dkishore1

Using the following tree of semiannual interest rates what is the value of a 5% 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) 99.01.

B) 97.17.

C) 98.29.

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The callable bond price tree is as follows:

		100.00
A → 98.67
98.29		100.00
	99.00
		100.00

As an example, the price at node A is obtained as follows:
PriceA = min[(prob × (Pup + (coupon / 2)) + prob × (Pdown + (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.

dkishore1

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% coupon rate that pays semiannually?

	7.59%
6.35%
	5.33%

A) 99.21.

B) 98.65.

C) 98.26.

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The callable bond price tree is as follows:

		100.00
	98.75
98.26		100.00
	99.00
		100.00

The formula for the price at each node is:
Price = min{(prob × (Pup + coupon/2) + prob × (Pdown + coupon/2)) / (1 + rate/2), call price}.
Up Node at t = 0.5: min{(0.5 × (100 + 2.5) + 0.5 × (100 + 2.5)) / (1 + 0.0759/2), 99} = 98.75.
Down Node at t = 0.5: min{(0.5 × (100 + 2.5) + 0.5 × (100 + 2.5)) / (1 + 0.0533/2), 99} = 99.00.
Node at t = 0.0: min{(0.5 × (98.75 + 2.5) + 0.5 × (99 + 2.5)) / (1 + 0.0635/2), 99} = 98.26.

dkishore1

A callable bond with an 8.2% annual coupon will mature in two years at par value. The current one-year spot rate is 7.9%. For the second year, the yield-volatility model forecasts that the one-year rate will be either 6.8% or 7.6%. 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.

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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:
V1,U =[(100+8.2)/1.076+(100+8.2)/1.076]/2 = 100.558
V1,L =[(100+8.2)/1.068+(100+8.2)/1.068]/2= 101.311
Since V1,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.

dkishore1

Which of the following is the appropriate "nodal decision" within the backward induction methodology of the interest tree framework for a callable bond?

A) Min(call price, discounted value).

B) Min(par value, discounted value).

C) Max(call price, discounted value).

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When valuing a callable bond using the backward induction methodology, the relevant cash flow to use at each nodal period is the coupon to be received during that nodal period plus the computed value or the call price, whichever is less.

dkishore1

For a callable bond, the value of an embedded option is the price of the option-free bond:

A) minus the price of a callable bond of the same maturity, coupon and rating.

B) plus the price of a callable bond of the same maturity, coupon and rating.

C) plus the risk-free rate.

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The value of the option embedded in a bond is the difference between that bond and an option-free bond of the same maturity, coupon and rating. The callable bond will have a price that is less than the price of a non-callable bond. Thus, the value of the embedded option is the option-free bond’s price minus the callable bond’s price.

dkishore1

Suppose that the value of an option-free bond is equal to 100.16, the value of the corresponding callable bond is equal to 99.42, and the value of the corresponding putable bond is 101.72. What is the value of the call option?

A) 0.74.

B) 0.64.

C) 0.21.

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The call option value is just the difference between the value of the option-free bond and the value of the callable bond. Therefore, we have:
Call option value = 100.16 – 99.42 = 0.74.

dkishore1

How is the value of the embedded call option of a callable bond determined? The value of the embedded call option is:

A) the difference between the value of the option-free bond and the callable bond.

B) equal to the amount by which the callable bond value exceeds the option-free bond value.

C) determined using the standard Black-Scholes model.

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The callable bond is equivalent to the option-free bond except that the issuer has the option to call the bond at the call price before maturity. Therefore, for the holder of the bond, the bond is worth the same as the option-free bond reduced by the value of the option.

dkishore1

Which of the following is equal to the value of the putable bond? The putable bond value is equal to the:

A) option-free bond value minus the value of the put option.

B) callable bond plus the value of the put option.

C) option-free bond value plus the value of the put option.

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The value of a putable bond can be expressed as Vputable = Vnonputable + Vput.

dkishore1

The value of a callable bond is equal to the:

A) option-free bond value minus the value of the call option.

B) callable bond value minus the value of the put option minus the value of the call option.

C) callable bond plus the value of the embedded call option.

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The value of a bond with an embedded call option is simply the value of a noncallable (Vnoncallable) bond minus the value of the option (Vcall). That is: Vcallable = Vnoncallable – Vcall.

dkishore1

How does the value of a callable bond compare to a noncallable bond? The bond value is:

A) lower.

B) lower or higher.

C) higher.

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Since the issuer has the option to call the bonds before maturity, he is able to call the bonds when their coupon rate is high relative to the market interest rate and obtain cheaper financing through a new bond issue. This, however, is not in the interest of the bond holders who would like to continue receiving the high coupon rates. Therefore, they will only pay a lower price for callable bonds.