spartan1

Which of the following is a correct statement concerning the backward induction technique used within the binomial interest rate tree framework? From the maturity date of a bond:

A) the corresponding interest rates are weighted by the bond's duration to discount the value of the bond.

B) the corresponding interest rates and interest rate probabilities are used to discount the value of the bond.

C) a deterministic interest rate path is used to discount the value of the bond.

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For a bond that has N compounding periods, the current value of the bond is determined by computing the bond’s possible values at period N and working “backwards” to the present. The value at any given node is the probability-weighted average of the discounted values of the next period’s nodal values.

spartan1

With respect to interest rate models, backward induction refers to determining:

A) convexity from duration.

B) one portion of the yield curve from another portion.

C) the current value of a bond based on possible final values of the bond.

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Backward induction refers to the process of valuing a bond using a binomial interest rate tree. For a bond that has N compounding periods, the current value of the bond is determined by computing the bond’s possible values at period N and working "backwards."

spartan1

For an option-free bond where the coupons and maturity value are known and assuming constant interest rate volatility, which of the following sets of information will allow an analyst to construct the entire tree? The:

A) beginning interest rate at the root only.

B) lowest interest rate in each nodal period.

C) interest rate at the root and in the final nodal period.

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Given the lowest interest rate in each nodal period, the interest rates at the other nodes can be calculated. The interest rate at any node above the lowest is larger than the one below it by a factor of e2 × σ. Neither of the other sets of information are sufficient for constructing the tree.

spartan1

A binomial model or any other model that uses the backward induction method cannot be used to value a mortgage-backed security (MBS) because:

A) the prepayments occur linearly over the life of an interest rate trend (either up or down).

B) the cash flows for an MBS only depend on the current rate, not the path that rates have followed.

C) the cash flows for the MBS are dependent upon the path that interest rates follow.

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A binomial model or any other model that uses the backward induction method cannot be used to value an MBS because the cash flows for the MBS are dependent upon the path that interest rates have followed.

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.