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LOS h: Illustrate how effective duration and effective convexity are calculated using the binomial model.

Which of the following correctly explains how the effective duration is computed using the binomial model. In order to compute the effective duration the:

A)	yield curve has to be shifted upward and downward in a parallel manner and the binomial tree recalculated each time.

B)	binomial tree has to be shifted upward and downward by the same amount for all nodes.

C)	the nodal probabilities are shifted upward and downward and the binomial tree recalculated each time.

Apply parallel shifts to the yield curve and use these curves to compute new forward rates in the interest rate tree. The resulting bond values are then used to compute the effective duration.

Which of the following most accurately explains how the effective convexity is computed using the binomial model. In order to compute the effective convexity the:

A)	binomial tree has to be shifted upward and downward by the same amount for all nodes.

B)	yield curve has to be shifted upward and downward in a parallel manner and the binomial tree recalculated each time.

C)	volatility has to be shifted upward and downward and the binomial tree recalculated each time.

Apply parallel shifts to the yield curve and use these curves to compute new forward rates in the interest rate tree. The resulting bond values are then used to compute the effective convexity.

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 yield curve.

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

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

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.

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)	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.

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

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

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?

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%

A CFA charter holder observes a 12-year 7 ? 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.

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