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Solutions to Selected Problems at the End of Chapter 5 - Interest Rates
Prepping for Exams
1. a.
2. c.
3. b.
4. a.
5. c.
6. c.
7. c.
8. c.
9. c.
10. b.
Problems
1. Periodic interest rates. In the following table, fill in the periodic rates and the effective annual rates.
Period
APR
Compounding
Per Year
Semi-Annual
8%
2
Quarterly
9%
4
Monthly
7.5%
12
Daily
4.25%
365
Periodic
Rate
Effective
Annual Rate
ANSWER
Period
APR
Compounding
Per Year
Periodic
Rate
Effective
Annual Rate
Semi-Annual
8%
2
4.0%
8.16%
Quarterly
9%
4
2.25%
9.31%
Monthly
7.5%
12
0.625%
7.76%
Daily
4.25%
365
0.01164%
4.34%
Periodic Rate = APR / (C/Y) = 0.08 / 2 = 0.04 = 4.0%
Periodic Rate = APR / (C/Y) = 0.09 / 4 = 0.0225= 2.25%
Periodic Rate = APR / (C/Y) = 0.075 / 12 = 0.00625= 0.625%
Periodic Rate = APR / (C/Y) = 0.0425 / 365 = 0.0001164384 = 0.01164%
EAR = (1 + Periodic Rate)C/Y – 1 = 1.042 – 1 = 1.0816 – 1 = 0.0816 = 8.16%
EAR = (1 + Periodic Rate)C/Y – 1 = 1.02254 – 1 = 1.0931 – 1 = 0.0931= 9.31%
EAR = (1 + Periodic Rate)C/Y – 1 = 1.0062512 – 1 = 1.0776 – 1 = 0.0776 = 7.76%
EAR = (1 + Periodic Rate)C/Y – 1 = 1.0001164365 – 1 = 1.0434 – 1 = 0.0434 = 4.34%
2.
3. EAR. What is the effective annual rate of a mortgage rate that is advertised at 7.75% (APR) over the next
twenty years and paid with monthly payments?
ANSWER
Periodic Rate = 0.0775 / 12 = 0.0064583333
EAR = (1 + Periodic Rate)C/Y – 1 = 1.0064583312 – 1 = 1.0803 – 1 = 0.0803 = 8.03%
4.
5. Present value with periodic rates. Let’s follow up with Sam Hinds, the dentist, from Chapter 4 and his
remodeling project (Problem 12). The cost of the equipment for the project is $18,000, and the purchase will
be financed with a 7.5% loan over six years. Originally, the loan called for annual payments. Redo the
payments based on quarterly payments (four per year) and monthly payments (twelve per year). Compare
the annual cash outflow of the two payments. Why does the monthly payment plan have less total cash
outflow each year?
ANSWER
Quarterly Payment = $18,000 / (1 – 1/[1 + (0.075/4)]6 x 4 ) / (0.075/4)
Quarterly Payment = $18,000 / 19.1845 = $938.26
Monthly Payment = $18,000 / (1 – 1/[1 + (0.075/12)]6 x 12 ) / (0.075/12)
Monthly Payment = $18,000 / 57.8365 = $311.22
Annual Cash Outflow Quarterly Payment = $938.26 x 4 = $3,753.04
Annual Cash Outflow Monthly Payment = $311.22 x 12 = $3,734.64
Difference of $18.04
It is lower for the monthly payment because each payment reduces some of the principal and so over the three
months between the quarterly payments the average borrowed amount is lower so that the accumulated interest
expense is lower.
6.
7. Future value with periodic rates. Matt Johnson delivers newspapers and is putting away $15.00 every
month from his paper route collections. Matt is eight years old and will use the money when he goes to
college in ten years. What will be the value of Matt’s account in ten years with his monthly payments if he
is earning 6% (APR), 8% (APR) or 12% (APR)?
ANSWER
FV at 6% APR = $15.00 x [(1 + 0.06/12)10 x 12 – 1] / (0.06/12)
FV at 6% APR = $15.00 x 163.8793 = $2,458.19
FV at 8% APR = $15.00 x [(1 + 0.08/12)10 x 12 – 1] / (0.08/12)
FV at 8% APR = $15.00 x 182.9460 = $2,744.19
FV at 12% APR = $15.00 x [(1 + 0.12/12)10 x 12 – 1] / (0.12/12)
FV at 12% APR = $15.00 x 230.0387 = $3,450.58
8.
9. Payments with periodic rates. What payment does Denise (from problem 8) need to make at the end of each
month over the coming forty-four years at 6% to reach her retirement goal of $1,000,000?
ANSWER
Payment = $1,000,000 / [(1 + 0.06/12)44 x 12 -1 ] / (0.06/12)
Payment = $1,000,000 / 2,584.2652 = $386.96
10.
11. Amortization schedule with periodic payments. Moulton Motors is advertising the following deal on a new
Honda Civic: “Monthly Payments of $400.40 for the next 60 months and this beauty can be yours!” The
sticker price of the car is $18,000. If you bought the car, what interest rate would you be paying in both
APR and EAR terms? What is the amortization schedule of these sixty payments?
ANSWER
The periodic or monthly interest rate, r, is the solution to the equation
PV = Payment x (1 – 1/(1+r)n) / r
$18,000 = $400.40 x (1 – 1/(1+r)60) / r
Using an iterative process you will eventually get to a periodic or monthly interest rate of 1%.
The annual percentage rate is 12%, periodic rate times 12, 1% x 12 = 12%
Or you can solve with TVM keys on a calculator:
P/Y = 12; C/Y=12
INPUT
60
Keys
N
OUTPUT
?
I/Y
12.0
-18,000
PV
400.40
PMT
0
FV
and the EAR is
EAR = 1.0112 – 1 = 12.68%.
Amortization Schedule (Can be done effectively on a spread sheet)
Cell A1 is Beginning Balance for month 1
Cell B1 is the Monthly Payment
Cell C1 is the Monthly Interest Expense and is the periodic or monthly interest rate times the beginning balance:
A1 * 0.01 (formula for the cell)
Cell D1 is the amount of the monthly payment that is applied to the principal and is the payment minus the
interest expense: B1 – C1 (formula for the cell)
Cell E1 is the ending balance after the applying of the monthly payment to interest and principal. It is the
beginning balance minus the principal reduction: A1 – D1 (formula for the cell).
Cell A2 is the ending balance from the previous month or the value in Cell E1.
Then for cells B2 through E2 copy the formulas down from the row above.
A
B
C
D
E
1
$18,000.00
$400.40
$180.00
$220.40
$17,779.60
2
$17,779.60
$400.40
$177.80
$222.60
$17,557.00
3
$17,557.00
$400.40
$175.57
$224.83
$17,332.17
…
…
57
$1,562.35
$400.40
$15.62
$384.78
$1,177.57
58
$1,177.57
$400.40
$11.78
$388.62
$788.94
59
$788.94
$400.40
$7.90
$392.50
$396.44
60
$396.44
$400.40
Repeat this for the sixty months…
$3.96
$396.44
$0.00
12.
13. Inflation, nominal interest rates, and real rates. Given the information below, estimate the nominal rate
with the approximation nominal interest rate equation and the true nominal interest rate equation.
Real Rate
Inflation Rate
3%
5%
8%
15%
1%
4%
2.5%
3.5%
Approximate
Nominal Rate
True Nominal Rate
ANSWER
Real Rate
Inflation Rate
Approximate
Nominal Rate
True Nominal Rate
3%
5%
8%
8.15%
8%
15%
23%
24.20%
1%
4%
5%
5.04%
2.5%
3.5%
6.0%
6.09%
Approximate Nominal Rate = 3% + 5% = 8%
Approximate Nominal Rate = 8% + 15% = 23%
Approximate Nominal Rate = 1% + 4% = 5%
Approximate Nominal Rate = 2.5% + 3.5% = 6%
True Nominal Rate = 1.03 x 1.05 – 1 = 1.0815 – 1 = 0.0815 or 8.15%
True Nominal Rate = 1.08 x 1.15 – 1 = 1.2420 – 1 = 0.2420 or 24.20%
True Nominal Rate = 1.01 x 1.04 – 1 = 1.0504 – 1 = 0.0504 or 5.04%
True Nominal Rate = 1.025 x 1.035 – 1 = 1.0609 – 1 = 0.0609 or 6.09%
14.
15. Inflation, nominal interest rates, and real rates. Given the information below estimate the inflation rate
with the approximation formula and the true inflation with the Fisher Effect formula.
Nominal Rate
Real Rate
11%
5%
8%
2%
21%
14%
5.5%
1.25%
Approximate
Inflation
True Inflation
ANSWER
Nominal Rate
Real Rate
Approximate
Inflation
True Inflation
11%
5%
6%
5.71%
8%
2%
6%
5.88%
21%
14%
7%
6.14%
5.5%
1.25%
4.25%
4.20%
Approximate Inflation = 11% – 5% = 6%
Approximate Inflation = 8% – 2% = 6%
Approximate Inflation = 21% – 14% = 7%
Approximate Inflation = 5.5% – 1.25% = 4.25%
True Inflation = 1.11 /1.05 – 1 = 1.0571 – 1 = 0.0571 or 5.71%
True Inflation = 1.08 /1.02 – 1 = 1.0588 – 1 = 0.0588 or 5.88%
True Inflation = 1.21 /1.14 – 1 = 1.0614 – 1 = 0.0614 or 6.14%
True Inflation = 1.0.55 /1.0125 – 1 = 1.0420 – 1 = 0.0420 or 4.20%
17. Inflation, nominal interest rates, and real rates. The Minister of Finance for the State of Tranquility has
just estimated the expected inflation rate for the coming year at 6.75%. If the real rate for the coming year is
3%, what should the nominal interest rates at the central bank of the State of Tranquility be for the coming
year?
ANSWER
Approximate Nominal Rate = 3% + 6.75% = 9.75%
True Nominal Rate = 1.03 x 1.0675 – 1 = 1.0995 – 1 = 0.0995 or 9.95%
18.
19. Negative inflation (deflation), nominal interest rates, and real rates. The Republic of Northern Lights, a
small stable country in the North Atlantic, is experiencing a negative inflation (deflation) at this time. The
annual inflation rate is -4%. If the nominal rate of interest is 6%, what is the real interest rate that the
Northern Lightians are getting as a reward for waiting?
ANSWER
Approximate Real Rate = 6% – (-4%) = 10%
True Real Rate = 1.06 / 0.96 – 1 = 1.1042 – 1 = 0.1042 or 10.42%
20.
21. Interest premium. The U.S. government offers two bonds: one selling to yield 6.5% and the other to yield
8.5%. Why would one bond sell for a lower yield if the originator is the same on both bonds?
ANSWER
If the bonds have different maturity dates the difference in yields is a reflection of the maturity premium where
bonds with longer maturities have higher rates.
22.
23. Interest premium. Ben has just purchased a long-term government bond and expects to make a 7% return.
Donna has just purchased a stock in a new start-up company but expects to make a 20% return. Why is
Donna expecting a higher return?
ANSWER
If the investments have the same investment horizon, but different issuers then the riskier asset will have a
higher expected return to compensate for the additional risk. Ben’s investment in government bonds is probably
less risky than Donna’s investment in the new start-up company stock
24.
25. Historical interest rates. Refer to Figure 5.5 in the text. For the risk-free rate, what decade experienced the
highest interest rates? The lowest?
ANSWER
The decade of the 80s had the highest interest rates and the decade of the 50s had the lowest rates.
26.
27. Historical interest rates. Refer to Table 5.6 for the average interest rates for the 2000 to 2010 period and
estimate the default premium using the average Treasury Bond rate and the AAA corporate bond rate.
ANSWER
Default premium (2000 -2010) = AAA Corp. Bond rate – Treasury Bond rate
= 5.66% – 4.12% = 1.54%
28.
29. Challenge question I. Michael is shopping for a special automobile. He finds the exact car he wants, a 1966
dark blue Pontiac GTO. This car is currently the property of a neighbor, so in order to buy the car for the
agreed-upon price of $35,000; Michael must secure his own financing. Michael visits four different
financial institutions and gets the following available loans:
Bank 1: 60 monthly payments of $726.54
Bank 2: 48 monthly payments of $870.97
Bank 3: 156 weekly payments of $256.20
Bank 4: 24 quarterly payments of $1,115.81
Which loan should Michael take? (Hint: what loan has the lowest EAR?)
ANSWER
Bank ONE’s EAR; first find the APR with the TVM keys or a spreadsheet:
TVM Keys:Mode P/Y = 12 and C/Y = 12
INPUT
60
?
-35,000
KEYS
N
I/Y
PV
OUTPUT
9.00
726.54
PMT
0
FV
Periodic Rate = 0.09/12 = 0.0075
EAR = 1.007512 – 1 = 9.38%.
Bank TWO’s EAR, first find the APR with the TVM keys or a spreadsheet:
TVM Keys:Mode P/Y = 12 and C/Y = 12
INPUT
48
?
-35,000
KEYS
N
I/Y
PV
OUTPUT
9.00
870.97
PMT
0
FV
256.20
PMT
0
FV
1,115.81
PMT
0
FV
Periodic Rate = 0.09/12 = 0.0075
EAR = 1.007512 – 1 = 9.38%.
Bank THREE’s APR is
TVM Keys:Mode P/Y = 52 and C/Y = 52
INPUT
156
?
-35,000
KEYS
N
I/Y
PV
OUTPUT
9.00
Periodic Rate = 0.09/52 = 0.0017308
EAR = 1.001730852 – 1 = 9.41%.
Bank FOUR’s APR is
TVM Keys:Mode P/Y = 4 and C/Y = 4
INPUT
24
?
-35,000
KEYS
N
I/Y
PV
OUTPUT
8.16
Periodic Rate = 0.0816/4 = 0.0204
EAR = 1.02044 – 1 = 8.41%.
Bank Four has the lowest EAR and all else equal Michael should take the quarterly payment choice.