Question

The current price of a 6-month zero coupon bond with a face value of $100 is B1. If a 9-month strip with a face value of $100 is currently trading for B2, find the forward interest rate for the 6 to 9 month period. Solve by both continuous compounding and quarterly compounding. Write your answers for the following:

14. Nine-month spot interest rate for continuous compounding.

15. Forward rate (6 to 9 months) for continuous compounding.

16. What is the guaranteed fair price of a 3-month T-Bill to be delivered at 6 months from now, assuming quarterly compounding?

17. What is the guaranteed fair price of a 3-month T-Bill to be delivered at 6 months from now, assume continuous compounding?

Answer 1

Q-14

Let S_9M = 9 month annual spot rate under continuous compounding

A 9-month strip with a face value of $100 is currently trading for B2. This implies,

100e^{-S_9M x T} = B₂

T = 9 months = 9 / 12 = 0.75 year

Hence, 100e^{-S_9M x 0.75} = B₂ or, e^{-S_9M x 0.75} = B₂ / 100

Take natural log (ln) on both the sides,

-0.75S_9M = ln (B₂ / 100) Or, 0.75S_9M = - ln (B₂ / 100) = ln (100 / B₂)

Hence, S_9M = [ln (100 / B₂)] / 0.75 = 4 / 3 x ln (100 / B₂)

Under quarterly compounding the equation will be 100 x (1 + S_9M / 4)^-3 = B₂

Hence, S_9M = [(100 / B₂)^1/3 - 1] x 4

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Q - 15

Let S_6M = 6 month annual spot rate under continuous compounding

The current price of a 6-month zero coupon bond with a face value of $100 is B₁. This implies,

100e^{-S_6M x T} = B₁

T = 6 months = 6 / 12 = 0.5 year

Hence, 100e^{-S_6M x 0.5} = B₁ or, e^{-S_6M x 0.5} = B₁ / 100

Take natural log (ln) on both the sides,

-0.5S_6M = ln (B₁ / 100) Or, 0.5S_6M = - ln (B₁ / 100) = ln (100 / B₁)

Hence, S_6M = [ln (100 / B₁)] / 0.5 = 2 x ln (100 / B₁)

Under quarterly compounding the equation will be 100 x (1 + S_6M / 4)^-2 = B₁

Hence, S_6M = [(100 / B₁)^1/2 - 1] x 4

Let's now assume F_6,9 = is the Forward rate (6 to 9 months) for continuous compounding then,

e^{S_6M x t₁} x e^{F_6,9 x t₂} = e^{S_9M x (t₁ + t₂)}

or, e^{(S_6M x t₁+ F_6,9 x t₂)} = e^{S_9M x (t₁ + t₂)}

where t₁ = 6 months = 0.5 year and t₂ = Period between 6 to 9 months = 3 months = 3 / 12 = 0.25 year

Taking natural log (ln) on both the sides,

(S_6M x t₁ + F_6,9 x t₂) = S_9M x (t₁ + t₂)

0.5S_6M + 0.25F_6,9 = S_9M x (0.5 + 0.25)

Hence, F_6,9 = 3S_9M - 2S_6M = 3 x 4 / 3 x ln (100 / B₂) - 2 x 2 x ln (100 / B₁) = 4 x [ ln (100 / B₂) - ln (100 / B₁)]

= 4 x ln (B₁ / B₂)

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16. What is the guaranteed fair price of a 3-month T-Bill to be delivered at 6 months from now, assuming quarterly compounding?

This is a security that will deliver a face value of $ 100 at the end of 9 months from now. This is same as a 9-month strip with a face value of $100 is currently trading for B₂. Hence, it's guaranteed price should be B₂.

17. What is the guaranteed fair price of a 3-month T-Bill to be delivered at 6 months from now, assume continuous compounding?

This is a security that will deliver a face value of $ 100 at the end of 9 months from now. This is same as a 9-month strip with a face value of $100 is currently trading for B₂. Hence, it's guaranteed price should be B₂.