Question

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

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?

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