BUS 424
Homework 3 – Due 10/31/2012
Chapter 22: 10, 12
Chapter 23: 10
Chapter 24: 5, 25 (a through g)
Chapter 22: 10
10. Below are two portfolios with a market value of $500 million. The bonds in both
portfolios are trading at par value. The dollar duration of the two portfolios is the same.
Issue
A
B
C
D
E
F
G
Years to Maturity
Par Value (in millions)
Bonds Included in Portfolio I
2.0
$120
2.5
$130
20.0
$150
20.5
$100
Bonds Included in Portfolio II
9.7
$200
10.0
$230
10.2
$ 70
Answer the below questions.
(a) Which portfolio can be characterized as a bullet portfolio?
In a bullet strategy, the portfolio is constructed so that the maturities of the securities in the
portfolio are highly concentrated at one point on the yield curve. Thus, Portfolio II can be
characterized as a bullet portfolio because the maturities of its securities are concentrated around
one maturity (ten years).
(b) Which portfolio can be characterized as a barbell portfolio?
In a barbell strategy, the maturities of the securities included in the portfolio are concentrated at
two extreme maturities. Thus, Portfolio I can be characterized as a barbell portfolio because the
maturities of its securities are concentrated at two extreme maturities (two years and twenty years).
(c) The two portfolios have the same dollar duration; explain whether their performance
will be the same if interest rates change.
Even if the yield curve shifts in a parallel fashion due to changes in interest rates, two portfolios
with the same dollar duration will not give the same performance if they have differences in dollar
convexity. Although with all other things equal it is better to have more convexity than less, the
market charges for convexity in the form of a higher price or a lower yield. But the benefit of the
greater convexity depends on how much yields change. As can be seen from the illustration in
the second column of Exhibit 22-9, if market yields change by less than 100 basis points (up or
down), the bullet portfolio, which has less convexity, will provide a better total return that the
barbell portfolio.
The last two columns Exhibit 22-9 illustrate the relative performance of a bullet portfolio and
a barbell portfolio for a nonparallel shift of the yield curve. Specifically, the first nonparallel
shift column assumes that if the yield on the bullet portfolio (consisting of the intermediate-term
bond) changes by the amount shown in the first column, the short-term bond in the barbell
portfolio will change by the same amount plus 25 basis points, whereas the long-term bond in the
barbell portfolio will change by the same amount shown in the first column less 25 basis points.
Measuring the steepness of the yield curve as the spread between the long-term yield and the
short-term yield, the spread has decreased by 50 basis points. Such a nonparallel shift means
a flattening of the yield curve. As can be seen in the exhibit, for this assumed yield curve shift,
the barbell outperforms the bullet.
In the last column of Exhibit 22-9, the nonparallel shift assumes that for a change in the
intermediate bond’s yield, the yield on the short-term will change by the same amount less 25
basis points, whereas that on the long-term bond will change by the same amount plus 25 points:
Thus, the spread between the long-term yield and the short-term yield has increased by 50 basis
points, and the yield curve has steepened. In this case the bullet portfolio outperforms the barbell
portfolio as long as the yield on the intermediate bond does not rise by more than 250 basis
points or fall by more than 325 basis points.
(d) If they will not perform the same, how would you go about determining which would
perform best assuming that you have a six-month investment horizon?
To determine which portfolio would have the superior performance, we would want to look at
the total return for the six-month investment horizon given expectations about change in yields
and how the yield curve will shift. More details are given below.
It is important to note that measures such as yield (yield to maturity or some type of portfolio
yield measure), duration, or convexity tell us little about performance over some investment
horizon, because performance depends on the magnitude of the change in yields and how the
yield curve shifts. Therefore, when a manager wants to position a portfolio based on expectations
as to how he might expect the yield curve to shift, it is imperative to perform total return
analysis. For example, in a steepening yield curve environment, it is often stated that a bullet
portfolio with the same duration as a barbell portfolio would perform better that the barbell
portfolio. However, as can be seen from Exhibit 22-9, it is not the case that a bullet portfolio
would outperform a barbell portfolio. Whether the bullet portfolio outperforms the barbell
depends on how much the yield curve steepens. An analysis similar to that in Exhibit 22-9 based
on total return for different degrees of steepening of the yield curve clearly demonstrates to
a manager whether a particular yield curve strategy will be superior to another. The same
analysis can be performed to assess the potential outcome of a ladder strategy.
Chapter 22: 12. What is a laddered portfolio?
A ladder portfolio is constructed to have approximately equal amounts of each maturity. So, for
example, a portfolio might have equal amounts of securities with one year to maturity, two years
to maturity, and so on.
5. 23.7. Answer the below questions.
(a) Compute the tracking error from the following information:
Month 2001
Portfolio A’s Return (%)
Lehman Aggregate Bond Index Return (%)
January
February
March
April
May
June
July
August
September
October
November
December
2.15
0.89
1.15
–0.47
1.71
0.10
1.04
2.70
0.66
2.15
–1.38
–0.59
1.65
–0.10
0.52
–0.60
0.65
0.33
2.31
1.10
1.23
2.02
–0.61
–1.20
The tracking error is the standard deviation of the active returns where an active return is the portfolio A’s
return minus the benchmark’s return for each month. The below exhibit has each active return in the
“Active Return” column. [Note that when subtracting a negative index return from a portfolio return, the
negative return is actually added to the portfolio return to get the active return (i.e., for February, we have
0.89% – –0.10% = 0.89% + 0.10% = 0.99%).] To compute the standard deviation of these active returns,
we subtract the average (or mean) active return from each active return, and then square each difference.
Each difference squared value is given in the exhibit below in the “Differences Squared” column. We
then divided this sum by 12 – 1 = 11. We then multiply by 100 to convert to basis points. One basis point
equals 0.0001 or 0.01%. We can then annualize the monthly basis points by multiplying by the square
root of 12. At the bottom of the below table we list details including the mean active return, variance,
standard deviation or tracking error (in terms of both percentage and basis points), and the annualized
tracking error (in terms of basis points).
Month 2001
January
February
March
April
May
June
July
August
Portfolio A’s
Return
Lehman Aggregate
Bond Index Return
Active Return
Differences
Squared
2.15%
0.89%
1.15%
–0.47%
1.71%
0.10%
1.04%
2.70%
1.65%
–0.10%
0.52%
–0.60%
0.65%
0.33%
2.31%
1.10%
0.50%
0.99%
0.63%
0.13%
1.06%
-0.23%
-1.27%
1.60%
0.0707(%2)
0.5713(%2)
0.1567(%2)
0.0109(%2)
0.6820(%2)
0.2155(%2)
2.2625(%2)
1.8655(%2)
September
October
November
December
0.66%
2.15%
–1.38%
–0.59%
1.23%
2.02%
–0.61%
–1.20%
-0.57%
0.13%
-0.77%
0.61%
Mean Active Return =
0.2342%
Variance (sum of differences squared / 11) =
Standard Deviation = Tracking Error =
Tracking error in basis points =
Tracking error in basis points annualized =
0.6467(%2)
0.0109(%2)
1.0084(%2)
0.1413(%2)
0.6947(%2)
0.8335%
83.35
288.74
(b) Is the tracking error computed in part (a) backward-looking or forward-looking tracking error?
The tracking error just computed is based on the actual active returns observed for a portfolio.
Calculations computed for a portfolio based on a portfolio’s actual active returns reflect the portfolio
manager’s decisions during the observation period with respect to the factors that affect tracking error.
We call tracking error calculated from observed active returns for a portfolio backward-looking
tracking error. It is also called the ex-post tracking error and the actual tracking error.
(c) Compare the tracking error found in part (a) to the tracking error found for Portfolios A and B in
Exhibit 19-1. What can you say about the investment management strategy pursued by this portfolio
manager?
The tracking error found for our problem is greater especially compared to Portfolio A in Exhibit 19-1. A
greater tracking error means greater deviation from the benchmark. This is seen if we compare active
return values from our exhibit with the greater active return values found in Exhibit 19-1. For our
problem, it appears the manager may be employing a high-risk strategy to enhance the indexed portfolio’s
return. This strategy is commonly referred to as enhanced indexing or indexing plus.
1. Chapter 25: 5
5. Suppose that the present value of the liabilities of some financial institution is $600
million and the surplus $800 million. The duration of the liabilities is equal to 5. Suppose
further that the portfolio of this financial institution includes only bonds and the duration
for the portfolio is 6.
Answer the following questions.
(a) What is the market value of the portfolio of bonds?
We rearrange the below equation:
economic surplus = market value of assets – market value of liabilities.
Doing this, we get:
market value of assets = economic surplus + market value of liabilities.
Inserting our given values, we get:
market value of assets = $800 million + $600 million = $1.4 billion.
(b) What does a duration of 6 mean for the portfolio of assets?
Duration is a measure of the responsiveness of cash flows to changes in interest rates. Duration
can be calculated for liabilities in the same way in which it is calculated for assets. Because the
duration of the assets is 6, the market value of the assets will change by about 6% for a 100 basis
points change in interest rates. Thus, the increase in the market value of the assets is about
0.06($1.4 billion) = $84 million for a 100 basis points decrease in interest rates. Similarly, the
decrease in the market value of the assets is about 0.06($1.4 billion) = $84 million for a 100
basis points increase in interest rates.
(c) What does a duration of 5 mean for the liabilities?
If the duration of the liabilities is 5, the present value of the liabilities will increase by 5% or
0.05($600 million) = $30 million for a 100 basis point decrease in interest rates. Similarly, it will
decrease by $30 million for every 100 basis point increase in interest rates.
(d) Suppose that interest rates increase by 50 basis points; what will be the approximate
new value for the surplus?
The decrease in assets will be
will be
0.06  $1.4 billion 
= $42 million. The decrease in liabilities
2
0.05  $600 million 
= $15 million. Thus, the net change in surplus value is:
2
decrease in assetsdecrease in liabilities = ($42 million)($15 million) = $27 million.
Thus, the surplus will experience a negative change where the new value for surplus is:
old surplus value +change in surplus value=$800 million  $27 million) = $773 million.
(e) Suppose that interest rates decrease by 50 basis points; what will be the approximate
new value for the surplus?
The increase in assets will be
0.06  $1.4 billion 
= $42 million. The increase in liabilities will be
2
0.05  $600 million 
= $15 million. Thus, the net change in surplus value is:
2
increase in assets  increase in liabilities = $42 million  $15 million = $27 million.
Given the net change in surplus value of $27 million, the surplus will experience a positive
change where the new value for surplus is:
old surplus value + change in surplus value = $800 million + $27 million = $827 million.
25. Suppose that a life insurance company sells a five-year guaranteed investment contract
that guarantees an interest rate of 7.5% per year on a bond-equivalent yield basis (or
equivalently, 3.75% every six months for the next 10 six-month periods). Also suppose that
the payment made by the policyholder is $9,642,899.
Consider the following three investments that can be made by the portfolio manager:
Bond X: Buy $9,642,899 par value of an option-free bond selling at par with a 7.5%
yield to maturity that matures in five years.
Bond Y: Buy $9,642,899 par value of an option-free bond selling at par with a 7.5%
yield to maturity that matures in 12 years.
Bond Z: Buy $10,000,000 par value of a six-year 6.75% coupon option-free bond selling
at 96.42899 to yield 7.5%.
Answer the below questions.
(a) Holding aside the spread that the insurance company seeks to make on the invested
funds, demonstrate that the target accumulated value to meet the GIC obligation five years
from now is $13,934,413.
To compute the target accumulated value, we need to take the future value of the annuity
resulting from the policyholder’s payment and add it to this payment due at the end of the last
period. We have:
 (1 + y ) n  1 
 annual coupon rate 
target accumulated value = 
(P)


 +P
y

2



where the semiannual coupon rate = annual payment  2 = 7.5%  2 = 3.75%; the payment or
price of bond (P) = $9,642,899; the yield = y = 7.5%  2 = 3.75%; and, the total number of
periods (n) = 5(2) = 10. Inserting these values into our target accumulated formula, we get:
 (1.0375)10  1 
 0.075 
target accumulated value = 
 ($9,642,899) 
 + $9,642,899
 2 
 0.0375 
= $361,608.71[11.86783847] + $9,642,899 = $4,291,513.79 + $9,642,899 = $13,934,412.79.
Thus, the target accumulated value is about $13,934,413.
(b) Complete Table A assuming that the manager invests in bond X and immediately
following the purchase, yields change and stay the same for the five-year investment horizon.
For the each row, we have six columns. Below we show how all values are gotten for the first
row of 11.00%. The same process can be repeated to get values for the remaining rows.
Column One gives the new yield which is 11% for the first row. This means the semiannual yield
will be 5.5%. This value changes for each row with each new yield provided for each row.
Column Two provides the coupon which is the total interest paid for each of the ten periods. Thus,
the interest paid is ten times the semiannual coupon payment. The semiannual coupon payment is
the annual coupon rate divided by two (i.e., 0.075  2 = 0.0375) times the par value purchased of
$9,642,899. We have: 10[(0.0375)($9,642,899]) = 10[$361,608.71] = $3,616,087.13 or about
$3,616,087. This value is the same for each row since the coupon rate of 7.5% does not change.
Column Three provides interest on interest. To get interest on interest, we compute (i) the future
value of semiannual coupon payment annuity for ten periods at the yield of 11%  2 = 5.5%, and
(ii) the total interest paid which is the number of periods (10) times the semiannual coupon
payment (0.0375 × $9,642,899). For (iii), we take the value in (i) minus the value in (ii).
n
 (1.055)10  1 
 annual coupon rate   (1 + y )  1   0.075 
For (i), we have: 
 (P) 
 ($9,642,899) 
 =
 =
y

2
 
  2 
 0.055 
$361,608.71[12.87535379] = $4,655,840.11.
For (ii), we have: n(semiannual coupon payment) = 10[(0.0375)($9,642,899]) = 10[$361,608.71] =
$3,616,087.13. Note that this value is the value also computed in column two.
For (iii), we have: $4,655,840.11 – $361,608.71 = $1,039,752.98 or about $1,039,753.
This third column value changes for each row because it is a function of the new yield which
changes for each row. For example, for the second row we repeat the above process using
10% / 2 = 5.0% to compute the future value of semiannual coupon payment annuity.
Column Four provides the maturity value of the bond, which is $9,642,899. This value is the
same for each row since the investment horizon is the same as the maturity. When the maturity
value increases then the value will not equal $9,642,899.
Column Five is the total accumulated value. This is computed by adding (i) the future value of
semiannual coupon payment annuity for ten periods at the yield of 11%  2 = 5.5% and (ii) the
maturity value given in the fourth column. The value for (i) was given previously as
$4,655,840.11 when computing the interest on interest. This value is also given by adding the
second and third column values. The value for (ii) is $9,642,899 as given in the fourth column.
For the accumulated value, we have:
total accumulated value = $4,655,840.11 + $9,642,899 = $14,298,739.11.
This value changes for each row as it is a function of the new yield which changes for each row.
Column Six, the last column, reports the total return which is given as:
 accumulated value 1/ n 
total return = 2 
  1
 policyholder payment 

where accumulated value is the fifth column value of $14,298,739.11, the policyholder payment
is the value of $9,642,899, and n is the number of periods which is 10. Inserting in these values,
we have:
 $14,298,739.11 1 / 10 
2 
  1 = 2[(1.482825767)0.1 – 1] = 2[1.040181 – 1] = 0.0803625 or
 $9,642,899 

about 8.04%. This value changes for each row because it is a function of the new yield which
changes for each row.
We repeat the above process to get all values for each row. These values are given below with
Table A filled in.
Table A
Accumulated Value and Total Return After Five Years:
Five-Year 7.5% Bond Selling to Yield 7.5%
Investment horizon (years): 5
Coupon rate: 7.50%
Maturity (years): 5
Yield to maturity: 7.50%
Price: 100.00000
Par value purchased: $9,642,899
Purchase price: $9,642,899
Target accumulated value: $13,934,413
After Five Years
New Yield
11.00%
10.00%
9.00%
8.00%
7.50%
7.00%
6.00%
5.00%
4.00%
Coupon
$3,616,087
$3,616,087
$3,616,087
$3,616,087
$3,616,087
$3,616,087
$3,616,087
$3,616,087
$3,616,087
Interest on
Interest
$1,039,753
$ 932,188
$ 827,436
$ 725,426
$ 675,427
$ 626,089
$ 529,352
$ 435,153
$ 343,427
Price of
Bond
$9,642,899
$9,642,899
$9,642,899
$9,642,899
$9,642,899
$9,642,899
$9,642,899
$9,642,899
$9,642,899
Accumulated
Value
$14,298,739
$14,191,175
$14,086,423
$13,984,412
$13,934,413
$13,885,073
$13,788,338
$13,694,139
$13,602,414
Total Return
8.04%
7.88%
7.73%
7.57%
7.50%
7.43%
7.28%
7.14%
7.00%
(c) Based on Table A, under what circumstances will the investment in bond X fail to satisfy
the target accumulated value?
Given that the target accumulated value is $13,934,413, we see any new yield below 7.50% will
give an accumulated value which is less than the target. This is due to reinvestment rate risk which
works to the disadvantage of the life insurance company since the coupon must be invested at a
rate below 7.5% which is the guaranteed rate of return.
(d) Complete Table B, assuming that the manager invests in bond Y and immediately
following the purchase, yields change and stay the same for the five-year investment horizon.
For the each row, we have six columns. Below we show how all values are gotten for the first
row of 11.00%. The same process can be repeated to get values for the remaining rows in the
table.
Column One gives the new yield which is 11% for the first row. This means the semiannual yield
will be 5.5%. This value changes for each row with each new yield provided for each row.
Column Two provides the coupon which is the total interest paid for each of the ten periods.
Thus, the interest paid is ten times the semiannual coupon payment. The semiannual coupon
payment is the semiannual coupon rate (0.075  2 = 0.0375) times the par value purchased of
$9,642,899. We have: 10[(0.0375)($9,642,899]) = 10[$361,609.71] = $3,616,087.13 or about
$3,616,087. This value is the same for each row since the coupon rate of 7.5% does not change.
The values for this column for Table B are the same values found in Table A.
Column Three provides interest on interest. To get interest on interest, we compute (i) the future
value of semiannual coupon payment annuity for ten periods at the yield of 11%  2 = 5.5%, and
(ii) the total interest paid which is the number of periods (10) times the semiannual coupon
payment (0.0375 × $9,642,899). For (iii), we take the value in (i) minus the value in (ii).
For
(i),
we
have:
 (1 + y ) n  1 
 annual coupon rate 
(P)




y

2



=
 0.075 

 ($9,642,899)
 2 
 (1.055 )10  1

 = $361,608.71[12.87535379] = $4,655,840.11.
 0.055 
For (ii), we have: n(semiannual coupon payment) = 10[(0.0375)($9,642,899]) = 10[$361,608.71]
= $3,616,087.13. Note that this value is the value also computed in column two.
For (iii), we have: $4,655,840.11 – $361,608.71 = $1,039,752.98.
This third column value changes for each row because it is a function of the new yield which
changes for each row. For example, for the second row we repeat the above process using
10% / 2 = 5% to compute the future value of semiannual coupon payment annuity. The values for
this column for Table B are the same values found in Table A.
Column Four provides the maturity value of the bond, which is $8,024,638.89. This value is
computed by taking the present value of the bond value at the end of five years. Because the
maturity has seven remaining years after five years, this means we compute (i) the value at the
end of five years for an annuity composed of 7(2) = 14 semiannual coupon payments of
$361,608.71 discounted at a yield of 5.5% and (ii) the value at the end of five years for the bond
value of $9,642,899 received in 14 periods and discounted at a yield of 5.5%.
  1  
n  
1  
 annual coupon rate    1 + y     0.075 
For (i), we get: 
=
($9,642,899)
 (P) 
  2 

2
 
y
 
1

14  
1   

  1 .055   = $361,608.71[9.5896479] = $3,467,700.23.
0.055


For (ii), we get:
P
1 y 
n
=
$9, 642,899
= $9,642,899(0.4725693866) = $4,556,938.66.
1.055 14
Adding (i) and (ii), we get the value of the bond price at the beginning of period 11 (or at the end
of period 10) as $3,467,700.23 + $4,556,938.66 = $8,024,638.89 or about $8,024,639. This value
changes for each row because it is a function of the new yield which changes for each row.
Column Five is the total accumulated value. This is computed by adding (i) the future value of
semiannual coupon payment annuity for ten periods at the yield of 11%  2 = 5.5% and (ii) the
maturity value given in the fourth column. The value for (i) was given previously as
$4,655,840.11 when computing the interest on interest. This value is also given by adding the
second and third column values. The value for (ii) is $8,024,638.89 as given in the fourth
column. For the accumulated value, we have:
total accumulated value = $4,655,840.11 + $8,024,638.89 = $12,680,479.00.
This value changes for each row as it is a function of the new yield which changes for each row.
Column Six, the last column, reports the total return which is given by
 accumulated value 1/ n 
total return = 2 
  1
 policyholder payment 

where accumulated value is the fifth column value of $12,680,479.00, the policyholder payment
is the value of $9,642,899, and n is the number of periods which is 10. Inserting in these values,
we have:
 $12,680,479 1 / 10 
2 
  1 = 2[(1.315006929)0.1 – 1] = 2[1.0277626 – 1] = 0.0555252 or about

 $9,642,899 
5.55%. This value changes for each row because it is a function of the new yield which changes
for each row.
We repeat the above process for each row to get all values. These values are given below with
Table B filled in.
Table B
Accumulated Value and Total Return After Five Years:
Twelve-Year 7.5% Bond Selling to Yield 7.5%
Investment horizon (years): 5
Coupon rate: 7.50%
Maturity (years): 12
Yield to maturity: 7.50%
Price: 100.00000
Par value purchased: $9,642,899
Purchase price: $9,642,899
Target accumulated value: $13,934,413
After Five Years
New Yield
11.00%
10.00%
9.00%
8.00%
7.50%
7.00%
6.00%
5.00%
4.00%
Coupon
$3,616,087
$3,616,087
$3,616,087
$3,616,087
$3,616,087
$3,616,087
$3,616,087
$3,616,087
$3,616,087
Interest on
Interest
$1,039,753
$ 932,188
$ 827,436
$ 725,426
$ 675,427
$ 626,089
$ 529,352
$ 435,153
$ 343,427
Price of
Bond
$8,024,639
$8,449,753
$8,903,566
$9,388,251
$9,642,899
$9,906,163
$10,459,851
$11,052,078
$11,685,837
Accumulated
Value
$12,680,479
$12,998,030
$13,347,090
$13,729,764
$13,934,413
$14,148,337
$14,605,289
$15,103,318
$15,645,352
Total Return
5.55%
6.06%
6.61%
7.19%
7.50%
7.82%
8.48%
9.18%
9.92%
(e) Based on Table B, under what circumstances will the investment in bond Y fail to
satisfy the target accumulated value?
Given that the target accumulated value is $13,934,413, we see any new yield above 7.50% will
give accumulated value which is less than the target.
This is due to fact that the increase in interest rates lowers the value of the coupon and principal
payments which are discounted at a higher yield in the future.
This works to the disadvantage of the life insurance company since they have chosen to match
their liability with a longer term twelve-year bond instead of a five-year bond.
One should note that when the market yield increases interest on interest will be greater;
however, the market price of the bond decreases.
The net effect is that the accumulated value is less than the target accumulated value. The reverse
is true when the market yield decreases.
The change in the price of the bond will more than offset the decline in the interest on interest,
resulting in an accumulated value that exceeds the target accumulated value.
(f) Complete Table C, assuming that the manager invests in bond Z and immediately
following the purchase, yields change and stay the same for the five-year investment
horizon.
For the each row, we have six columns. Below we show how all values are gotten for the first
row of 11.00%. The same process can be repeated to get values for the remaining rows in the
table.
Column One gives the new yield which is 11% for the first row. This means the semiannual yield
will be 5.5%. This value changes for each row with each new yield provided for each row.
Column Two provides the coupon which is the total interest paid for each of the ten periods.
Thus, the interest paid is ten times the semiannual coupon payment. The semiannual coupon
payment is the semiannual coupon rate (0.0675  2 = 0.03375) times the par value purchased of
$10,000,000. We have: 10[(0.03375)($10,000,000]) = 10[$337,500] = $3,375,000. This value is
the same for each row since the coupon rate of 6.75% does not change.
Column Three provides interest on interest. To get interest on interest, we compute (i) the future
value of semiannual coupon payment annuity for ten periods at the yield of 11%  2 = 5.5%, and
(ii) the total interest paid which is the number of periods (10) times the semiannual coupon
payment (0.03375 × $10,000,000). For (iii), we take the value in (i) minus the value in (ii).
 (1 + y )n  1  0.0675 
 annual coupon rate 
For (i), we have: 
 (P) 
 ($10,000,000)
= 
y

2


  2 
 (1.055 )10  1

 = $337,500[12.87535379] = $4,345,431.90.
 0.055 
For (ii), we have: n(semiannual coupon payment) = 10[(0.03375)($ 10,000,000]) = 10[$337,500]
= $3,375,000. Note that this value is the value also computed in column two.
For (iii), we have: $4,345,431.90 – $3,375,000 = $970,431.90.
This third column value changes for each row because it is a function of the new yield which
changes for each row. For example, for the second row we repeat the above process using 10% 
2 = 5% to compute the future value of semiannual coupon payment annuity.
Column Four provides the maturity value of the bond, which is $9,607,657.06. This value is
computed by taking the present value of the bond value at the end of five years. Because the
maturity has one remaining year after five years, this means we compute (i) the value at the end
of five years for an annuity composed of two semiannual coupon payments of $337,500
discounted at a yield of 5.5% and (ii) the value at the end of five years of the bond value of
$10,000,000 received in two periods and discounted at a yield of 5.5%.
For (i), we get:
  1  
 
1
 
n  
1  
1   
2  
 annual coupon rate 
 1 + y    =  0.0675  ($10,000,000)   1 .055   

 (P) 


  2 
0.055

2

y

= $337,500[1.8463197] = $623,132.90.
For (ii), we get:
P
1 y 
n
=
$10,000,000
(1.055)
2
= $10,000,000(0.898452416) = $8,984,524.16.
Adding (i) and (ii), we get the value of the bond price at the beginning of period 11 (or at the end
of period 10) as $623,132.90 + $8,984,524.16 = $9,607,657.06 or about $9,607,657. This value
changes for each row because it is a function of the new yield which changes for each row.
Column Five is the total accumulated value. This is computed by adding (i) the future value of
semiannual coupon payment annuity for ten periods at the yield of 11%  2 = 5.5% and (ii) the
maturity value given in the fourth column. The value for (i) was given previously as
$4,345,431.90 when computing the interest on interest. This value is also given by adding the
second and third column values. The value for (ii) is $9,607,657.06 as given in the fourth
column. For the accumulated value, we have:
total accumulated value = $4,345,431.90 + $9,607,657.06 = $13,953,088.96.
This value changes for each row as it is a function of the new yield which changes for each row.
Column Six, the last column, reports the total return which is given by
 accumulated value 1/ n 
total return = 2 
  1
 policyholder payment 

where accumulated value is the fifth column value of $13,953,088.96, the policyholder payment
is the value of $9,642,899, and n is the number of periods which is 10. Inserting in these values,
we have:
 $13,953,089 1/10 
 1 = 2[(1.446980723)0.1 – 1] = 2[1.037638971 – 1] = 0.075278 or about
2 

 $9,642,899 

7.53%. This value changes for each row because it is a function of the new yield which changes
for each row.
We repeat the above process for each row to get all values. These values are given below with
Table C filled in.
Table C
Accumulated Value and Total Return After Five Years:
Six-Year 6.75% Bond Selling to Yield 7.5%
Investment horizon (years): 5
Coupon rate: 6.75%
Maturity (years): 6
Yield to maturity: 7.5%
Price: 96.42899
Par value purchased: $10,000,000
Purchase price: $9,642,899
Target accumulated value: $13,934,413
After Five Years
New Yield
11.00%
10.00%
9.00%
8.00%
7.50%
7.00%
6.00%
5.00%
4.00%
Coupon
$3,375,000
$3,375,000
$3,375,000
$3,375,000
$3,375,000
$3,375,000
$3,375,000
$3,375,000
$3,375,000
Interest on
Interest
$970,432
$870,039
$772,271
$677,061
$630,395
$584,345
$494,059
$406,141
$320,531
Price of
Bond
$ 9,607,657
$ 9,789,325
$ 9,882,119
$ 9,929,017
$ 9,976,254
$10,071,755
$10,168,650
$10,266,965
$10,266,965
Accumulated
Value
$13,953,089
$13,942,885
$13,936,596
$13,934,180
$13,934,413
$13,935,599
$13,940,814
$13,949,791
$13,962,495
Total Return
7.53%
7.51%
7.50%
7.50%
7.50%
7.50%
7.51%
7.52%
7.54%
(g) Based on Table C, under what circumstances will the investment in bond Z fail to
satisfy the target accumulated value?
There is only one yield in the table that gives a value below the target of $13,934,413 and that is
for 8.00% yield where the target value is $13,934,180 which is $133 off.
(h) What is the modified duration of the liability?
Modified duration is a measure of the sensitivity of a bond’s price to interest-rate changes,
assuming that the expected cash flow does not change with interest rates. One modified duration
expression we can use is:
 n 100  C / y 
C 
1
1



n
n 1
y 2  1  y  
1  y 
modified duration =
P
where C is the semiannual coupon payment, y is the semiannual yield, n is the number of
semiannual periods, and P is the bond quote in 100’s.
For our bond (expressing numbers in terms of a $100 bond quote), we have: C = $3.75, y =
0.0375, n = 10, and P = $96.43. Inserting these values in our modified duration formula, we can
solve as follows:
C
y2
=

 n 100  C / y 
1

1 
n 
n 1
1

y


1  y 


P
=
$3.75
 0.3752

 10  $100  $3.75 / 0.0375
1

10 
1  
1 .037511
1 .0375 

$96.43
$2,666.67 (0.3079795)  $0
$821.28
=
= 8.52.
$96.43
$96.43
Converting to annual number by dividing by two gives a modified duration for the liabilities of
4.26. The Macaulay duration is 4.13 while the modified duration using the Macaulay relationship
is 3.98.
(i) Complete the following table for the three bonds assuming that each bond is trading to
yield 7.5%:
Bond
Modified Duration
5-year, 7.5% coupon, selling at par
12-year, 7.5% coupon, selling at par
6-year, 6.75% coupon, selling for 96.42899
Using the formula in part (h) for modified duration, we get the below values as given in the
completed table:
Bond
5-year, 7.5% coupon, selling at par
12-year, 7.5% coupon, selling at par
6-year, 6.75% coupon, selling for 96.42899
Modified Duration
4.11
7.82
4.33
(j) For which bond is the modified duration equal to the duration of the liability?
The 6-year, 6.7%% coupon, selling for $96.43 per $100 at the end of 5 years gives the closest
duration to the liability. Rounding off to the nearest tenth both are equal at 4.3.
(k) Why in this example can one focus on modified duration rather than effective duration?
Modified duration is a measure of the sensitivity of a bond’s price to interest-rate changes,
assuming that the expected cash flows do not change with changes in interest rates. Modified
duration is not an appropriate measure for securities where the projected cash flows change as
interest rates change. The effective duration computation allows for changing cash flow when
interest rates change.