Question

In January 2020, the term-structure of spot rates is as follows
(with continuous compounding):

Maturity (years) Zero-rate(%)
1 2.0
2 3.0
3 4.0

(a) A 3-year zero-coupon bond has the face value of $1,000. Consider a 1-year forward
contract on the zero coupon bond. What should be the forward price?

(b)Suppose that an investor takes a long position in the above forward contract. One year
later, in January 2021, the term-structure turns out to be as follows:

Maturity (years) Zero-rate(%)
1 3.0
2 4.0
3 5.0

What is the value (in January 2021) of the long position in the forward contract?

Answer 1

Bond price = Face value * e^{-(3 year zero rate * Time to Maturity)}

Bond price = $1000 * e^{-(4% * 3)}

Bond price = $886.92

Future price = Bond price * e^{(1 year zero rate * Time to Maturity)}

Future price = $886.92 * e^{(2% * 1)}

Future price = $904.84

2)

Zero coupon Bond price after 1 year after the change in term structure

Bond price = Face value * e^{-(2 year zero rate * Time to Maturity)}

Bond price = $1000 * e^{-(4% * 2)}

Bond price after 1 year = $923.12

Value of Long position in Future contract = Future price - Bond price after 1 year

Value of Long position in Future contract = $923.12 - $904.84

Value of Long position in Future contract = $18.28