.
de la Grandville, Olivier (2001): Bond pricing and portfolio analysis. The MIT Press.
Hull, John C. (2000): Options, futures, and other derivative securities. Prentice-Hall
International.
Jorion, Philippe (1996): Value at risk: The new benchmark for controlling market risk.
Irwin Professional Publishing.
Sundaresan, Suresh M. (1997): Fixed income markets and their derivatives. SouthWestern.
Tuckman, Bruce (1996): Fixed income securities: Tools for today’s markets. John Wiley
& Sons.