Reading 57: LOS d ～ Q6- 8

6.A callable bond with an 8.2 percent annual coupon will mature in two years at par value. The current one-year spot rate is 7.9 percent. For the second year, the yield-volatility model forecasts that the one-year rate will be either 6.8 or 7.6 percent. The call price is 101. Using a binomial interest rate tree, what is the current price?

A)   100.558.

B)   100.279.

C)   101.000.

D)   99.759.

The correct answer was C)

The tree will have three nodal periods: 0, 1, and 2. The goal is to find the value at node 0. We know the value for all the nodes in nodal period 2: V₂=100. In nodal period 1, there will be two possible prices:

V_1,U =[(100+8.2)/1.076+(100+8.2)/1.076]/2 = 100.558

V_1,L =[(100+8.2)/1.068+(100+8.2)/1.068]/2= 101.311

Since V_1,L is greater than the call price, the call price is entered into the formula below:

V₀=[(100.558+8.2)/1.079)+(101+8.2)/1.079)]/2 = 101.000.

7.Using the following tree of semiannual interest rates what is the value of a callable bond that has one year remaining to maturity, a call price of 99 and a 5 percent coupon rate that pays semiannually?

         7.59%
6.35%
         5.33%

A)   98.26.

B)   98.65.

C)   99.21.

D)   99.98.

The correct answer was A)

The callable bond price tree is as follows:

As an example, the price at node A is obtained as follows:

Price_A = min{(prob * (P_up + coupon/2) + prob * (P_down + coupon/2))/(1 + rate/2), call price} = min{(0.5 * (100 + 2.5) + 0.5 * (100 + 2.5))/(1 + 0.0759/2),99} = 98.75. The bond values at the other nodes are obtained in the same way.

8.Using the following tree of semiannual interest rates what is the value of a 5 percent callable bond that has one year remaining to maturity, a call price of 99 and pays coupons semiannually?

        7.76%
6.20%
        5.45%

A)   97.17.

B)   98.29.

C)   99.01.

D)   99.71.

The correct answer was B)

The callable bond price tree is as follows:

As an example, the price at node A is obtained as follows:

Price_A = min{(prob * (P_up + coupon/2) + prob * (P_down + coupon/2))/(1 + rate/2), call price} = min{(0.5 * (100 + 2.5) + 0.5 * (100 + 2.5))/(1 + 0.0776/2),99} = 98.67. The bond values at the other nodes are obtained in the same way.