ANSWERS TO QUESTIONS FOR CHAPTER 2
(Questions are in bold print followed by answers.)
2. Suppose that a life insurance company has guaranteed a payment of $14 million to a
pension fund 4.5 years from now. If the life insurance company receives a premium of $10.4
million from the pension fund and can invest the entire premium for 4.5 years at an annual
interest rate of 6.25%, will it have sufficient funds from this investment to meet the $14
million obligation?
To determine the future value of any sum of money invested today, we can use the future value
equation, which is: Pn = P0 (1 + r)n where n = number of periods, Pn = future value n periods from
now (in dollars), P0 = original principal (in dollars)and r = interest rate per period (in decimal
form). Inserting in our values, we have: P4.5 = $10,400,000(1.0625)4.5 =
$10,400,000(1.313651676) = $13,661,977.43. Thus, it will be short $13,661,977.43 –
$14,000,000 = –$338,022.57.
4. Suppose that a portfolio manager purchases $10 million of par value of an eight-year
bond that has a coupon rate of 7% and pays interest once per year. The first annual coupon
payment will be made one year from now. How much will the portfolio manager have if she
(1) holds the bond until it matures eight years from now, and (2) can reinvest all the annual
interest payments at an annual interest rate of 6.2%?
At the end of year eight, the portfolio manager’s amount is given by the following equation, which
adjusts for annual compounding.
We have:
 1 + r n  1 
Pn  A 
 + Par Value
r


where A = coupon rate times par value. Inserting in our values, we have:
 (1 + 0.062)8  1
P8 = 0.07($10,000,000) 
 + $10,000,000 = $700,000[9.9688005] + $10,000,000 =
0.062


$6,978,160.38 + $10,000,000 = $16,978,160.38.
6. The pension fund obligation of a corporation is calculated as the present value of the
actuarially projected benefits that will have to be paid to beneficiaries. Why is the interest
rate used to discount the projected benefits important?
The present value increases as the discount rate decreases and decreases as the discount rate
increases. Thus, in order to project the benefits accurately, we need an accurate estimate of the
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discount rate. If we underestimate the discount rate then we will be projecting more available
pension funds than we will actually have.
8. Calculate for each of the following bonds the price per $1,000 of par value assuming
semiannual coupon payments.
Bond Coupon Rate (%) Years to Maturity Required Yield (%)
A
8
9
7
B
9
20
9
C
6
15
10
D
0
14
8
A. Consider a 9-year 8% coupon bond with a par value of $1,000 and a required yield of 7%. Given
C = 0.08($1,000) / 2 = $40, n = 2(9) = 18 and r = 0.07 / 2 = 0.035, the present value of the coupon
payments is:
1
1




1

1  1.035 18 
1  1  r n 
1  1.85748920
P= C
 = $40
 = $40 

r

0.035


0.035

1  0.53836114  
$40 
 = $40 13.1896817 = $$527,587.
0.035


The present value of the par or maturity value of $1,000 is:
M
1+ r 
n


 =

=
$1, 000
$1, 000
=
80
15.6757375
1.035 
= $538,361. Thus, the price of the bond (P) = present value of coupon payments + present value of
par value = $527,587 + $538,361 = $1,065.95.
B. Consider a 20-year 9% coupon bond with a par value of $1,000 and a required yield of 9%.
Given C = 0.09($1,000) / 2 = $45, n = 2(20) = 40 and r = 0.09 / 2 = 0.045, the present value of the
coupon payments is:
1

1  1  r n
P= C

r
1


1  1.045 40

 = $45 

0.045


1


1  5.81863645
 = $45 

0.045



1  0.1719287 
 = $45 
 =
0.045


$45[18.401584] = $828.071.
The present value of the par or maturity value of $1,000 is:
M
1+ r 
n
=
$1, 000
$1, 000
=
40
5.81863645
1.045 
= $171.929. Thus, the price of the bond (P) = $828.071+ $171.929= $1,000.00. [NOTE. We
already knew the answer would be $1,000 because the coupon rate equals the yield to maturity.]
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C. Consider a 15-year 6% coupon bond with a par value of $1,000 and a required yield of 10%.
Given C = 0.06($1,000) / 2 = $30, n = 2(15) = 30 and r = 0.10 / 2 = 0.05, the present value of the
coupon payments is:
1

1  1  r n
P= C

r
1


1  1.05 30

 = $30 

0.05


1

1



 = $30 4.3219424

0.05



1  0.2313774 
 = $30 
 =
0.05


$30[15.372451] = $461.174.
The present value of the par or maturity value of $1,000 is:
M
1+ r 
n
=
$1, 000
$1,000
=
=
30
4.3219424
1.05 
231.377. Thus, the price of the bond (P) = $461.174+ $231.377= $692.55.
D. Note – This calculation is done assuming semiannual compounding because of the
directions given in the problem. In general, this assumption is not used for zero coupon
bonds. Consider a 14-year 0% coupon bond with a par value of $1,000 and a required yield of 8%.
Given C = 0($1,000) / 2 = $0, n = 2(14) = 28 and r = 0.08 / 2 = 0.04, the present value of the coupon
payments is:
1

1

 1  r n
P=C 

r


 = $0

1

1  (1.04) 28

0.04



 = $0

1

1  2.998703319

0.055



1  0.33477471 
 = $0 
 =
0.055


$0[16.66306322] = $0. [NOTE. We already knew the answer because the coupon rate is zero.]
The present value of the par or maturity value of $1,000 is:
M
1+ r 
n
=
$1, 000
$1, 000
=
=
28
2.99870332
1.04 
$333.48. Thus, the price of the bond (P) = $0 + $333.48 = $333.48.
10. Suppose that you purchased a debt obligation three years ago at its par value of $100,000
and nine years remaining to maturity. The market price of this debt obligation today is
$90,000. What are some reasons why the price of this debt obligation could have declined
from the time you purchased it three years ago?
The price of a bond will change for one or more of the following three reasons:
(i) There is a change in the required yield owing to changes in the credit quality of the issuer.
(ii) There is a change in the price of the bond selling at a premium or a discount, without any
change in the required yield, simply because the bond is moving toward maturity.
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(iii) There is a change in the required yield owing to a change in the yield on comparable bonds
(i.e., a change in the yield required by the market).
The first and third reasons are the likely reasons for the situation where the bond has plummeted
from $100,000 to $90,000. The bond has plummeted in value because the credit quality of the
issuer has fallen and/or the bond has plummeted because the yield on comparable bonds has
increased.
12. What is the maximum price of a bond?
Consider an extreme case of a 100-year 20% coupon bond with a par value of $1,000 that after one
year falls so that the required yield is 1%. Given C = 0.2 ($1,000) / 2 = $100, n = 2(99) = 198 and
r = 0.01 / 2 = 0.005, the present value of the coupon payments is:
1

1  1  r n
P=C 

r
1


1  (1.005)198

 = $100 

0.005

1  0.3724944
$100 
0.005


1


1  2.684604
 = $100 
0.005




 =


 = $100[125.50112] = $12,550.112.
The present value of the par value of $1,000 is:
M
1+ r 
n
=
$1, 000
$1, 000
=
= $372.494.
198
2.684604
1.005 
Thus, the price of the bond (P) = $12,550.112 + $372.494 = $12,922.61.
This is a percent increase of ($12,922.6 – $1,000) / $1,000 = 11.92606 or about 1,192.61%.
If the required yield falls to 0.001%, then the bond price would increase to $20,778.33, which
would be a percent increase of about 1,977.83%.
If the required yield falls to 0.00001%, then the bond price would increase to $20,778.33, which
would be a percent increase of about 1,977.83%.
If the required yield falls to 0.0000000001%, then the bond price would increase to $20,801.76,
which would be a percent increase of about 1,980.18%.
Thus, we see that even for these extreme numbers (that are highly unlikely), we find there appears
to be a limit on how high a bond price might rise assuming that rates do not reach negative
numbers.
If the required yield is a negative number then there would be no limit to how high a bond price
might rise. For example, if the required yield becomes a negative 1%, then the bond price would
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increase to $70,468.18. If it becomes a negative 10%, then the bond price becomes
$2,296,218.049,925.23.
14. Explain why you agree or disagree with the following statement: “The price of a floater
will always trade at its par value.”
One would disagree with the statement: “The price of a floater will always trade at its par value.”
First, the coupon rate of a floating-rate security (or floater) is equal to a reference rate plus some
spread or margin. For example, the coupon rate of a floater can reset at the rate on a three-month
Treasury bill (the reference rate) plus 50 basis points (the spread).
Next, the price of a floater depends on two factors: (1) the spread over the reference rate and (2)
any restrictions that may be imposed on the resetting of the coupon rate. For example, a floater
may have a maximum coupon rate called a cap or a minimum coupon rate called a floor. The price
of a floater will trade close to its par value as long as (1) the spread above the reference rate that the
market requires is unchanged and (2) neither the cap nor the floor is reached.
However, if the market requires a larger (smaller) spread, the price of a floater will trade below
(above) par. If the coupon rate is restricted from changing to the reference rate plus the spread
because of the cap, then the price of a floater will trade below par.
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