1. Annual returns for two mutual funds, Fidelity Low Priced Stock Fund and
Vanguard 500 Index Fund, are given in the following table. Assume that you
invest $1,000 in each and leave the money for 4 years.
a. How much money do you accumulate in each fund at the end of 4 years –
assuming that all income is reinvested?
Year
1 (2000)
2 (2001)
3 (2002)
4 (2003)
FLPSX
18.8%
26.7%
-6.2%
40.9%
VFINX
-9.1%
-12%
-22.2%
28.5%
FLPSX = 1989.33; VFINX = 799.7040
b. What is the single rate that properly measures the average return for each
fund over the four-year period?
FLPSX: 18.7618%; VFINX: -5.43%
c. What is your overall return at the end of four years?
FLPSX: 98.93%; VFINX: -20.02%
2. In 1991, a Connecticut woman wrote a letter to the newspaper columnist, Ann
Landers, revealing that she had secretly stashed $35,000 in her home during her
46-year marriage and was afraid to tell her husband. Ann advised her, “Money
belongs in a bank where it is safe from fire and theft, and can generate interest.”
If the woman had deposited $35,000/46 each year in a bank account paying 5
percent interest after taxes, how much would she have had at the end of 46 years?
(Assume that the deposits occur at the end of the year.)
PMT = 760.87, i=5%, n=46, FV=128,347.41
3. Suppose you are given the choice of two annuities:
a. $10,000 payable at the end of each of the next 6 years and zero thereafter;
b. $10,000 forever, but payments do not begin until 10 years from now.
Which annuity do you choose if the annual interest rate is 5%? Does your answer
change if the interest rate is 10%? Explain why or why not?
5%: PV(a)=50,756.92; PV(b)=128,921.78 (PV of the perpetuity is 200,000,
then discount back 9 years) Choose b.
10%: PV(a)=43,552,61;PV(b)=42,409.76 (PV of the perpetuity is 100,000, then
discount back 9 years) Choose a.
4. Suppose you bought a five-year zero-coupon bond for $800. The face value of
the bond is $1,000.
a. What is the yield to maturity on the bond?
PV = -800, FV = 1000, n=5, i=?=4.56%
Since this is a zero-coupon bond, if you hold the bond to maturity this is your annual
return.
b. If market rates on comparable instruments increases to 7% immediately
after you purchase the bond and remains there, what is your annual return
if you sell the bond after one year? The interest rate increased – what
happened to the bond price? Why?
Interest rates have increased (7%>4.56%), so the price should decrease.
FV = 1000, n=4, i=7%, PV = $762.90.
Annual return for one year: (762.90-800)/800 = -4.64%
Or FV = 762.90, PV=-800, n=1, i=?=-4.64%
Since interest rates increased, the price decreased and you lost money.
c. If the rate remains at 7%, calculate your annual return if you sell the bond
after two years. What happened to the bond price? Why?
FV = 1000, n=3, i=7%, PV = ?=816.30
Annual return if you paid 800 at time 0 and sold for 816.30 at time 2:
PV=-800, FV=816.30, n=2, i=?=1.01%
The bond price increased because the time to maturity decreased. As the bond nears
maturity, the price will converge to the face value.
d. If rates go to 8% and you sell the bond after 3 years, what is your annual
return? The interest rate increased – what happened to the bond
price? Why?
FV = 1000, i=8, n=2, PV = ? = 857.34
The interest rate increased and the price also increased. This is because the time to
maturity decreased.
Annual return: FV = 857.34, PV = -800, n=3, i=2.33%
e. Suppose that after 3 years, market rates drop to 3% and remain there.
What is your annual return if you sell the bond after four years?
Price = 970.87
Annual return = 4.96%.
5. Suppose the yield to maturity on a one-year pure discount bond is 8%. The yield
to maturity on a two-year pure discount bond is 10%. According to the pure
expectations theory, what is the expected one-year rate in the marketplace for year
2?
Expectations theory says that the two year rate is the average of the two one year rates:
(8+x)/2=10; x=12%
6. For each of the bonds and reinvestment rates listed below calculate the amount of
money accumulated from a $1,000 initial investment:
a. Invest $1000 in a 5-year zero coupon bond with a yield to maturity of 9
percent.
1000(1.09)5 = $1538.62
PMT=90, n=5, i=9, FV=?
Annual return = ytm = 9%
b. Buy a 5-year 9% coupon annual bond at par ($1000) and reinvest the
annual coupons at 9%.
Annual coupons = .09*1000 = $90
FV of 1000 = 1000
FV of coupons = 538.62
FV of all payments = 1538.62
FV = 1538.62, PV = 1000, n=5, i=? Annual return
c. Same as (b) but reinvest the annual coupons at 12%.
FV of the payments = 1571.76
If interest rates increase after you buy the bond, and you hold the bond to maturity your
realized yield will be greater than the ytm when you bought the bond (9%). This is
because you can reinvest the coupons at the higher rate.
Annual return = 9.47%
d. Same as (b) but reinvest the annual coupons at 6%.
FV of the payments = 1507.34
If interest rates go down after you buy the bond, and you hold the bond to maturity, your
realized yield will be less than 9%.
Annual return = 8.55%
e. For (a) through (d) calculate the annual return. What can you conclude
about the relationship between yield to maturity and annual return?
See above answers.