Question

A $1,000 United States Treasury bond with a maturity date 20 years from now and a coupon rate of 5 percent has a required rate of return of 5 percent. Calculate the value of the bond if annual interest rate payment are in force. Show your data inputs.

Strip the bond into an interest only bond and a face value only bond. That is, create a bond that has only the present value of the interest payments and one that is only the present value of the Face value. Which bond has more interest rate risk?

Which bond has more default risk?Now say that instead of annual payments, the Treasury makes payments every six months. Recalculate the value of your stripped bonds. Explain the results. Show your data inputs.

Answer 1

A $1,000 United States Treasury bond with a maturity date 20 years from now and a coupon rate of 5 percent has a required rate of return of 5 percent. Calculate the value of the bond if annual interest rate payment are in force. Show your data inputs.

I/Y = 5%; n = 20; PMT = 5% x FV = 5% x 1000 = 50; FV = 1000;

Value of the bond = PV = ?? = PV (I/Y, n, PMT, FV) = PV (5%, 20, 50, 1000) = 1000

This is expected because a coupon with required rate of return same as its coupon rate will trade at par.

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Strip the bond into an interest only bond and a face value only bond. That is, create a bond that has only the present value of the interest payments and one that is only the present value of the Face value. Which bond has more interest rate risk?

The bond that has only the present value of the interest payments:

I/Y = 5%; n = 20; PMT = 5% x 1000 = 50; FV = 0;

Value of the bond = PV = ?? = PV (I/Y, n, PMT, FV) = PV (5%, 20, 50, 0) = 623.11

The bond that has only the present value of the Face value:

I/Y = 5%; n = 20; PMT = 0; FV = 1000;

Value of the bond = PV = ?? = PV (I/Y, n, PMT, FV) = PV (5%, 20, 0, 1000) = 376.89

The bond that has only the present value of the Face value, has the higher interest rate risk, as there is no intermediate payment from the bond. The duration of this bond is higher and hence it's more sensitive to interest rate risk.

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Which bond has more default risk?Now say that instead of annual payments, the Treasury makes payments every six months. Recalculate the value of your stripped bonds. Explain the results. Show your data inputs.

The bond that has only the present value of the Face value, has the higher default risk, as there is no intermediate payment from the bond. There is just one bullet payment at the end of 20 years.

If bonds make payments semi annually, period will be doubled and rate will be halved.

The bond that has only the present value of the interest payments:

I/Y = 5% / 2 = 2.5%; n = 20 x 2 = 40; PMT = 5%/2 x 1000 = 25; FV = 0;

Value of the bond = PV = ?? = PV (I/Y, n, PMT, FV) = PV (2.5%, 40, 25, 0) = $ 627.57

The payments are now being received earlier in time. A yearly payment of $ 50 is now split in two payments of $ 25 every six months. Hence, the present (time) value of the future coupon payments improve and the value increases.

The bond that has only the present value of the Face value:

I/Y = 2.5%; n = 40; PMT = 0; FV = 1000;

Value of the bond = PV = ?? = PV (I/Y, n, PMT, FV) = PV (2.5%, 40, 0, 1000) = 372.43

The interest rates are now compounded semi annually and hence effective interest rate increases and hence the present (time) value of the bullet payment at the end of 20 years decreases and hence the value of the bond decreases.