Question

The current term-structure of spot rates is as follows (with continuous compounding): Maturity (years) Zero-rate(%) 1 3.0 2 4.5 3 5.5

What is the implied forward rate r0(2, 3)?

(a) 6.00% (b) 6.75% (c) 7.50% (d) 7.53%

A bank offers a special bond A through which investors can borrow (lend) $100 in year 2 and repay (receive) $100×e 0.07 in year 3. Is there an arbitrage? If so, what is the arbitrage’s net cash flow in year 0? (Consider an arbitrage strategy where we use one unit of bond A and the net cash flows are zero from years 1 through 3)

(a) 0.392 (b) 0.456 (c) 0.499 (d) 0.538

Answer 1

1.
=(3*5.5%-2*4.5%)/(3-2)
=7.50000%

2.
Buy one 2 year bond at t0. CF=-100*e^(-4.5%*2)

Sell e^(0.07) unit of 3 year bond at t0. CF=100*e^(-0.055*3)*e^(0.07)

Total cash flow at t0=-100*e^(-4.5%*2)+100*e^(-0.055*3)*e^(0.07)=-0.4558

Receive 100 from 2 year bond at t2
Invest 100 at 7% from t2 to t3
Get 100*e^(0.07) at t3

Pay back 100*e^(0.07) for 3 year bond at t3

Option B