Reading 54: Valuing Bonds with Embedded Options-LOS i 习题精 annual, current, interest, face, 2010

土豆妮

Session 14: Fixed Income: Valuation Concepts
Reading 54: Valuing Bonds with Embedded Options

LOS i: Calculate the value of a putable bond by using an interest rate tree.

 

 

 

A putable bond with a 6.4% annual coupon will mature in two years at par value. The current one-year spot rate is 7.6%. For the second year, the yield volatility model forecasts that the one-year rate will be either 6.8% or 7.6%. The bond is putable in one year at 99. Using a binomial interest rate tree, what is the current price?

A) 98.885.

B) 98.246.

C) 98.190.

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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 at all nodes in nodal period 2: V₂=100. In nodal period 1, there will be two possible prices:

V_i,U = [(100 + 6.4) / 1.076 + (100+6.4) / 1.076] / 2 = 98.885

V_i,L = [(100 + 6.4) / 1.068 + (100 + 6.4) / 1.068] / 2 = 99.625.

Since 98.885 is less than the put price, V_i,U = 99

V₀ = [(99 + 6.4) / 1.076) + (99.625 + 6.4) / 1.076)] / 2 = 98.246.

土豆妮

Using the following tree of semiannual interest rates what is the value of a putable bond that has one year remaining to maturity, a put price of 99, coupons paid semiannually with payments based on a 5% annual rate of interest?

         7.59%
6.35%
         5.33%

A) 98.75.

B) 99.00.

C) 97.92.

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The putable bond price tree is as follows:

		100.00
A → 99.00
99.00		100.00
	99.84
		100.00

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

Price_A = max[(prob × (P_up + coupon / 2) + prob × (P_down + (coupon / 2)) / (1 + (rate / 2)), put price] = max[(0.5 × (100 + 2.5) + 0.5 × (100 + 2.5)) / (1 + (0.0759 / 2)) ,99] = 99.00. The bond values at the other nodes are obtained in the same way.

The calculated price at node 0 =

[0.5(99.00 + 2.5) + 0.5(99.84 + 2.5)] / (1 + (0.0635 / 2)) = $98.78 but since the put price is $99 the price of the bond will not go below $99.

土豆妮

Dawn Adams, CFA, along with her recently hired staff, have responsibilities that require them to be familiar with backward induction methodology as it is used with a binomial valuation model. Adams, however, is concerned that some of her staff, particularly those not enrolled in the CFA program, are a little weak in this area. To assess their understanding of the binomial model and its uses, Adams presented her staff with the first two years of the binomial interest rate tree for an 8% annually compounded bond (shown below). The forward rates and the corresponding values shown in this tree are based on an assumed interest rate volatility of 20%.

A member of Adams' staff has been asked to respond to the following:

Compute V_1L, the value of the bond at node 1L.

A) $103.58.

B) $101.05.

C) $95.99.

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V_1L = (?)[(V_2LU + C) / (1 + r_1L)] + [(V_2,LL + C) / (1 + r_1L)]

V_1L = (?)[(99.455 + 8) / (1 + 0.05331)] + [(102.755 + 8) / (1 + 0.05331)] = $103.583

(Study Session 14, LOS 55.i)

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Compute V_1U, the value of the bond at node 1U.

A) $99.13.

B) $91.72.

C) $99.01.

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V_1U = (?)[(V_2,UU + C) / (1 + r_1U)] + [(V_2,UL + C)/(1 + r_1U)]

V_1U = (?)[(98.565 + 8) / (1 + 0.079529)] + [(99.455 + 8) / (1 + 0.079529)] = $99.127

(Study Session 14, LOS 55.i)

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Compute V₀, the value of the bond at node 0.

A) $104.76.

B) $99.07.

C) $101.35.

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V₀ = (?)[(V_1U + C) / (1 + r₀)] + [(V_1L + C) / (1 + r₀)]

From the previous question the value for V_1U was determined to be $99.127

V₀ = (?)[(99.127 + 8) / (1 + 0.043912)] + [(103.583 + 8)/(1 + 0.043912)] = $104.755

(Study Session 14, LOS 55.i)

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Assume that the bond is putable in one year at par ($100) and that the put will be exercised if the computed value is less than par. What is the value of the putable bond?

A) $103.04.

B) $105.17.

C) $95.38.

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The relevant value to be discounted using a binomial model and backward induction methodology for a putable bond is the value that will be received if the put option is exercised or the computed value, whichever is greater.

In this case, the relevant value at node 1U is the exercise price ($100.000) since it is greater than the computed value of $99.127. At node 1L, the computed value of $103.583 must be used.

Therefore, the value of the putable bond is:

V₀ = (?)[(100.00 + 8) / (1 + 0.043912)] + [(103.583 + 8) / (1 + 0.043912)] = $105.17314

(Study Session 14, LOS 55.i)

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Assume that the bond is putable in one year at par ($100) and that the put will be exercised if the computed value is less than par. What is the value of the put option?

A) $0.42.

B) $1.86.

C) $3.70.

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V_putable = V_nonputable + V_put

Rearranging, the value of the put can be stated as:

V_put = V_putable ? V_nonputable

V_putable was computed to be $105.173 in the previous question, and V_nonputable was determined to be $104.755 in the question prior to that. So the value of the embedded put option for the bond under analysis is:

$105.173 ? 104.755 = $0.418

(Study Session 14, LOS 55.e, i)

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Which of the following statements regarding the option adjusted spread (OAS) is least accurate?

A) The OAS is equal to the Z-spread plus the option cost.

B) The OAS for a corporate bond must be calculated using a binomial interest rate model.

C) The OAS is the spread on a bond with an embedded option after the embedded option cost has been removed.

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The OAS is equal to the Z-spread minus the option cost. Both of the other choices are true statements. (Study Session 14, LOS 55.g)

土豆妮

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

A) Max(put price, discounted value).

B) Max(par value, discounted value).

C) Min(put value, discounted value).

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When valuing a putable 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 exercise price, whichever is greater.

土豆妮

 

Using the following tree of semiannual interest rates what is the value of a putable semiannual bond that has one year remaining to maturity, a put price of 98 and a 4% coupon rate? The bond is putable today.

         7.59%
6.35%
         5.33%

A) 98.75.

B) 98.00.

C) 97.92.

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The putable bond price tree is as follows:

		100.00
A ==> 98.27
98.00		100.00
	99.35
		100.00

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

Price_A = max{(prob × (P_up + coupon/2) + prob × (P_down + coupon/2))/(1 + rate/2), putl price} = max{(0.5 × (100 + 2) + 0.5 × (100 + 2))/(1 + 0.0759/2),98} = 98.27. The bond values at the other nodes are obtained in the same way.

The price at node 0 = [0.5 × (98.27+2) + 0.5 × (99.35+2)]/ (1 + 0.0635/2) = $97.71 but since this is less than the put price of $98 the bond price will be $98.