Econ 422
Fall 2013
HW 3 – Answer Key
1. Upon retirement, a university professor finds that he has $1 million in his retirement
account at TIAA-CREF. He wants to continue receiving a monthly salary for the next 20
years, so he decides to purchase an annuity product from TIAA-CREFF. He has the
following options:
a). Receive fixed monthly payments for the next 20 years starting next month.
b). Receive monthly payments for the next 20 years starting next month that grow at an
estimated inflation rate of 0.0025 per month (3% per year). Assume that the first payment
is not inflation adjusted. That is, the inflation adjustment starts in month 2.
What are his monthly payment under the two options if the annual rate on the annuity is
6% (0.06/12 = 0.005 per month)?
Here we want to calculate the payment associated with a finite annuity and a finite
growing annuity. The appropriate formulas are
Finite annunty: PV = C*PVA(r,T), PVA(r,T) = (1/r)[1 - 1/(1 + r)T]
Finite growing annuity : PV = C/(r – g)*[1 - [(1+g)/(1 + r)] T]
In each case we have monthly compounding so the appropriate interest rate is 0.005 and
the number of periods is 12*20 = 240.
Under option 1, we compute
C = PV/PVA(r,T) = $1M/PVA(0.005, 240) = $7,164.31
Under options 2, we compute
C = PV/PGVA(r,g,T) = $1M/PVGA(0.005,0.0025,240) = $5,556.09
2. EFFECTIVE ANNUAL RATES
Consider the following credit card offer from a hypothetical bank. The card advertises an
annual percentage rate (APR) of 19.34%. However, the bank actually compounds interest
charges on a daily basis.
a) Assuming 365 days in the year, what is the daily periodic interest rate being charged?
The periodic rate is APR/365 = 0.1934/365 = 0.00053 per day
b). If you carried a $1,000 balance throughout the year, how much would you owe at the
end of the year?
FV = $1,000*(1 + 0.00053)365 = $1213.306
Which implies an effective annual rate of 21.33%.
3. EFFECTIVE ANNUAL RATES
Consider “payday loans”. Payday loans are short-term loans made to consumers, often for
less than two weeks, and are offered by companies such as AmeriCash Advance and
National Payday. The loan works like this: You write a check today that is postdated and
the company gives you an amount of money less than the check. When the check date
arrives, you go to the store and pay cash for the check, or the company cashes the check.
For example, suppose AmeriCash Advance allows you to write a postdated check for
$125 for 15 days later in return for a $100 loan today. What is the annual percentage rate
(APR) on this loan and what is the effective annual rate (EAR)? Note: the answer to this
question will help you understand why there was recent Washington legislation to limit
the interest rate charged on payday loans.
The 15 day rate of return is
r15day 
125  100
 0.25
100
The APR associated with this periodic rate is
APR  0.25 
365
 6.083 or 608.3%
15
The effective annual rate is computed using
rEAR  (1.25)365/15  1  227.45 or 22,745%
PCBR (Interest Rates) pages 102 – 106
Problems 4, 9, 12, 14, 25, 27, 29 - 32
5-4.
You have found three investment choices for a one-year deposit: 10% APR compounded
monthly, 10% APR compounded annually, and 9% APR compounded daily. Compute the
EAR for each investment choice. (Assume that there are 365 days in the year.)
For a $1 invested in an account with 10% APR with monthly compounding you will have
 1  0.1   $1.10471


 12 
12
So the EAR is 10.471%.
For a $1 invested in an account with 10% APR with annual compounding you will have
1  0.1  $1.10
So the EAR is 10%.
For a $1 invested in an account with 9% APR with daily compounding you will have
 1  0.09   1.09416


 365 
365
So the EAR is 9.416%.
5-9.
Suppose you invest $100 in a bank account, and five years later it has grown to $134.39.
a.
What APR did you receive, if the interest was compounded semiannually?
b.
What APR did you receive if the interest was compounded monthly?
The EAR can be calculated as follows:
 f
 p 
a)
1/5
 1  1.3439   1  6.0897%
1/5
Using the formula for EAR, we can calculate the APR for semi-annual compounding.




APR  2   EAR  1  1  2  1.06897   1  6%
12
12
b) Similarly we can calculate the APR for monthly compounding

APR  12   EAR  1
5-12.
1 12


 1  12  1.06897 
1 12

 1  5.926%
Capital One is advertising a 60-month, 5.99% APR motorcycle loan. If you need to borrow
$8000 to purchase your dream Harley Davidson, what will your monthly payment be?
Timeline:
0
1
2
3
4
60
–8,000
C
C
C
C
C
5.99 APR monthly implies a discount rate of
5.99
 0.499167%
12
Using the formula for computing a loan payment
C
8, 000


1
1 
60 
0.00499167 
1.00499167  
1
 $154.63
5-14.
You have decided to refinance your mortgage. You plan to borrow whatever is outstanding
on your current mortgage. The current monthly payment is $2356 and you have made every
payment on time. The original term of the mortgage was 30 years, and the mortgage is
exactly four years and eight months old. You have just made your monthly payment. The
mortgage interest rate is 63⁄8% (APR). How much do you owe on the mortgage today?
Timeline:
56
0
57
1
58
2
360
304
2,356
2,356
2,356
To find out what is owed compute the PV of the remaining payments using the loan interest rate to
compute the discount rate:
Discount Rate 
6.375
 0.53125%
12
PV 
5-25.


1
 $354, 900
1 
304 
0.0053125 
1.0053125  
2, 356
In 1975, interest rates were 7.85% and the rate of inflation was 12.3% in the United States.
What was the real interest rate in 1975? How would the purchasing power of your savings
have changed over the year?
rr 
r  i 7.85%  12.3%

 3.96%
1 i
1.123
The purchasing power of your savings declined by 3.96% over the year.
5-27.
Can the nominal interest rate available to an investor be significantly negative? (Hint:
Consider the interest rate earned from saving cash “under the mattress.”) Can the real
interest rate be negative? Explain.
By holding cash, an investor earns a nominal interest rate of 0%. Since an investor can always
earn at least 0%, the nominal interest rate cannot be negative. The real interest rate can be
negative, however. It is negative whenever the rate of inflation exceeds the nominal interest rate.
5-29.
Suppose the term structure of risk-free interest rates is as shown below:
a.
Calculate the present value of an investment that pays $1000 in two years and $2000 in
five years for certain.
b.
Calculate the present value of receiving $500 per year, with certainty, at the end of the
next five years. To find the rates for the missing years in the table, linearly interpolate
between the years for which you do know the rates. (For example, the rate in year 4
would be the average of the rate in year 3 and year 5.)
c.
Calculate the present value of receiving $2300 per year, with certainty, for the next 20
years. Infer rates for the missing years using linear interpolation. (Hint : Use a
spreadsheet.)
a.
Timeline:
0
1
2
3
4
5
1,000
2,000
Since the opportunity cost of capital is different for investments of different maturities, we
must use the cost of capital associated with each cash flow as the discount rate for that cash
flow:
PV 
b.
1, 000
1.0241
2

2, 000
 $2, 652.15.
1.0332 5
Timeline:
0
1
2
3
4
5
500
500
500
500
500
Since the opportunity cost of capital is different for investments of different maturities, we
must use the cost of capital associated with each cash flow as the discount rate for that cash
flow. Unfortunately, we do not have a rate for a 4-year cash flow, so we linearly interpolate.
r4 
1
1
 2.74    3.32   3.03
2
2
PV 
c.
500
500
500
500
500




 $2, 296.43
1.0199 1.02412 1.0274 3 1.03034 1.0332 5
Timeline:
0
1
2
3
20
2,300
2,300
2,300
2,300
Since the opportunity cot of capital is different for investments of different maturities, we
must use the cost of capital associated with each cash flow as the discount rate for that cash
flow. Unfortunately, we do not have a rate for a number of years, so we linearly interpolate.
1
1
 2.74    3.32 
2
2
 3.03
r4 
1
1
 3.32    3.76 
2
2
 3.54
r6 
2
1
 3.76    4.13
3
3
 3.883
r8 
1
2
 3.76    4.13
3
3
 4.0067
r9 
9
1
 4.13   4.93
10
10
 4.21
r11 
8
2
 4.13   4.93
10
10
 4.29
r12 
r13  4.37
r14  4.45
r15  4.53
r16  4.61
r17  4.64
r18  4.77
r19  4.85
PV =
2,300 2,300
2,300
2,300
+
+
+ ... +
20
1 + r1 (1 + r2 )2 (1 + r3 )3
(1 + r20 )
=
2,300 2,300 2,300
2,300
+
+
+ ... +
1.0199 1.0241 1.0274
(1.0493)20
= $30,636.56
5-30.
Using the term structure in Problem 29, what is the present value of an investment that pays
$100 at the end of each of years 1, 2, and 3? If you wanted to value this investment correctly
using the annuity formula, which discount rate should you use?
PV = 100 / 1.0199 + 100 / 1.02412 + 100 / 1.02743 =$285.61.
To determine the single discount rate that would compute the value correctly, we solve the
following for r:
PV = 285.61 = 100/(1 + r) + 100 / (1 + r) 2 + 100/(1 + r)3 = $285.61.
This is just an IRR calculation. Using trial and error or the annuity calculator, r = 2.50%. Note that
this rate is between the 1, 2, and 3-yr rates given.
5-31.
What is the shape of the yield curve given the term structure in Problem 29? What
expectations are investors likely to have about future interest rates?
The yield curve is increasing. This is often a sign that investors expect interest rates to rise in the
future.
5-32.
Suppose the current one-year interest rate is 6%. One year from now, you believe the
economy will start to slow and the one-year interest rate will fall to 5%. In two years, you
expect the economy to be in the midst of a recession, causing the Federal Reserve to cut
interest rates drastically and the one-year interest rate to fall to 2%. The one-year interest
rate will then rise to 3% the following year, and continue to rise by 1% per year until it
returns to 6%, where it will remain from then on.
a.
If you were certain regarding these future interest rate changes, what two-year interest
rate would be consistent with these expectations?
b.
What current term structure of interest rates, for terms of 1 to 10 years, would be
consistent with these expectations?
c.
Plot the yield curve in this case. How does the one-year interest rate compare to the 10year interest rate?
a.
The one-year interest rate is 6%. If rates fall next year to 5%, then if you reinvest at this rate
over two years you would earn (1.06)(1.05) = 1.113 per dollar invested. This amount
corresponds to an EAR of (1.113)1/2 – 1 = 5.50% per year for two years. Thus, the two-year
rate that is consistent with these expectations is 5.50%.
b.
We can apply the same logic for future years:
Year
Future Interest Rates
FV from reinvesting
1
6%
1.0600
2
5%
1.1130
3
2%
1.1353
4
3%
1.1693
5
4%
1.2161
6
5%
1.2769
7
6%
1.3535
8
6%
1.4347
9
6%
1.5208
10
6%
1.6121
c.
EAR
6.00%
5.50%
4.32%
3.99%
3.99%
4.16%
4.42%
4.62%
4.77%
4.89%
We can plot the yield curve using the EARs in (b); note that the 10-year rate is below the 1year rate (yield curve is inverted).
PCBR (Valuing Bonds) pages 196-202
Problems 1-4, 10, 13, 17-19, 28
8-1.
A 30-year bond with a face value of $1000 has a coupon rate of 5.5%, with semiannual
payments.
a.
What is the coupon payment for this bond?
b.
Draw the cash flows for the bond on a timeline.
a.
The coupon payment is:
CPN 
b.
Coupon Rate  Face Value
0.055  $1000

 $27.50.
Number of Coupons per Year
2
The timeline for the cash flows for this bond is (the unit of time on this timeline is six-month
periods):
0
1
2
3
$27.50
$27.50
$27.50
60
$27.50 +
$1000
P  100/(1.055) 2  $89.85
8-2.
8-3.
Assume that a bond will make payments every six months as shown on the following timeline
(using six-month periods):
a.
What is the maturity of the bond (in years)?
b.
What is the coupon rate (in percent)?
c.
What is the face value?
a.
The maturity is 10 years.
b.
(20/1000) x 2 = 4%, so the coupon rate is 4%.
c.
The face value is $1000.
The following table summarizes prices of various default-free, zero-coupon bonds (expressed
as a percentage of face value):
a.
Compute the yield to maturity for each bond.
b.
Plot the zero-coupon yield curve (for the first five years).
c.
Is the yield curve upward sloping, downward sloping, or flat?
a.
Use the following equation.
1/ n
 FV 
1  YTMn   n 
 P 
1/1
 100 
1  YTM1  

 95.51 
 YTM1  4.70%
1/ 2
 100 
1  YTM1  

 91.05 
 YTM1  4.80%
1/ 3
 100 
1  YTM 3  

 86.38 
 YTM 3  5.00%
1/ 4
 100 
1  YTM 4  

 81.65 
 YTM 4  5.20%
1/ 5
 100 
1  YTM 5  

 76.51 
b.
 YTM 5  5.50%
The yield curve is as shown below.
Yield to Maturity
Zero Coupon Yield Curve
5.6
5.5
5.4
5.3
5.2
5.1
5
4.9
4.8
4.7
4.6
0
2
4
Maturity (Years)
c.
The yield curve is upward sloping.
6
8-4.
8-10.
Suppose the current zero-coupon yield curve for risk-free bonds is as follows:
a.
What is the price per $100 face value of a two-year, zero-coupon, risk-free bond?
b.
What is the price per $100 face value of a four-year, zero-coupon, risk-free bond?
c.
What is the risk-free interest rate for a five-year maturity?
a.
P  100(1.055) 2  $89.85
b.
P  100/(1.0595) 4  $79.36
c.
6.05%
Suppose a seven-year, $1000 bond with an 8% coupon rate and semiannual coupons is
trading with a yield to maturity of 6.75%.
a.
Is this bond currently trading at a discount, at par, or at a premium? Explain.
b.
If the yield to maturity of the bond rises to 7% (APR with semiannual compounding),
what price will the bond trade for?
a.
Because the yield to maturity is less than the coupon rate, the bond is trading at a premium.
b.
40
40


(1  .035) (1  .035)2
Given:
Solve For PV:
8-13.

NPER
14
40  1000
 $1, 054.60
(1  .035)14
Rate
3.50%
PV
(1,054.60)
PMT
40
FV
1,000
Excel Formula
=PV(0.035,14,40,1000)
Consider the following bonds:
a.
What is the percentage change in the price of each bond if its yield to maturity falls from
6% to 5%?
b.
Which of the bonds A–D is most sensitive to a 1% drop in interest rates from 6% to 5%
and why? Which bond is least sensitive? Provide an intuitive explanation for your
answer.
a.
We can compute the price of each bond at each YTM using Eq. 8.5. For example, with a 6%
YTM, the price of bond A per $100 face value is
P(bond A, 6% YTM) 
100
 $41.73.
1.0615
The price of bond D is
P(bond D, 6% YTM)  8 
1 
1  100
 $114.72.
1 

.06  1.0610  1.0610
One can also use the Excel formula to compute the price: –PV(YTM, NPER, PMT, FV).
Once we compute the price of each bond for each YTM, we can compute the % price change
as
Percent change =
 Price at 5% YTM    Price at 6% YTM 
.
 Price at 6% YTM 
The results are shown in the table below.
Bond
Coupon Rate
(annual payments)
0%
0%
4%
8%
A
B
C
D
b.
8-17.
Maturity
(years)
15
10
15
10
Price at
6% YTM
$41.73
$55.84
$80.58
$114.72
Price at
5% YTM
$48.10
$61.39
$89.62
$123.17
Percentage Change
15.3%
9.9%
11.2%
7.4%
Bond A is most sensitive, because it has the longest maturity and no coupons. Bond D is the
least sensitive. Intuitively, higher coupon rates and a shorter maturity typically lower a bond’s
interest rate sensitivity.
What is the price today of a two-year, default-free security with a face value of $1000 and an
annual coupon rate of 6%? Does this bond trade at a discount, at par, or at a premium?
P
CPN
CPN
CPN  FV
60
60  1000

 ... 


 $1032.09
2
N
1  YTM1 (1  YTM 2 )
(1  .04) (1  .043) 2
(1  YTM N )
This bond trades at a premium. The coupon of the bond is greater than each of the zero coupon
yields, so the coupon will also be greater than the yield to maturity on this bond. Therefore it
trades at a premium
8-18.
What is the price of a five-year, zero-coupon, default-free security with a face value of
$1000?
The price of the zero-coupon bond is
P
8-19.
FV
1000

 $791.03
(1  YTM N ) N (1  0.048)5
What is the price of a three-year, default-free security with a face value of $1000 and an
annual coupon rate of 4%? What is the yield to maturity for this bond?
The price of the bond is
P
CPN
CPN
CPN  FV
40
40
40  1000

 ... 



 $986.58.
1  YTM1 (1  YTM 2 )2
(1  YTM N ) N (1  .04) (1  .043) 2 (1  .045)3
The yield to maturity is
P
CPN
CPN
CPN  FV

 ... 
1  YTM (1  YTM ) 2
(1  YTM ) N
$986.58 
40
40
40  1000


 YTM  4.488%
2
(1  YTM ) (1  YTM )
(1  YTM )3
The following table summarizes the yields to maturity on several one-year, zero-coupon securities:
a.
What is the price (expressed as a percentage of the face value) of a one-year, zerocoupon corporate bond with a AAA rating?
b.
What is the credit spread on AAA-rated corporate bonds?
c.
What is the credit spread on B-rated corporate bonds?
d.
How does the credit spread change with the bond rating? Why?
a.
The price of this bond will be
P
100
 96.899.
1  .032
b.
The credit spread on AAA-rated corporate bonds is 0.032 – 0.031 = 0.1%.
c.
The credit spread on B-rated corporate bonds is 0.049 – 0.031 = 1.8%.
d.
The credit spread increases as the bond rating falls, because lower rated bonds are riskier.