Fin 473 Excercise

Finance 473 Debt and Money Market

Exercise set 4

Andrei Simonov

Duration

1. Consider a portfolio consisting of 4 different bonds with market values as in the Table below.

Bond                                 A         B         C         D

Mod. Duration (years)                 2         7         8         14

Market value($)                 13         27         60         40

a. What is the modified duration of the portfolio? Show that the modified duration of a portfolio of bonds is the (portfolio) weighted average of the individual bond durations.

b. Suppose interest rates falls by 50 bp for all maturities. Compute an approximate percentage change in the value of the portfolio.

Answer:

1. Suppose there are n bonds in the portfolio. Let xi denote the number of bonds of type i with price Pi that is held in the portfolio. Hence the price of the portfolio is

V=Σ i=1…N  xiPi

Denote the individual durations by Di = | (1/P i)(dP i /dy|. The portfolio duration may then be calculated by

[pic 1]

where wi =xi Pi/V  are the weights.

Thus, MD=2*13/140+7*27/140 + 8*60/140 + 14*40/140=8.9643

B: dP/P=-MD dy=-8.9643*(-.005)=4.48%

2. Consider the two semi-annual bonds E and F described in the table below.

Bond                         E                 F

Coupons                 8%                 9%

Yield to maturity         8%                 8%

Maturity                 2                 5

Face value                 $100                 $100

Price                         $100                 $104.055

Now assume interest rates increases by 100 bp.

a. Calculate the price of the bonds.

b. Approximate the price using duration.

c. Approximate the price using both duration and convexity.

d. Comment on the accuracy and explain why one of the approximations are better than the other.

e. Without calculations, indicate whether the duration of the two bonds would be higher or lower if the yield to maturity is 10% rather than 8%.

ANSWER:

a. PA = $98.206. PB = $100.

b. First note that if the yearly yield increases by 100bp then the semiannual

yield increases by 50bp, hence _y = 0.005. First order Taylor

expansion gives (note that we are here using dollar duration)

[pic 2]

and hence PdurA= PoldA+ΔPA=98.185, and PdurB= 99.899.

c. From a second order Taylor expansion we get

ΔPA ≈ (∂P/∂y) Δy + (1/2) (∂2P/∂y2) (Δy)2

Hence, we need to compute the ‘dollar convexity’ term, (∂2P/∂y2). Straightforward

computations gives

PAdur+conv= PAdur+(1/2) 1710.934*(0.005)2 =98.206 and PBdur+conv=  = 100.001.

d. For large changes in yield the tangent can not approximate the curve very well. Adding convexity gives a better approximation.

e. Since the duration is related to the slope of the tangent and the bond price curve as a function of yield is flatter for high yields the duration will decrease.

3. Consider a bond with par value $100, coupon rate 6% and 10 years to maturity.

a. Using Microsoft Excel produce a printout of a table and a graph of the PVBP corresponding to required yields in the range 1%-16% with intervals of 1% (place yield on the horizontal axis of the graph).