Chapter 4 Recommended End-of-Chapter Problems and Solutions
3.
What is the "Fisher effect"? How does it affect the nominal rate of interest?
The Fisher effect is Irving Fisher’s hypothesis that expected inflation is embodied in current
nominal interest rates. Assuming the ability to forecast expected inflation, nominal rates should vary
directly with expected inflation.
4.
The one-year real rate of interest is currently estimated to be 4 percent. The
current annual rate of inflation is 6 percent, and market forecasts expect the annual rate of
inflation to be 8 percent. What is the current one-year nominal rate of interest?
Assuming the Fisher effect, the current 1-year nominal rate should be roughly 12 percent, the sum
of the real rate (4%) plus the expected inflation rate (8%), an approximate but illustrative way of
estimating the answer. The correct way to deal with compounding rates is to multiply (1+.04)(1+.08) - 1
= 12.32%.
5.
The following annual inflation rates have been forecast for the next 5 years:
Year 1
Year 2
Year 3
Year 4
Year 5
3%
4%
5%
5%
4%
Use the average annual inflation rate and a 3% real rate to calculate the appropriate contract rate
for (a) a one-year and (b) a five-year loan.
•
•
A lender would require compensation for both opportunity cost and loss of purchasing power.
The sum of the real rate, 3%, plus the expected rate of inflation, 3%, would be roughly 6%.
Or more accurately: .03 + .03 + .03*.03 = .0609 or 6.09%
The sum of the real rate, 3%, plus the expected average rate of inflation (assume geometric
mean 4.197%) would be roughly 7.2%. Or more accurately:
.03 + .04197
+.03*.04197 = .073229 or 7.3229%
9.
An investor purchased a one-year Treasury security with a promised yield of 10
percent. The investor expected the annual rate of inflation to be 6 percent; however, the actual rate
turned out to be 10 percent. What were the expected and the realized real rate of interest for the
investor?
Approx: The expected real rate is re = i - ΔPe = 10% - 6% = 4%; the realized real rate is rr = i ΔPa = 10% - 10% = 0%.
Exact: The expected real rate is re = (i - ΔPe)/(1 + ΔPe) = 3.7736%; the realized real rate is rr = (i
– ΔPa)/(1 + ΔPa) = 0%.
10.
If the realized real rate of return turns out to be positive, would you rather have
been a borrower or a lender? Explain in terms of the purchasing power of the money used to repay
a loan?
The answer depends upon whether the lender (borrower) earns (pays) their expected real rate. If
inflation were originally underestimated, borrowers would benefit at the cost of lenders—their actual cost
of borrowing would be less. If inflation were overestimated, lenders would benefit at the cost of
borrowers—actual returns (borrowing costs) would higher than originally anticipated.
• Suppose a mutual fund advertises that its performance over the past four years was
+25%, -50%, +25%, +40%. (a) What is the average annual rate of return? (b) If you
invested $10,000 with the fund four years ago, what would your account balance show
today? (c) What is the problem here?
•
Most likely the mutual fund would claim in its advertisements an average annual
rate of return of 10%.
•
It would show, unfortunately, that after four years your account balance had only
grown to $10,937.50.
•
$10,937.50 represents only a 2.2656% annual compounded return. The problem
is that you should not use the arithmetic mean to average cumulative percentages, you should
use the geometric mean.
• Consider a loan for which the real interest rate is to be 3% and there is to be full
protection against inflation which id forecast to be 4%. Using the proper version of the
Fisher equation, what is to be the nominal interest rate?
.03 + .04 + (.03)(.04) = .0712 or 7.12%
• A one-year project is over and inflation was 5%. If the nominal interest rate was 12%,
what was the real rate?
(.12 -.05)/(1 + .05) = .0667
•
or
6.67%
Consider a bank with: Reserves=80, Loans=240, Investments= 100, Premises=20,
Deposits=400, Capital=40. If the bank suffers 30 in loan losses, specify its balance sheet?
If the bank suffers yet another 20 in loan losses, specify its balance sheet. Is each case is
the bank solvent or insolvent?
80
210
100
20
400
10
solvent
80
190
100
20
400
-10
insolvent