Bonds 1:
Bond price equation
Price-yield relationship: A fundamental property of a bond is that its price changes in the
opposite direction from the change in the required yield.
Measuring Yield: Yield to maturity, Holding-Period Return, Yield to call
Bonds 2: Price Volatility characteristics of bonds
Convex shape of the price-yield relationship
There are two characteristics of a bond that determine its price volatility: coupon and
term to maturity
Duration is the average maturity of a cash flow stream associated with a bond
Equivalently it is: - %change in price/%change in (1+ytm)
Modified duration gives approximate percentage change in price for a given change in
yield. For small changes in the required yield, modified duration gives a good
approximation of the percentage change in price.
Immunization is a technique designed to achieve a specified return target in the face of
changes in interest rates.
Horizon Analysis
Selection of a holding period for analysis and consideration of possible yield structures at
the end of the horizon.
return ($ or %) = time + yield change + coupons + interest on
effect
effect
received coupons
1
Problem 8
Consider a bond selling at its par value of $1,000, with six years to maturity and a 7%
coupon rate (with annual interest payments) Calculate the bond’s duration.
Answer: 2.8 years
Problem 9
What is the modified duration of the bond in Problem 8?
Answer: 2.6 years
Problem 24
Consider a bond with $1,000 face value, ten years to maturity, and $80 annual coupon
interest payments. The bond sells so as to produce a 10% yield-to-maturity. That yield is
expected to decline to 9% at the end of four years. Interest income is assumed to be
invested at 9.5%. Calculate the bond’s four-year holding period return and the four
components of that return.
Answer: Overall return= 4.1% (time change ) + 4.8% (yield change) +36.5% (coupon) +
5.5% (interest on coupon)= 50.9%
Quiz question
You are planning to offset a single-payment liability with an immunized bond portfolio.
Your liability is due in three years and amounts to $1,200,000. You have decided to form
your portfolio by using bonds A and B. Bond A and B have six and two year time-tomaturity, respectively. Bond A is a pure-discount bond while Bond B has a coupon rate
of 8% and makes annual coupon payments. If both bonds have face values of $1,000 and
yields-to-maturity of 12%, how many units of each bond should you purchase to form
your immunized portfolio? In other words, you need to find your total dollar
investment today and the way you need to allocate this amount between bonds A and
B by finding how many units of each bond you should buy today.
Bonds 3: Turkish Treasury securities
Variable coupon bonds
Bond with variable coupon indexed to 3-month reference auction
Identifier for bonds/bills
Eurobonds
2
Yield to Maturity and Spot Rates
Spot rate is YTM on a pure discount security (e.g. zero-coupon bond)
PV of a riskless security that pays C1 in one year and C2 in two years
A future interest rate calculated from spot rates is called a forward rate.
The relationship between the yields on otherwise comparable securities with different
maturities is called the term structure of interest rates.
The graphical depiction of this relationship is known as the yield curve.
es1,2 = f1,2
Unbiased Expectations Theory
f1,2 = es1,2 + L1,2
where L1,2 > 0
Liquidity Preference Theory
f1,2 = es1,2 + L1,2
where L1,2 can be + or -
Preferred Habitat Theory
3
Measuring Portfolio Performance
Multi period return with cash inflows and outflows:
Dollar weighted return
Time weighted return (geometric and arithmetic)
Risk-adjusted measure of performance
There are two possible measures of risk that can be used:
total risk or systematic risk.
Measures based on ex-post SML
Jensen’s alpha
Find the return of the benchmark portfolio by using the ex-post SML.
avg return of benchmark portfolio: arbp = arf + (arm – arf) p
Compare the benchmark portfolio return to arp
p : ex post alpha or differential return
p = arp – arbp

for risk-adjustment
The second measure based on the ex-post SML is the reward to Volatility Ratio (RVOLp)
RVOLp is closely related to p
RVOLp =
arp  arf
p
portfolio excess return over 
The line drawn by using two data points: (0, arf) and (p, arp) has the slope of RVOLp
Benchmark to compare RVOLp is slope of ex post SML.
ex post SML line drawn by using two data points (0, arf) and (1, arm) has
slope =
arm  arf
1
4
Measures based on ex-post CML
The first measure is the Sharpe Ratio (SRp)
Both p and RVOLp measure return relative to market risk of portfolio.
Sharpe ratio expresses returns by using the ex post CML (total risk of portfolio).
Ex post CML line drawn by using two data points (0, arf) and (m, arm)
arm  arf
arx = arf +
SRp =
m
x
for any portfolio x on CML
arp  arf
portfolio excess return over its total risk
P
SRp is the slope of line drawn by using two data points (0, arf) and (p, arp)
Performance is measured by comparing SRp to the slope of ex-post CML
The second measure based on ex-post CML is the M2
Combine rf and portfolio to create a portfolio that has the same total risk as market
portfolio.
Rcomb = (1-Wp) arf + Wp Rp
comb = Wp p
since comb =m then Wp =
m
P
plug in the weight into the expression for Rcomb, you get Mp2
Mp2 = arf +
(arp  arf )
p
m
Performance is measured by comparing Mp2 to arm
5
Problem 28
Consider the following annual returns produced for Minifund, a mutual fund
investing in small stocks.
Referring to Table 1.1, use the Treasury bill returns as the risk free return and the
common stock returns as the market return, and calculate the following riskadjusted return measures for the small stock mutual fund:
a. Ex-post alpha
b. Reward-to-volatility ratio
c. Sharpe ratio
Year MiniFund
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
p =
16.50%
4.43%
-30.90%
-19.95%
52.82%
57.38%
25.38%
23.46%
43.46%
39.88%
13.88%
28.01%
39.67%
-6.67%
24.66%
6.85%
-9.30%
22.87%
10.18%
-21.56%
T-Bill Common
Stocks
4.39%
14.31%
3.84%
18.98%
6.93% -14.66%
8.00% -26.47%
5.80%
37.20%
5.08%
23.84%
5.12%
-7.18%
7.18%
6.56%
10.38%
18.44%
11.24%
32.42%
14.71%
-4.91%
10.54%
21.41%
8.80%
22.51%
9.85%
6.27%
7.72%
32.16%
6.16%
18.47%
5.47%
5.23%
6.35%
16.81%
8.37%
31.49%
7.81%
-3.17%
1
(er pt  er p )(er mt  er m )
(Ter pt er mt )  (er pt er mt )
T

2
(Ter 2 mt )  (er mt ) 2
(er mt  er m )
T
Ex-post beta of the portfolio
6
Year MiniFund
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
16.50%
4.43%
-30.90%
-19.95%
52.82%
57.38%
25.38%
23.46%
43.46%
39.88%
13.88%
28.01%
39.67%
-6.67%
24.66%
6.85%
-9.30%
22.87%
10.18%
-21.56%
MiniFund
average
std dev.
SML
Rbp
16.05%
24.16%
=
For MiniFund
beta
=
T-Bill Common MiniFund
Stocks
Excess
4.39% 14.31% 12.11%
3.84% 18.98%
0.59%
6.93% -14.66% -37.83%
8.00% -26.47% -27.95%
5.80% 37.20% 47.02%
5.08% 23.84% 52.30%
5.12%
-7.18% 20.26%
7.18%
6.56% 16.28%
10.38% 18.44% 33.08%
11.24% 32.42% 28.64%
14.71%
-4.91%
-0.83%
10.54% 21.41% 17.47%
8.80% 22.51% 30.87%
9.85%
6.27% -16.52%
7.72% 32.16% 16.94%
6.16% 18.47%
0.69%
5.47%
5.23% -14.77%
6.35% 16.81% 16.52%
8.37% 31.49%
1.81%
7.81%
-3.17% -29.37%
Sumproduct
Common
Excess
0.663
MiniFund
Excess
Common
Excess
Common
Stocks
7.69% 12.49%
2.61% 16.61%
0.620
T-Bill
7.69% +
1.02
Rbp
=
12.56%
alpha
=
RVOL
=
Sharpe ratio
=
Common
Excess
9.92%
15.14%
-21.59%
-34.47%
31.40%
18.76%
-12.30%
-0.62%
8.06%
21.18%
-19.62%
10.87%
13.71%
-3.58%
24.44%
12.31%
-0.24%
10.46%
23.12%
-10.98%
Sum
beta
MiniFund
Excess
167.31%
Common
Excess
95.97%
4.80%
For Common
beta
=
1
Rbp
=
12.49%
3.49%
alpha
=
0.00%
8.24%
RVOL
=
4.80%
Sharpe ratio
=
28.89%
34.62%
7
Timing ability
Timing ability can be measured by fitting a quadratic curve to the performance data
An alternative way to analyze market timing is to fit two separate lines.
8
Problem 29
A portfolio managed by Keynes had the following returns:
An article concluded that Keynes demonstrated superior investment abilities.
They did not, however, distinguish between his market timing and security
selection skills. Using the quadratic regression and dummy variable techniques,
evaluate Keynes’s market timing skills.
Year Keynes's
Return
1928
-3.40%
1929
0.80%
1930 -32.40%
1931 -24.60%
1932
44.80%
1933
35.10%
1934
33.10%
1935
44.30%
1936
56.00%
1937
8.50%
1938 -40.10%
1939
12.90%
1940 -15.60%
1941
33.50%
1942
-0.90%
1943
53.90%
1944
14.50%
1945
14.60%
Market Riskfree
Return
Return
7.90%
4.20%
6.60%
5.30%
-20.30%
2.50%
-25.00%
3.60%
-5.80%
1.60%
21.50%
0.60%
-0.70%
0.70%
5.30%
0.60%
10.20%
0.60%
-0.50%
0.60%
-16.10%
0.60%
-7.20%
1.30%
-12.90%
1.00%
12.50%
1.00%
0.80%
1.00%
15.60%
1.00%
5.40%
1.00%
0.80%
1.00%
9
Year Keynes's
Return
1928
-3.40%
1929
0.80%
1930 -32.40%
1931 -24.60%
1932 44.80%
1933 35.10%
1934 33.10%
1935 44.30%
1936 56.00%
1937
8.50%
1938 -40.10%
1939 12.90%
1940 -15.60%
1941 33.50%
1942
-0.90%
1943 53.90%
1944 14.50%
1945 14.60%
average
13.06%
Sumproduct Market
Excess
Keynes's
0.46678
Excess
Market
Excess
Market Riskfree Keynes's Market
Return Return
Excess Excess
7.90%
4.20%
-7.60%
3.70%
6.60%
5.30%
-4.50%
1.30%
-20.30%
2.50% -34.90% -22.80%
-25.00%
3.60% -28.20% -28.60%
-5.80%
1.60% 43.20% -7.40%
21.50%
0.60% 34.50% 20.90%
-0.70%
0.70% 32.40% -1.40%
5.30%
0.60% 43.70%
4.70%
10.20%
0.60% 55.40%
9.60%
-0.50%
0.60%
7.90% -1.10%
-16.10%
0.60% -40.70% -16.70%
-7.20%
1.30% 11.60% -8.50%
-12.90%
1.00% -16.60% -13.90%
12.50%
1.00% 32.50% 11.50%
0.80%
1.00%
-1.90% -0.20%
15.60%
1.00% 52.90% 14.60%
5.40%
1.00% 13.50%
4.40%
0.80%
1.00% 13.60% -0.20%
-0.11%
1.57%
beta
1.77
SML
Rbp
=
Rbp
=
-1.40%
alpha
=
14.46%
1.57% + beta
-1.67%
0.28714
Sum
Keynes's
Excess
206.80%
Market
Excess
-30.10%
10
Regress excess Keynes on excess market
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.766777465
R Square
0.587947682
Adjusted R Square
0.562194412
Standard Error
0.197557515
Observations
18
ANOVA
df
1
16
17
SS
0.891032229
0.624463549
1.515495778
MS
0.891032
0.039029
F
22.83002
Coefficients
0.144608038
1.777224877
Standard Error
0.046978328
0.371953999
t Stat
3.078186
4.778077
P-value
0.007201
0.000205
Regression
Residual
Total
Intercept
X Variable 1
Significance F
0.000205306
Regress excess Keynes on excess market and excess market squared
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.76918035
R Square
0.591638411
Adjusted R Square
0.5371902
Standard Error
0.203120697
Observations
18
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
X Variable 2
2
15
17
SS
0.896625514
0.618870263
1.515495778
MS
0.448313
0.041258
F
10.86608
Coefficients
0.157937339
1.699455146
-0.917109604
Standard Error
0.06036196
0.436880312
2.490817578
t Stat
2.616504
3.889979
-0.3682
P-value
0.019449
0.001451
0.717871
Significance F
0.001210143
11
Regress excess Keynes on excess market and (dummy * excess market )
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.767147086
R Square
0.588514652
Adjusted R
Square
0.533649939
Standard Error
0.203896102
Observations
18
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
X Variable 2
2
15
17
SS
0.89189147
0.623604308
1.515495778
MS
0.445946
0.041574
F
10.72665
Coefficients
0.152830013
1.66374052
-0.180708664
Standard Error
0.074977755
0.877778239
1.256985943
t Stat
2.038338
1.8954
-0.14376
P-value
0.059549
0.077477
0.887601
Significance F
0.001281321
12
Performance attribution procedures
Asset allocation decision: Being in equities as opposed to fixed-income securities
when the stock market is performing well
Sector and security allocation decision: Choosing the relatively better sectors and
stocks.
Benchmark portfolio versus Managed portfolio
13
Quiz question
Consider the following information regarding the performance of a money manager in a recent
month. The table represents the actual return of each sector of the manager’s portfolio in column
1, the fraction of portfolio allocated to each sector in column 2, the benchmark sector allocations
in column 3, and the returns of sector indices in column 4.
Equity
Bonds
Cash
Actual Return
8.0%
2.0%
1.0%
Actual Weight
0.7
0.2
0.1
Benchmark Weight
0.6
0.3
0.1
Benchmark Index Return
10.0%
(S&P 500)
3.0% (Salomon Index)
1.0%
a. What was the manager’s return in the month? What was her overperformance or
underperformance?
b. What was the contribution of security selection to relative performance?
c. What was the contribution of asset allocation to relative performance? Confirm that
the sum of selection and allocation contributions equals her total “excess” return
relative to the benchmark.
Equity
Bonds
Cash
actual return
benchmark return
excess return
Equity
Bonds
Cash
Actual Return Actual Weight
8.0%
2.0%
1.0%
0.7
0.2
0.1
Benchmark Weight Index Return
0.6
0.3
0.1
6.10%
7.00%
-0.90% part a
Actual Weight Benchmark Weight Index Return
Contribution
0.7
0.6
10.0%
0.2
0.3
3.0%
0.1
0.1
1.0%
part c
Equity
Bonds
10.0%
3.0%
1.0%
Actual Weight Actual Return
Index Return
Contribution
0.7
8.0%
10.0%
0.2
2.0%
3.0%
part b
1.00%
-0.30%
0.00%
0.70%
-1.40%
-0.20%
-1.60%
14
Futures
A forward contract is an agreement to buy or sell an asset on a specified future date for a
specified price, referred to as the delivery price.
Delivery price is chosen so that the value of the forward contract to both parties is zero.
A forward contract is settled at maturity when the holder of the short position delivers the
asset to the holder of the long position in return for the delivery price.
Future contracts are standardized in terms of their specifications, such as maturity date
and delivery price
This feature allows future contracts to be traded on organized exchanges.
Margin requirements on both buyers and sellers
Mark to market the accounts of buyers and sellers every day
Futures prices will deviate from forward price when marking to market gives a
systematic advantage to either the long or short position.
Whenever there is a positive correlation between interest rates and changes in futures
prices, futures price will exceed the forward price.
We can generalize the relation between futures (forward) prices and current spot prices as
F = Ps + I – B + C
Buying the futures contract means postponing the purchase of the underlying asset.
In doing this, one earns interest on the amount not spent today, loses the benefits of
ownership and avoids costs of ownership.
F Futures price
Ps Current spot price
I The dollar amount of interest corresponding to the period of time from the present to
the delivery date. Since, today the asset is not purchased, interest can be earned from
current spot price of the asset until the maturity.
B Value of the benefits of the ownership.
C Cost of ownership
B, I, C are values at the delivery date of futures contract
15
Problem 21
Assume that the S&P 500 currently has a value of 200 (in “index” terms). The dividend
yield on the underlying stocks in the index is expected to be 4% over the next six months.
New-issue six month treasury bills now sell for a six-month yield of 6%.
What is the theoretical value of a six-month futures contract on the S&P 500?
16