CHAPTER 1
3.
Assume that the inflation-free rate of interest is 3 percent and that the inflation rate is 10
percent with complete certainty and no taxes. Determine the nominal interest rate.
i = nominal rate
r = inflation-free rate
p = inflation rate
i = r + p + rp
r = 3%; p = 10%; i = ?
i = 0.03 + 0.1 + (0.03)(0.1)
i = 0.133 = 13.3%
4.
In a world of certainty with no taxes, the nominal interest rate is 10 percent and the
inflation-free interest rate is 5 percent. What is the inflation rate?
i = 10%; r = 5%; p = ?
i = r + p + rp
0.1 = 0.05 + p + 0.05p
0.05 = 1.05p
p = 0.0476 = 4.76%
5.
Assume no taxes. Suppose the inflation-free interest rate is 5 percent.
forecasts a deflation rate of 15 percent. What is the nominal interest rate?
r = 5%; p = -15%; i = ?
The market
i = r + p + rp
i = 0.05 + (-0.15) + 0.05(-0.15)
i = -0.1075 = -10.75%
*The nominal interest rate cannot be negative; no one will invest at a negative rate.
CHAPTER 2
4.
The Treasury announces an auction of $10 billion par value of 52-week Treasury bills. $2
billion of noncompetitive bids are received. The competitive bids are as follows:
Price per $1 of par
0.9200
0.9194
0.9188
0.9180
0.9180
0.9178
Par value
$3 billion
$3 billion
$4 billion
$2 billion
$2 billion
$6 billion
Compute the price per dollar of par paid by noncompetitive bidders with a one-price
auction. How would this price compared to the price paid by noncompetitive bidders
with a discriminatory auction?
$10 billion auction
$2 billion on non competitive bids
$8 billion of competitive bids
In a one-price auction, everyone would pay .9188. In a discriminatory auction, the
noncompetitive bidders would pay
.9200(3) + .9194(3) + .9188(2)
= .9195
8
5.
The Treasury auctions $12 billion of 52-week Treasury bills. Three billion dollars of
noncompetitive bids are received. The competitive bids are:
Price per $1 of par
0.9200
0.9180
0.9170
0.9160
Par value
$5 billion
$4 billion
$3 billion
$2 billion
What is the price paid per dollar of par (to 4 decimals) by noncompetitive bidders with a
single-price (Dutch) auction?
$12 billion auction
$3 billion noncompetitive bids
$9 billion competitive bids
Everyone would pay .9180.
6.
Suppose that the interest rate on taxable Treasury securities is 5 percent and that the tax
rate for all investors is 20 percent. What should the interest rate be on a tax-free municipal bond
assuming no default risk and no difference in liquidity?
ym = yt(1 – t)
= 0.05(1 – 0.20) = 0.04 = 4%
7.
The interest rate on fully taxable Treasury securities is 4 percent and the interest rate on
tax-free municipal bonds is 2.75 percent. What is the tax rate of investors assuming all
investors have the same tax rate, no default risk, and no difference in liquidity risk?
ym = yt(1 – t)
0.0275 = 0.04(1 – t)
1 – t = 0.0275/0.04 = 0.6875
t = 31.25%
CHAPTER 4
1.
Suppose you have $20 and are going to invest it for one year at 8 percent. What is the
future value? What is the future value if you invest for two years? Decompose the future
value into original principal, simple interest, and compound interest.
FV = PV(1 + I)N
One Year Investment:
FV = 20(1.08) = $21.60
Two Year Investment:
FV = 20(1.08)2 = $23.328
Principal
20
2.
Simple Interest
20(0.08 + 0.08) = 3.20
Compound Interest
3.328 – 3.20 = 0.128
Suppose you are going to receive $50 in 28 years. What is its present value at 8 percent
interest? Compute the answer using the present value table and your calculator.
FV = 50; N = 28; I/YR = 8; PV = ?
PV 
3.
PV 
50
 $5.80
(1.08) 28
Suppose you will receive $50 one year from now and $80 two years from now. What is
the present value at 8 percent?
PV 
4.
FV
(1  YR ) N
50
80

 $114.88
(1.08) (1.08) 2
Repeat problem 3 using a 10 percent interest rate. Is the present value higher or lower?
Explain why.
PV 
50
80

 $111.57
(1.10) (1.10) 2
The present value is smaller because the interest rate is larger.
5.
You will receive an annuity of $12 per year for two years. For an 8 percent interest rate,
compute the present value of this annuity three ways. Use the tables, use your calculator,
and use the formula for the present value of an annuity.
PV 
12
12

 $21.40
(1.08) (1.08) 2
On Financial Calculator:
FV = 0; I/YR = 8; N = 2; PMT = 12; PV = ?
PV = $21.40
6.
A 2-year bond pays $12 coupons annually and has a $100 par value. Compute its price if
the yield to maturity is 10 percent.
PV 
7.
12
112

 $103.47
(1.10) (1.10) 2
In problem 6, compute the stated yield and the current yield. What is the relationship
between the current yield and the yield to maturity and why?
Stated Yield =
coupon 12

 12%
par
100
Current Yield =
coupon
12

 11.6%
price
103.47
Premium Bond: y 
8.
c
c

P par
Using the information in problem 6, what is the holding period yield over the next year
(a) if the yield to maturity remains at 10 percent, and (b) if the yield to maturity declines
to 8 percent at time 1? In each case, decompose the holding period return into the current
yield and the capital gain or loss.
a. HPR 
P1  P0  C P1  P0 C


 Capital Gain (Loss)  Current Yield
P0
P0
P0
YTM = 10%
P0 
12
112

 $103.47
1.10 (1.10) 2
P1 
112
 $101.82
(1.10)
101.82  103.47
12

103.47
103.47
= -0.016
+ 0.1160
= -1.6%
+ 11.60%
|__________|
|___________|
Capital Loss
Current Yield
HPR =
b. P0 
P1 
= 10%
12
112

 $103.47
(1.10) (1.10) 2
112
 $103.70
(1.08)
103.70  103.47
12

103.47
103.47
= 0.0022
+ 0.1160
= 0.22%
+ 11.60%
|__________|
|___________|
Capital Gain
Current Yield
HPR =
9.
= 11.82%
A 2-year bond with annual coupons of $12 and par value of $100 has a current price of
$109.04. (a) What is its yield to maturity? (b) What is the current yield? (c) What
would the yield to maturity be if the maturity of this bond is ten years?
a. N = 2; PMT = 12; FV = 100; PV = -109.04; I/YR = ?
109.04 
12
112

; y = 7%
1  y (1  y) 2
b. Current Yield =
c
12

 11%
P 109.04
c. In financial calculator:
N = 10; PV = -109.04; PMT = 12; FV = 100; I/YR = ?
I/YR = YTM = 10.50%
10.
In problem 9 part (a), suppose the holding period return over the next year is 8 percent.
What would the yield to maturity at time 1 have to be for this to occur?
HPR = 8%
HPR 
P1  P0  c
P0
P1  (HPR )( P0 )  P0  c
P1  P0 (1  HPR )  c
P1  109.04(1.08) 12  $105.76
11.
P1 
c  par
1 y
y
c  par
1
P1
1 y 
c  par
P1
y
12  100
 1  0.059  5.9%
105.76
A Treasury strip with 10-year maturity and $100 par value has a current price of $55.84.
Compute its yield to maturity.
In financial calculator:
N = 10; PV = -55.84; PMT = 0; FV = 100; I/YR = ?
I/YR = YTM = 6%
or
P
12.
par
(1  y)10
55.84 
par
(1  y)10
y = 6%
In problem 11, suppose that the interest rate drops over the next year by 1 percent. What is
the price of this strip in one year and what is the holding period return on this strip over the
next year? What would the yield on this strip have to become in one year for the holding
period return to be 5 percent higher than the initial yield to maturity in problem 11?
I = 6% - 1% = 5%
P1 
100
 64.46
(1.05) 9
HPR =
P0  $55.84
64.46  55.84
= 0.1544 = 15.44%
55.84
HPR = 6% + 5% = 11%
HPR 
P1  P0
P0
P1  (HPR )P0  P0
.
P1  P0 (1  HRP )
P1  55.84(1.11)  $61.98
In financial calculator:
N = 9; PV = 61.98; PMT = 0; FV = 100; I/YR = ?
I/YR = 5.46%
13.
A 2-year bond with a $12 annual coupon and $100 par value pays half of the coupon
every six months. The yield to maturity (annualized semiannually compounded yield to
maturity) is 10 percent. Find the bond’s price.
N = (2)(2) = 4
i = annual yield
PMT = c/2 = 12/2 = 6
FV = Par = 100
I/YR = i/2 = 10/2 = 5
PV = ? = $103.55
14.
A 2-year bond with an $8 annual coupon pays half of the coupon semiannually. Par
value is $100 and the current price is $96.45. Compute the annualized semiannually
compounded yield to maturity.
N = (2)(2) = 4
PMT = 8/2 = c/2 = 4
FV = par = 100
PV = - $96.45
I/YR = 5  2 = 10%
15.
Suppose a bond has an annual coupon of $8, paid in two equal installments of $4. The par
value is $100. Assume a semiannual period is 182 days and there are 100 days until the
next coupon. Compute the accrued interest. Besides the next coupon in 100 days there
will be three more coupons of $4 and a par value of $100 at final maturity. Compute the
bond price if the annualized semiannually compounded yield to maturity is 9 percent.
Semiannual coupon = 4; par = 100
Semiannual period = 182 days; 100 days left
i = 9%
 c  j 
Accrued Interest =   
 2  k 
j = day since last coupon
k = days for whole period
 82 
Accrued Interest = (4)
 = $1.80
 182 
FV = 100; PMT = 4; N = 4; I/YR= 4.5; PV= 98.21
j/k
i

P + Accrued Interest = 1  
[value at last coupon]
 2
P + 1.80 = (1 +0.09/2)82/182 [98.21]
P + 1.80 = 1.020(98.21)
P = $98.37.
16.
A 20-year bond pays $8 coupons annually and has a $100 par value. Compute its price if
the yield to maturity is 10 percent.
In financial calculator:
N = 20; PMT = 8; FV = 100; I/YR = 10; PV= 82.97
17.
A 20-year bond pays $8 coupons annually and has a $100 par value. Compute its price if
the yield to maturity is 6 percent.
In financial calculator:
N = 20; PMT = 8; FV = 100; I/YR = 6; PV= 122.94
19.
Suppose an individual is going to receive a retirement annuity of $12,000 per year for 20
years with the first payment beginning 4 years from now. Assuming an interest rate of 5
percent, find the present value of this annuity.
PV = $12,000[PVA20, 5%]/(1.05)3
PV = $12,000[12.4622]/1.1576
PV = $129,183.80
20.
To save for retirement, you decide to invest $10,000 per year for 15 years with the first
investment beginning one year from now. Assuming an interest rate of 6 percent, what
will be the value of your investment at the end of the 15 years?
FV = $10,000[FVA15,6%] = $10,000[23.275970] = $232,759.70.
21.
Suppose you will receive $80 one year from now and $50 four years from now. What is
the interest rate if the present value is $122.09?
80
50

1  y (1  y) 4
y = 3%.
122.09 =
22.
You will receive an annuity of $15 per year for 20 years, with the first payment
beginning 6 years from now. For a 6% interest rate, compute the present value of this
annuity.
PV = 15[PVA20, 6%]/(1.06)5
PV = 15[11.4699]/1.3382
PV = 128.57.
23.
An annuity has a present value of $42.17. This annuity is a little different. There are 7
annual payments of $10 each beginning at time 10. What is the interest rate on this
annuity?
42.19 = 10[PVA16, y – PVA9, y]
4.219 = PVA16, y – PVA9, y
y is 4% using the present value of an annuity table.
24.
Suppose a 25-year bond has an annual coupon of c, par value of $100, a yield to maturity
of 5.50%, and current yield of 5.5322%. What is the bond’s coupon?
c
= 5.5322%
P
c
c[ PVA25,5.50% ]  100 /(1.055) 25
= 0.055322
c = $5.625.
25.
You will receive an annuity of $10 per period. The first payment is received immediately
and then there are 19 more payments of $10. You invest these all at the interest rate x%.
At time 19, the value of your investment is $511.60. What is the interest rate?
10(1 + x)19 + 10[FVA19, x] = 511.60
(1 + x)19 + FVA19, x = 51.16
x = 9%.
Use PV tables.
26.
Suppose we have a 6-year strip with interest rate x. If the interest rate instantly increases
by 1%, the price would drop by 3.61% of par. What is x?
x = 7%. Use PV tables.
27.
An investor named Colleen is given a choice of two bonds. Bond A is a 6-year bond with
annual coupon of $6 and par value of $100. Bond B is a 5-year bond with annual coupon
of $6.1724, par value of $100, and the same price as Bond A. If the interest rate is the
same on both bonds, what is the interest rate?
5% by trial and error.