he spot rate for three hypothetical zero-coupon bonds (zeros) with maturities of
ID: 2641218 • Letter: H
Question
he spot rate for three hypothetical zero-coupon bonds (zeros) with maturities of one, two and three years are given in the following table (based on annual compounding).
Maturity(T)
1
2
3
Spot rates
r(1)=11%
r(2)=10%
r(3)=9%
(a) Calculate the forward rate for a one-year zero in one year, f (1,1).
(b) Calculate the forward rate for a one-year zero in two years, f (2,1).
(c) Calculate the forward rate for a two-year zero in one year, f (1,2).
(d) Calculate the price for a 3-year treasury bond with par value of $1,000 and having a 5% coupon rate paid annually.
Maturity(T)
1
2
3
Spot rates
r(1)=11%
r(2)=10%
r(3)=9%
Explanation / Answer
Forward rates are calculated using the sopt rates. As in the above question the spot rates are given forward rates are:
A) Forward rate for a one-year zero in one year
Let the forward rate be = f
Spot rate = 11% for bond maturing in 1 year
Now forward rate for a one-year zero in one year will be 0% as one year rate are already provided
f = 0%
B) Forward rate for a one year zero in two year
Let forward rate be = f
Then, (1+11%) * (1+f) = (1+10%)^2
f = 9.01%
C) Forward Rate for two-year zero in one year
Let forward rate = f
Then,
(1+10%)^2 * (1+f) = (1+9%)^3
f = 7.027%
D) Price of 3-year treasury bond with par value $ 1,000 at 5% coupon rate annually paid
Price of bond = Present Value of the bond
Present Value of the bond = Present value of coupon payments + Present Value of maturity value of bond
Year
Inflows (A)
Present Value at 9% for $ 1 (B)
Total Amount (A*B)
1
$ 50
0.917431193
$ 45.87
2
$ 50
0.841679993
$ 42.08
3
$ 50
0.77218348
$ 38.61
3
$ 1,000
0.77218348
$ 772.18
Grand Total
$ 898.75
Price of Bond = $ 898.75
Year
Inflows (A)
Present Value at 9% for $ 1 (B)
Total Amount (A*B)
1
$ 50
0.917431193
$ 45.87
2
$ 50
0.841679993
$ 42.08
3
$ 50
0.77218348
$ 38.61
3
$ 1,000
0.77218348
$ 772.18
Grand Total
$ 898.75
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