Practice Problems
FIN 5210: Investments
Fall 2014
Problem Set 2: Inflation and the yield curve
1.
Suppose the interest rate in Germany is 4% and the expected inflation rate there is
1%. Meanwhile the interest rate in the United States is 3% and the expected U.S.
inflation rate is 2%. What direction will funds flow?
2.
Suppose the interest rate in Japan is 5% and the expected inflation rate there is 1%.
Meanwhile the interest rate in the United Kingdom is 4% and the expected U.K.
inflation rate is 3%. What direction will funds flow?
Note that the real interest rate must be equal in all countries, or capital will flow to the
highest real rate.
3.
Suppose the equilibrium real rate is 3% and the expected rate of inflation in the U.S.
is 4%. What is the equilibrium nominal interest rate?
Now let’s develop the yield curve.
4.
Suppose the interest rate on debt with a 1-year maturity is 6%, and the interest rate
on debt with a 2-year maturity is 6.5%. Theodore Jones expects that the 1-year
interest rate will be 7.5% next year. He could invest over a 2-year horizon by
purchasing the 2-year bond, or by purchasing the 1-year bond with the intention of
rolling over the money. Which bond will he choose this year, the 1-year maturity or
the 2-year maturity? What risks are involved with the different strategies?
5.
Suppose the interest rate on debt with a 2-year maturity is 7%. Which bond will
Theodore Jones choose this year, the 1-year maturity or the 2-year maturity? What
market pressures would there be?
6.
Assume the yield curve is based upon unbiased expectations. Given Theodore
Jones’ expectations, what would be the equilibrium interest rate on debt with a 2year maturity?
7.
Suppose the interest rate on debt with a 1-year maturity is 5%. The 1-year rate is
expected to rise to 6% next year and 7% the following year (in other words, the 1year forward rate is 6% and the 2-year forward rate is 7%). Assume the yield curve
is based upon unbiased expectations. What would be the equilibrium interest rate on
debt with a 3-year maturity?
8.
Suppose the interest rate on debt with a 1-year maturity is 6%. The 1-year rate is
expected to rise to 6.5% next year, 7% the following year, and 8% the year after that.
Assume the yield curve is based upon unbiased expectations. What would be the
equilibrium interest rate on debt with a 4-year maturity?
Now let’s do some more work with inflation.
page 1
Practice Problems
9.
FIN 5210: Investments
Fall 2014
A dollar at the end of 1976 could purchase about as much as 58 cents could at the
end of 1967. What was the average annually compounded inflation rate over that
period?
10. If it seems to you that everything costs four times as much now as it did twenty years
ago, are you surprised? What average annually compounded rate of inflation is
implied?
11. One hundred years ago an unskilled laborer earned $1 a day. Now such a laborer
earns $32 (at about $4 an hour for 8 hours). What is the average compound annual
rate of wage inflation?
12. If this rate of inflation continues, what will the daily wage be after another hundred
years?
13. If inflation averages 10% per year, what will a dollar stuffed into the mattress be
worth in 1 year? Hint: Remember that inflation is defined as the rate of growth in
the price level, so it will cost $1.10 next year to buy what you can get for $1.00
today.
14. If inflation averages 10% per year, what will a dollar stuffed into the mattress be
worth in 10 years?
15. What is the present value of $1,000 to be received 1 year from now if the required
real rate of return is 3% and the expected rate of inflation is 5%?
16. What is the equilibrium nominal 1-year interest rate if the required real rate of return
is 3% and the expected rate of inflation is 5%?
17. What is the present value of $1,000 to be received 10 years from now if the required
real rate of return is 3% compounded annually and the expected rate of inflation is
5% compounded annually?
18. What is the present value of $1,000 to be received 1 year from now if the required
real rate of return is 4% and the expected rate of inflation is 3%?
19. What is the equilibrium nominal 1-year interest rate if the required real rate of return
is 4% and the expected rate of inflation is 3%?
20. What is the present value of $1,000 to be received 10 years from now if the required
real rate of return is 4% compounded annually and the expected rate of inflation is
3% compounded annually?
Now let’s work with real performance of investments, adjusted forshifts in purchasing
power.
21. In terms of today's purchasing power (in real terms, that is) how much would you
expect to have on today’s date 25 years from now as the result of investing $100,000
page 2
Practice Problems
FIN 5210: Investments
Fall 2014
today, if the nominal interest rate is 9% compounded annually and the inflation rate
is expected to be 3% compounded annually?
22. In terms of today's purchasing power (in real terms, that is) how much would you
expect to have on today’s date 30 years from now as the result of investing $100,000
today, if the nominal interest rate after tax is 3% compounded annually and the
inflation rate is expected to be 4% compounded annually?
23. In terms of today's purchasing power (in real terms, that is) how much would you
expect to have on today’s date 20 years from now as the result of investing $100,000
today, if the nominal interest rate is 12% compounded annually and the inflation rate
is expected to be 3% compounded annually?
24. What is the equilibrium real interest rate if the nominal rate is 12% and the expected
rate of inflation is 3%?
Now let’s work with real return after inflation.
25. A rare painting purchased for $300,000 in June 1977 sold for $1,250,000 in June
2007. Assume that over this period inflation averaged 6% compounded annually.
What was the average compound annual real rate of change in the value of this work
of art?
26. A home purchased for $200,000 in September 1997 sold for $350,000 in September
2007. Assume that over this period inflation averaged 4% compounded annually.
What was the average compound annual real rate of change in the value of this
property?
Now let’s work with real return after inflation and tax.
27. Suppose you invest at a nominal interest rate of 4% in a certificate of deposit, and
you are in the 35% marginal income tax bracket. If you expect inflation will average
3% per year over the span of your involvement in this investment, what is the real
rate of interest you expect to earn, after tax?
28. Suppose you borrow money at a nominal interest rate of 6.9% for a home
improvement loan (a purpose that allows the entire nominal interest to be taxdeductible), and you are in the 35% marginal income tax bracket. You believe the
investment will be a good inflation hedge (that is, you think it will increase in value
at the same rate as the general price level). Also, you expect certain tax law
provisions will continue, allowing you to avoid paying any tax on a capital gain from
the future sale of your house. If you expect inflation in home prices will average 5%
per year over the span of your involvement in this investment, what is the real rate of
interest you expect to pay, after tax? (Hint: Remember, all of the nominal interest is
tax deductible.)
Now let’s work with retirement planning.
page 3
Practice Problems
FIN 5210: Investments
Fall 2014
29. Robert Anderson wants to retire right away, and would like to set aside enough
from the advance on his latest book to give himself an income of $3,000 per
month for 25 years. He also would like to be able to give himself monthly cost
of living increases to keep up with inflation. He expects inflation to average 6%
compounded monthly (.5% per month), and he can earn a nominal rate of return
of 12% with monthly compounding. How much will his retirement annuity
cost?
page 4
Practice Problems
FIN 5210: Investments
Solutions: Set 2
1.
To Germany, because it has the higher real interest
rate.
9.
FV is 1, PV is –0.58, P/YR is 1, N is 9, calculate
I/YR. Result is 6.24%
2.
To Japan, because it has the higher real interest
rate.
10.
FV is 4, PV is –1, P/YR is 1, N is 20, calculate
I/YR. Result is 7.18%
3.
R = r + i + ri
11.
FV is 32, PV is –1, P/YR is 1, N is 100, calculate
I/YR. Result is 3.53%
12.
Leave interest, N, and P/YR unchanged from
problem 11. Then PV is –32, calculate FV. Result
is $1,024
13.
Fat Value (FV is 1, P/YR is 1, N is 1, I/YR is 10,
PMT is 0, compute Puny Value (PV). Result is
90.91¢
14.
Fat Value (FV is 1, P/YR is 1, N is 10, I/YR is 10,
PMT is 0, compute Puny Value (PV). Result is
38.55¢
15.
This can be done in one step using the nominal
rate. Data input follows: FV is 1000, P/YR is 1,
I/YR is 8.15, N is 1, PMT is 0, compute PV.
Result is $924.64. (The negative sign for PV
recognizes the sign convention).
R = 0.03 + 0.04 + (0.03 x 0.04) = 7.12%
4.
If $100 were invested at 6% for the first year
followed by 7.5% for the second year, it would
grow to $113.95. Input this as FV, and –100 as
PV. This is an average compound annual rate of
6.75% (in order to find this, additional data inputs
are as follows: PMT is zero, P/YR is 1, and N is 2,
compute I/YR).
Since 6.75% > 6.5%, Theodore will choose the
rollover strategy. Risk is that interest rate next
year will be less than expected.
5.
Since 7% > 6.75%, Theodore will choose the 2year maturity. There would be pressure on the
price of the 2-year bond, pushing the 2-year rate
down.
6.
The 2-year rate is an average of the 1-year rate this
year and the expected rate next year. If someone
invests $100 for the first year at 6% and then rolls
over at 7.5% for the second year, the money would
grow to $113.95 in two years. The average
compound annual rate for the 2-year period then is
6.75% (PV is –100, FV is 113.95, N is 2, P/YR is
1, calculate I/YR).
7.
If $100 were invested at 5% for the first year
followed by 6% for the second year, and 7% for the
third year, it would grow to $119.09. Input this as
FV, and –100 as PV. This is an average compound
annual rate of 6.00% (in order to find this,
additional data inputs are as follows: PMT is zero,
P/YR is 1, and N is 3, compute I/YR).
8.
Alternatively, the calculation can be done in two
steps. First deflate the future amount, as follows:
Fat Value (FV) is 1000, P/YR is 1, I/YR is 5, N is
1, PMT is 0, compute Puny Value (PV).
Intermediate result is –924.64. Second step:
change sign to positive, input as FV, change I/YR
to 3, and compute PV. Final result is $924.64.
(The negative sign for PV recognizes the sign
convention).
16.
17. First, deflate the money to be received in future. So
then P/YR is 1, FV is 1000, I/YR is 5, N is 10,
calculate PV. Result is 613.91. Make sure the sign
of this is positive, then enter as FV. In the next step
I/yr is 3, then calculate PV. Result is 456.81.
If $100 were invested at 6% for the first year
followed by 6.5% for the second year, 7% for the
third year, and 8% for the fourth year, it would
grow to $130.46. Input this as FV, and –100 as
PV. This is an average compound annual rate of
6.87% (in order to find this, additional data inputs
are as follows: PMT is zero, P/YR is 1, and N is 4,
compute I/YR).
Prof. Kensinger
R = 0.03 + 0.05 + (.03*.05) = 8.15%
18.
Fall 2014
This can be done in one step using the nominal
rate. Data input follows: FV is 1000, P/YR is 1,
I/YR is 7.12, N is 1, PMT is 0, compute PV.
Result is $933.53. (The negative sign for PV
recognizes the sign convention).
page 1
Practice Problems
FIN 5210: Investments
Alternatively, the calculation can be done in two
steps. First deflate the future amount, as follows:
Fat Value (FV) is 1000, P/YR is 1, I/YR is 3, N is
1, PMT is 0, compute Puny Value (PV).
Intermediate result is –970.87. Second step:
change sign to positive, input as FV, change I/YR
to 4, and compute PV. Final result is $933.53.
(The negative sign for PV recognizes the sign
convention).
19.
20. First, deflate the money to be received in future. So
then P/YR is 1, FV is 1000, I/YR is 3, N is 10,
calculate PV. Result is 744.09. Make sure the sign
of this is positive, then enter as FV. In the next step
I/yr is 4, then calculate PV. Result is 502.68
21.
22.
Purchasing power = $74,837.03. Real rate is (3% 4%)/1.04, negative about -0.96%
23.
Purchasing power = $534,091.86. Real rate is
(12% - 3%)/1.03, about 8.74%
24.
(12% - 3%)/1.03, approximately 8.74%
25.
Adjusted for inflation, the selling price is
$217,637.66 in terms of dollars in the year 1977. So
the value did not keep up with inflation. Thus the
real return is negative, about -1.06%
26.
Adjusted for inflation, the selling price is
$236,447.46 in terms of dollars in the year 1997. So
the value gained a little more than inflation. Thus
the real return is positive, about 1.69%
27.
[4%(1-.35) – 3%] / 1.03 = -0.39%
28.
Class discussion (Topic 3). Cost of borrowing is
negative, so incentive is to borrow.
29.
PMT is 3000, N is 300, P/YR is 12, FV is zero,
MODE is END. Interest per month is (1%–
0.5%)/1.005. Calculate this and multiply times 12
to annualize it (result is about 5.97%). Input this
result directly as I/YR and calculate PV. Answer is
$466,942.12
R = (1.04 x 1.03) – 1 = 7.12%
Alternatively, you could leave the settings and data
from the previous problem in your calculator.
Then input 1000 as FV and compute I/YR.
Purchasing power = $411,843.13. Real rate is (9%
- 3%)/1.03, about 5.83%
Prof. Kensinger
Fall 2014
Solutions: Set 2
page 2