Question

Currently, in October 2020, the term-structure of spot rates is as follows (with continuous compounding):

Maturity (years)	Zero-rate (%)
1	1.0
2	2.0
3	3.0

(a) Consider a 2-year forward contract on a zero-coupon bond. This bond is risk-free and will pay a face value of $1,000 in year 3. What is the forward price? [6 points]

(b) Suppose that, in October 2020, an investor entered a long position in the forward found in (a). One year later, in October 2021, the term structure turns out to be as follows:

Maturity (years)	Zero-rate (%)
1	1.5
2	2.5
3	3.2

What is the current value of the long position that was started in October 2020? [6 points]

Answer 1

a)

Price of 3 year zero coupon bond = Face value * e^{-(3 year zero rate * Time to maturity)}

Price of 3 year zero coupon bond = $1000 * e^{-(3% * 3)}

Price of 3 year zero coupon bond = $913.93

2 year Forward price = Price of 3 year zero coupon bond * e^{(2 year zero rate * Time to Expiry)}

2 year Forward price = $913.93 * e^{(2% * 2)}

2 year Forward price = $951.23

b)

Price of 3 year zero coupon bond 1 year from now = Face value * e^{-(2 year zero rate * Time to maturity)}

Price of 3 year zero coupon bond 1 year from now = $1000 * e^{-(2.5% * 2)}

Price of 3 year zero coupon bond 1 year from now = $951.23

Value of Long position = (Price of 3 year zero coupon bond 1 year from now -  2 year Forward price * e^{-(1 year zero rate * Time to expiry)}

Value of Long position = $951.23 - $951.23 * e^{-(1.5% * 1)}

Value of Long position = $14.16