Question

Suppose that the following risk-free spot rates prevail in the market: r1=4%, r2=5%, r3=6%, r4=6.5%, where rT denotes the spot rate per annum on an investment with T years to maturity, quoted with annual compounding. You can borrow and lend at these rates. Let f(T1,T2) denote the annual forward rate, quoted with annual compounding, applicable to a forward rate agreement to borrow or lend money in T1 years from today until T2 years from today.

Questions:

1. Compute the forward rate f2,4 that prevails if there are no arbitrage opportunities.
2. You are expecting a USD 1,000 payment in two years from today. You want to invest this money for two years at the appropriate forward rate prevailing today. Unfortunately, you cannot find a market participant willing to enter into a forward loan agreement with you. However, you can borrow and lend at the prevailing spot rates given above. How can you ensure to earn the prevailing forward rate on your future investment?

Answer 1

1 year spot rate = 4%

2 year spot rate = 5%

3 year spot rate = 6%

4 year spot rate = 6.5%

1)Forward rate f2,4   implies 2 year forward rate 2 year from now

(1 + 4 year Spot rate) ⁴ = (1 + 2 year spot rate)² * (1 + Forward rate f2,4 )²

(1 + 6.5%)⁴ = (1 + 5%)² * (1 + Forward rate f2,4)²

(1 + Forward rate f2,4)² = (1 + 6.5%)⁴ / (1 + 5%)²

(1 + Forward rate f2,4)² = 1.2865 / 1.1025

(1 + Forward rate f2,4)² = 1.1669

(1 + Forward rate f2,4)= 1.1669

(1 + Forward rate f2,4)= 1.080214

Forward rate f2,4 = 8.0214%

2) Since I'm expecting USD 1000, two years from today

I will borrow USD 1000 / (1 + 2 year spot rate)²

Amount borrowed = USD 1000 / (1 + 5%)²

Amount borrowed = USD 907.0295

I will lend this amount borrowed at the 4 year spot rate = 6.5%

At the end of 4 years the amount I'll have

Amount at 4 year maturity = Amount borrowed * (1 + 4 year spot rate)⁴

Amount at 4 year maturity = USD 907.0295 * (1 + 6.5%)⁴

Amount at 4 year maturity = USD 1166.8629

At the end of 2 years I will payback the amount borrowed with the accrued interest = USD 1000 with the proceeds from USD 1000 expected at the end 2 years.

Amount at 2 year maturity = Amount borrowed * (1 + 2 year spot rate)²

Amount at 2 year maturity = USD 907.0295 * (1 + 5%)²

Amount at 2 year maturity = USD 1000

Through this borrowing today at 2 year spot rate & lending the same amount at 4 year spot rate I'll be able to capture the 2 year forward rate expected 2 years from now (Forward rate f2,4)

Illustrating how I've captured the Forward rate f2,4

Amount at 4 year maturity = Amount at 2 year maturity * (1 + interest rate)²

USD 1166.8629 = USD 1000 * (1 + interest rate)²

(1 + interest rate)² = USD 1166.8629 / USD 1000

(1 + interest rate)² = 1.1668629

(1 + interest rate) = 1.1668629

(1 + interest rate) = 1.080214

Interest rate = 0.080214 or 8.0214%

The interest rate captured through these borrowing & lending transactions equals the Forward rate f2,4