dkishore1

An analyst has constructed an interest rate tree for an on-the-run Treasury security. The analyst now wishes to use the tree to calculate the duration of the Treasury security. The usual way to do this is to estimate the changes in the bond’s price associated with a:

A) parallel shift up and down of the forward rates implied by the binomial model.

B) parallel shift up and down of the yield curve.

C) shift up and down in the current one-year spot rate all else held constant.

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The usual method is to apply parallel shifts to the yield curve, use those curves to compute new sets of forward rates, and then enter each set of rates into the interest rate tree. The resulting volatility of the present value of the bond is the measure of effective duration.

dkishore1

An analyst has constructed an interest rate tree for an on-the-run Treasury security. The analyst now wishes to use the tree to calculate the convexity of a callable corporate bond with maturity and coupon equal to that of the Treasury security. The usual way to do this is to calculate the option-adjusted spread (OAS):

A) compute the convexity of the Treasury security, and divide by (1+OAS).

B) compute the convexity of the Treasury security, and add the OAS.

C) shift the Treasury yield curve, compute the new forward rates, add the OAS to those forward rates, enter the adjusted values into the interest rate tree, and then use the usual convexity formula.

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The analyst uses the usual convexity formula, where the upper and lower values of the bonds are determined using the tree.

dkishore1

Steve Jacobs, CFA, is analyzing the price volatility of Bond Q. Q’s effective duration is 7.3, and its effective convexity is 91.2. What is the estimated price change for Bond Q if interest rates fall/rise by 125 basis points?

Fall

Rise

A) +7.70% −10.55%

B) +10.55% −7.70%

C) −10.55% +7.70%

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Estimated change if rates fall by 125 basis points:

(-7.3 × -0.0125) + (91.2)(0.0125)2 = 0.1055 or 10.55%


Estimated change if rates rise by 125 basis points:

(-7.3 × 0.0125) + (91.2)(0.0125)2 = -0.0770 or -7.70%

dkishore1

A CFA charter holder observes a 12-year 7 ¾ percent semiannual coupon bond trading at 102.9525. If interest rates rise immediately by 50 basis points the bond will sell for 99.0409. If interest rates fall immediately by 50 basis points the bond will sell for 107.0719. What are the bond's effective duration (ED) and effective convexity (EC).

A) ED = 8.031, EC = 2445.120.

B) ED = 40.368, EC = 7.801.

C) ED = 7.801, EC = 40.368.

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ED = (V- − V+) / (2V0(∆y))
= (107.0719 − 99.0409) / (2 × 102.9525 × 0.005) = 7.801
EC = (V- + V+ − 2V0) / (2V0(∆y)2)
= (107.0719 + 99.0409 − (2 × 102.9525)) / [(2 × 102.9525 × (0.005)2)] = 40.368

dkishore1

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) 98.246.

B) 98.885.

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:
Vi,U = [(100 + 6.4) / 1.076 + (100+6.4) / 1.076] / 2 = 98.885
Vi,L = [(100 + 6.4) / 1.068 + (100 + 6.4) / 1.068] / 2 = 99.625.
Since 98.885 is less than the put price, Vi,U = 99
V0 = [(99 + 6.4) / 1.076) + (99.625 + 6.4) / 1.076)] / 2 = 98.246.

dkishore1

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) 99.00.

B) 98.75.

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:
PriceA = max[(prob × (Pup + coupon / 2) + prob × (Pdown + (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.

dkishore1

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(par value, discounted value).

B) Min(put value, discounted value).

C) Max(put price, 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.

dkishore1

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) 98.00.

B) 98.75.

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:
PriceA = max{(prob × (Pup + coupon/2) + prob × (Pdown + 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.

dkishore1

For a convertible bond, which of the following is least accurate?

A) The issuer can decide when to convert the bonds to stock.

B) The conversion ratio times the price per share of common stock is a lower limit on the bond's price.

C) A convertible bond may be putable.

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All of these are true except the possibility of the issuer to force conversion. The bondholder has the option to convert.

dkishore1

Which of the following is equal to the value of a noncallable / nonputable convertible bond? The value of the corresponding:

A) callable bond plus the value of the call option on the stock.

B) straight bond.

C) straight bond plus the value of the call option on the stock.

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The value of a noncallable/nonputable convertible bond can be expressed as:
Option-free convertible bond value = straight value + value of the call option on the stock.