550.444
Introduction to
Financial Derivatives
Where we are
 Previously: Fundamentals of Forward and
Futures Contracts (Chapters 2-3, OFOD)
 This week: Introduction to Interest Rates and
the Value of Future Cash (Chapter 4, OFOD)
 Next week: Continuation of Interest Rates,
FRAs, Forward and Futures Prices (Chapters
4-5, OFOD)
Interest Rates
Week of September 24, 2012
1.1
Assignment

1.2
Assignment
For This Week (of September 24th )

 Read: Hull Chapter 4 (Interest Rates)
 Problems (Due September 24 - Today)
For Next Week
 Read: Hull Chapter 5 (FRAs, Forward & Futures Prices)
 Problems (Due October 1st )
 Chapter 3: 4, 7, 10, 17, 18, 20, 22; 29
 Chapter 3 (7e): 4, 7, 10, 17, 18, 20, 22; 26
 Chapter 4: 5, 8, 9, 11, 12, 14, 16, 22; 32
 Chapter 4 (7e): 5, 8, 9, 11, 12, 14, 16, 22; 27
 Problems (Due October 1st )
 Problems (Due October 8th )
 Chapter 4: 5, 8, 9, 11, 12, 14, 16, 22; 32
 Chapter 4 (7e): 5, 8, 9, 11, 12, 14, 16, 22; 27
 Chapter 5: 2, 4, 6, 7, 12, 16, 17, 20; 24
 Chapter 5 (7e): 2, 4, 6, 7, 12, 16, 17, 20; 24

1.3
Exams
 Final Exam: Dec 20th; 9:00am – Noon (Gilman 132)
1.4
 Midterm: October 17th (Tentative)
1
Interest Rates - Types of Rates
Plan for This Week

Interest Rates

 Present Value & Future Value
 Compounding frequency; Continuous
Compounding
 Spot Rate & Discount Function
 Yield measures
 The Yield Curve & Zero Curve
 Duration & Convexity




Treasury rates
Loan/Note/Bond rates
LIBOR rates
Repo rates
Risk Free Rate: LIBOR, Eurodollar
futures, and the swap market
1.5
Measuring Interest Rates


1.6
Measuring Interest Rates
The compounding frequency used for an
interest rate defines the units in which
an interest rate is measured
The difference between quarterly and
annual compounding is analogous to the
difference between miles and kilometers
 10 kilometers ≠ 10 miles
 10 KM = 6.214 miles

Future Value of a Present Dollar
FV(t0,T) = 1 x (1 + R)n
where T is a term of n years and R is the interest rate per annum,
on today, t0, with compounding once per annum
More generally,
FV(t0,T) = 1 x (1 + R/m)nm
where T is a term of n years and R is the interest rate per annum,
on t0, with compounding m times per annum
We might want to denote R as R(m) to emphasize compounding
frequency
For m=1, R = R(1), the equivalent annual interest rate
1.7
1.8
2
Measuring Interest Rates

Measuring Interest Rates
Equivalently, we think of the Present Value of a
Future Dollar


PV(t0,T) = 1 / (1 + R/m)nm
Where a Dollar is received on t0+T
Compounding Frequency
Annual (m=1)
Semiannual (m=2)
Quarterly (m=4)
Monthly (m=12)
Weekly (m=52)
Daily (m=365)
And the future unit of currency is discounted by
1 / (1 + R/m)nm

Note the effect of compounding frequency
Look at the FV of $100 deposited today @ 10%
In general, the PV of A dollars received on t0+T
PV(t0,T) = A / (1 + R/m)nm
FV(1 year)
$110.00
$110.25
$110.38
$110.47
$110.51
$110.52
1.9
Continuous Compounding


Continuous Compounding
In the limit as we compound more and more frequently
we obtain continuously compounded interest rates
FV of P dollars, invested today, t0, at rate R for n years,
and compounding k periods per year, is given by
FV(t0,n) = P x (1 + R/k)kn = P x [(1 + R/k)k]n = P [ f(k) ]n
where,
f(k) = [ 1 + R/k ]k

1.10



To find the limit of f(k) when the number of compounding
periods, k, increases without bound, take the natural
logarithm both sides of f(k) above,
ln [f(k)] = k x ln[ 1 + R/k ]
1.11

Let h = R/k
ln [f(k)] = k x ln[ 1 + R/k ] = R/h x ln(1+h) = R x ln(1+h)/h
So as, h->0, k->oo and as a consequence of L’Hopital’s rule
(when lim f(x) & lim g(x) are both 0
lim f(x)/g(x) = lim f’(x)/g’(x),
if RHS is finite) so,
As k->oo, the compounding frequency becomes arbitrarily often, and
lim ln[f(k)] = R x lim [ 1 / (1+h) ] = R => f(cc) = eR
(where the lhs lim is k->oo, and the rhs lim is h->0)
So from the previous slide, where there is continuous compounding
FV(t0,n) = P x (1 + R/k)kn -> P x eRn
as k -> oo and where R is the rate with continuous compounding
1.12
3
Conversion Formulas
Continuous Compounding

$100 grows to $100eRT when invested on t0 at a
continuously compounded rate R for time T
FV(t0,T) = 100eRT




$100 received at time T discounts to
t0 when the continuously compounded discount
rate is R
PV(t0,T) = 100e-RT
 R 
e Rc  1  m 
m

$100e-RT at
By convention we will use R for the continuously
compounded rate, RC
Define
Rc : continuously compounded spot rate
Rm: rate compounding m times per year
Defining equivalence as

m
Then we can show the conversion
R 

R c  m ln  1  m 
m 

R m  m e Rc / m  1



And we can determine R= Rc given any rate Rm
1.13
Continuous Compounding – Spot
Rate, Zero Yield, … , ? (terminology)





$100 grows to $100eRT when invested on t0 at a continuously
compounded (spot or zero) rate (yield) R for term T
FV(t0,T) = 100eRT
$100 received after term T discounts to $100e-RT at t0 when
the continuously compounded spot yield is R
PV(t0,T) = 100e-RT
By convention we will use R for the cc spot rate, Rc:
R = Rc = R(t0,T)
At each t0, and for every term, T, we have the cc spot rate
curve; sometimes called the Zero (yield) Curve.
We will always assume cc in our analysis, unless otherwise
stated
1.15
1.14
Zero Rate & Discount Function


Let R(t0,T) be today’s spot rate for term T; the rate of
interest earned on an investment that provides a payoff
only at time t0+T
More formally; for a cash flow, CF(t0+T), to be received on
t0+T, its present value is,
PV(t0,T) = CF(t0+T) e-R(t0,T)xT = CF(t0+T) x d(t0,T)


where d(t0,T) is the discount function ( d(t0,T) = e-R(t0,T)xT )
It is important to recognize that today’s discount function,
like the spot rate, is a function of term, T.
A cash flow, CF(t0+T), happens on a date, t0+T.
1.16
4
Bond Pricing
Example – Spot Rate & Discount
Maturity (Years)
Spot Rate (%)
Discount
PV of 6% 2-year
Bond (SA pay)
0.5
5.0
0.975310
2.925930
1.0
5.8
0.943650
2.830950
1.5
6.4
0.908464
2.725392
2.0
6.8
0.872843
89.902829
TOTAL =
98.385101


To calculate the cash “price” of a bond, discount
each cash flow at the appropriate zero rate
In our example, the theoretical price of a 2-year
bond providing a 6% coupon semiannually is
3e 0.05 0.5  3e 0.0581.0  3e 0.064 1.5
 103e  0.068 2 .0  98.39
1.17
Bond Yield

Bond Yield

The bond yield to maturity, y, is the single discount
rate for a “bullet” bond with term to maturity T
y(t0,T) = y
that makes the present value of the cash flows on the
bond equal to the market price of the bond,
n
n
i 1
i 1

PV (t0 )   CF (t0  Ti )  e  R ( t0 ,Ti )Ti   CF (t0  Ti )  e  yTi

1.18

c / 2, i  1,..., n  1
CF (t0  Ti )  
 100  c / 2, i  n
There is both a spot yield curve and a (term) yield
curve in this sense
The bond yield is the single discount rate that
makes the present value of the cash flows on
the bond equal to the market price of the bond
Suppose that the market price of the bond in our
example equals its theoretical price of 98.39
The bond yield (continuously compounded) is
given by solving (numerically)
3e  y  0.5  3e  y 1.0  3e  y 1.5  103e  y  2 .0  98.39
to get y = 0.0676 or 6.76%.
1.19
1.20
5
Par Yield or Par Coupon
Example – Bond Yield
Maturity (Years)
Yield (%)
Discount
PV of 6% 2-year
Bond (SA pay)
0.5
6.76
0.966765
2.900295
1.0
6.76
0.934634
2.803903
1.5
6.76
0.903572
2.710715
2.0
6.76
0.873541
89.974742
TOTAL =
98.389564


The par yield for a certain maturity is the
coupon rate that causes the bond price to
equal its par or face value.
In our example we solve (same spot curve)
c 0.050.5 c 0.0581.0 c 0.0641.5
e
 e
 e
2
2
2
c

 100  e 0.0682.0  100
2

to get c=6.87 (with s.a. compounding)
1.21
1.22
Example – Par Coupon (6.87SA)
- Yield = 6.7546 (cc)
Example – Par Coupon
Maturity (Years)
Spot Rate (%)
Discount
PV of 6.87% 2year Bond
Maturity (Years)
Par Coupon (%)
Discount
PV of 6.87% 2year Bond
0.5
5.0
0.975310
3.350190
0.5
3.435
0.96679
3.321
1.0
5.8
0.943650
3.241438
1.0
3.435
0.93476
3.211
1.5
6.4
0.908464
3.120574
1.5
3.435
0.90376
3.104
2.0
6.8
0.872843
90.282516
2.0
3.435
0.87387
90.389
TOTAL =
99.994718
TOTAL =
100.014
1.23
1.24
6
Par Yield continued


Par Yield or Par Coupon
In general if m is the number of coupon payments per
year, P is the present value of $1 received at maturity
and A is the present value of an annuity of $1 on each
coupon date
In our notation (where m = 2)
100 = c/m [d(.5)+d(1.0)+d(1.5)+d(2.0)] + 100(d(2.0)=P)
= c/m x A + 100 x P

n
c
100    e  R (t0 ,Ti )Ti  100  e  R (t0 ,Tn )Tn
2
i 1

so
or
(100  100 P ) m
A




And
c
(100  100 P ) m
A
1.25
Yield Curve

In general if m=2 is the number of coupon payments per year, P is
the present value of $1 received at maturity and A is the present
value of an annuity of $1 on each coupon date, Ti, then
c n
c

100    e  R ( t0 ,Ti )Ti   100  e  R ( t0 ,Tn )Tn   A  100  P
2  i 1
2

cA/m = 100 – 100P
c
The par yield or par coupon for a certain maturity is the coupon
rate that causes the bond price to equal its face value.
1.26
Yield Curve
The market expresses the current view on Par
Coupon instruments through the Yield Curve.
The yield curve is the curve generated by the set
of current instruments arranged by maturity.
For the US Government these instruments include
1, 3, 6, & 12 –month bills (zeros) and the 2, 3, 5,
7, 10, & 30 –year notes and bonds (SA pay)
These yields are tracked closely during market
hours and published daily by the financial media.
1.27

From the yield curve we may construct the zero
(spot) rate curve from which all other
instruments of like credit can be priced
 Spot curve gives us the discount function
 Fair prices are constructed by present valuing the
cash flows

As the US government is considered as having
no credit risk, all other bonds may be valued by
adding a “spread” to the Treasury yield curve
1.29
7
Bond Pricing

Sample Data for Constructing Zero Rates
Bond
Time to
Annual
Principal
Maturity
Coupon
Price
(dollars)
(years)
(dollars)
(dollars)
n
100
0.25
0
97.5
i 1
100
0.50
0
94.9
100
1.00
0
90.0
100
1.50
8
96.0
100
2.00
12
101.6
To calculate the cash price of a bond, discount each
cash flow at the appropriate zero rate, R(t0,Ti),
PV (t0 )   CF (t0  Ti )  e  R (t0 ,Ti )Ti
c / 2, i  1,..., n  1
CF (t0  Ti )  
 100  c / 2, i  n
Bond Cash
1.30
The Bootstrap Method



1.31
The Bootstrap Method (continued)
An amount 2.5 can be earned on 97.5 during 3 months.
97.5=100/(1+R/4) or R=4(100/97.5 – 1) = 10.256%
With quarterly compounding
This is 10.127% with continuous compounding
Similarly the 6 month and 1 year rates are 10.469% and
10.536% with continuous compounding
94.9=100/(1+R/2) or R=2(100/94.9 – 1) = 10.748%
90.0=100/(1+R) or R=(100/90 – 1) = 11.111%
Using the conversion formula,
Rc=m x ln(1 + Rm/m) = 2 x ln(1+.05374) = .10469
and
ln (1 + .11111) = .10536
1.32

To calculate the 1.5 year spot rate we solve
4e 0.10469 0.5  4e 0.10536 1.0  104 e  R1.5  96

to get R = 0.10681 or 10.681%
Similarly the two-year rate is 10.808%
1.33
8
Forward Rates
Zero Curve Calculated from the Data
12

Zero
Rate (%)
The forward rate is the future spot rate (yield) implied by
today’s term structure of interest rates
e R (T1 )T1  e F (T2 T1 )(T2 T1 )  e R (T2 )T2
11
10.808
10.681
10.469
10

10.536
10.127
e R ( t0 ,T1 )T1  e F (t0 ,T1 ,T2 T1 )(T2 T1 )  e R (t0 ,T2 )T2
Maturity (yrs)
9
0
0.5
1
1.5
2
More precisely, as we refer to today’s spot rate for term
T as, R(t0,T), we should recognize the T-term forward
rate for period T as, F(t0, T , T )
2.5
1.34
Calculation of Forward Rates
Calculation of Forward Rates

Using the convenient properties of the exponential
function,
Zero Rate for
Year (n )
Taking natural log, ln, of both sides,
ln e R (T 1)T 1 F (T 2T 1)(T 2T 1)  ln e R (T 2)T 2
R(T1 )  T1  F (T2  T1 )  (T2  T1 )  R(T2 )  T2

From which we can solve for the Forward Rate
R (T2 )T2  R (T1 )T1
F (T2  T1 ) 
T2  T1
Forward Rate
an n -year Investment for n th Year
e R (T 1)T 1 F (T 2T 1)(T 2T 1)  e R (T 2 )T 2

1.35
1.36
(% per annum)
(% per annum)
1
3.0
2
4.0
5.0
3
4.6
5.8
4
5.0
6.2
5
5.3
6.5
1.37
9
Formula for Forward Rates


Instantaneous Forward Rate
Suppose that the zero rates for time periods T1
and T2 are R1 and R2 with both rates continuously
compounded.
The forward rate for the period between times T1
and T2 is
R 2 T 2  R1T1
T1
 R 2  ( R 2  R1 )
T 2  T1
T 2  T1


The instantaneous forward rate for a maturity T
is the forward rate that applies for a very short
time period starting at T. It is
RT

where R is the T-year spot (zero) rate
1
And since P(0, T )  e RT  R   ln P(0, T ) 
1
R

T
T
1

T
Taking the limit as T2 approaches T1, the
common value being T gives
1.38
R
1

ln P (0, T )  2 ln P (0, T )
T
T

R
ln P (0, T ) 
T
T
1.39
Upward vs Downward Sloping
Yield Curve
Instantaneous Forward Rate

R
T
we get that
RF  R  T
R
T
R
 1 
 R  T 
ln P(0, T )  
T
T
T




ln P (0, T )

T
1.40

For an upward sloping yield curve:
Fwd Rate > Zero Rate

For a downward sloping yield curve
Zero Rate > Fwd Rate
1.41
10
Forward Rate Agreement
(continued)
Forward Rate Agreement

A forward rate agreement (FRA) is an
agreement that a certain rate will apply to
a certain principal during a certain future
time period


An FRA is equivalent to an agreement
where interest at a predetermined rate, RK
is exchanged for interest at the market
rate
An FRA can be valued by assuming that
the forward interest rate is certain to be
realized
1.43
Duration
Valuation Formulas






1.44
Value of FRA where a fixed rate RK will be received on a
principal L between times T1 and T2 is
L( RK  RF )(T2  T1 )e  R2T2
Value of FRA where a fixed rate is paid is
L( RF  RK )(T2  T1 )e  R2T2
RF is the forward rate for the period and R2 is the zero
rate for maturity T2
What compounding frequencies are used in these
formulas for RK, RM, and R2?
RK & RM are defined by the period of the forward T  T2  T1
R2 is continuously compounded
1.45


Bond Duration is the (value-weighted) average time to receipt of cash
Duration, D, of a bond that provides cash flow c i at time t i is
n
t
D 
n

i 1
 yti
n
dB
  ci t i e  yti
dy
i 1
B
 Dy
   B  D   y and
B
dB
y
dy
n

 c i e  yt i 


 B 
with price B   ci e
and (continuously compounded) yield y
i 1
Since a small change in yield, y, leads to a change in price as
B 

i
and
So B  y   ci ti e yt
i 1
The percentage change in price is negatively related to a change in
yield.
i
1.46
11
Duration Continued

When the yield y is expressed with compounding m times per
year
 D 
BDy
B  

The expression


c t
2  yt i
i i e
1  B
i 1

C
B y 2
B
so that
B
1
  Dy  C ( y ) 2
2
B
2
D
1 y m
defines D* and it is referred to as Modified Duration
Duration measures are important for risk management
B
  D * y
B

The convexity of a bond is defined as
n
 B
 y
1 y m
1  y / m 
D* 

Convexity

Which exhibits the well known characteristic for bonds that yield
and price move in opposite direction
Note that for continuous compounding, duration doesn’t have to
be “modified”.
1.47
A useful result to show the limitation of duration hedging
as yield changes become larger – the effect of the nonlinearity of the price-yield relationship
1.48
12