Duration of the Bond question.

[NNath]

Hi, I have the following question

The table below gives the closing prices and yields of a particular liquid bond over the past few days.

Monday's Price - 106.3 and Yield - 4.25%

Tuesday's Price - 105.8 and Yield - 4.20%

Wednesday's Price - 106.1 and Yield - 4.23%

What is the approximate duration of the bond?

A. 18.8
B. 9.4
C. 4.7
D. 1.9

How do you solve this. and how is that when the yield is increasing the price is also increasing and not decreasing ?

 

[David Harper CFA FRM]

Hi @NNath

Dollar duration is slope of the Price/yield curve, so you just need (change in price)/(change in yield) for the dollar duration; and multiply that by -1/Price for the effective duration (as an approximation to the modified duration). The slope of the price/yield like (while slightly variable depending) is given by:

[rise in Y axis]/[run in X axis] = (106.3 - 106.1)/(4.25% - 4.23%) = 1,000.00 dollar duration. Multiply by -1/P or -1/105.8 and you have: -1/105.8*(106.3 - 106.1)/(4.25% - 4.23%) ~= -9.425 years, which is the same exact concept as: effective duration = -1/P(0)*(P[shock up] - P[shock down])/(2*yieldShock); i.e., both are computing the slope of a secant which approximates the tangent slope (effective duration is a secant slope which approximates modified duration as the accurate tangent slope)

Re: how is that when the yield is increasing the price is also increasing and not decreasing ? Excellent question! It should not be! No doubt the question writer is testing the formulas without an intuitive understanding (maybe was confused by the negative sign). Strictly speaking, this instrument has a negative duration, but that would require an esoteric (synthetic) instrument.

 

[pint0]

Even though this threat is really old, i have a (hopefully not too stupid) question. The effective Duration, which is afaik mostly used for options on bonds is as following:

(P(Down by 1% yield) - P(upby 1% yield) ) / ( 2 x P x yield change)

in this case i calc. 4,725 Duration.

Why is this wrong? I dont understand why the denominator is slightly different.

 

[David Harper CFA FRM]

Hi @pint0 it's a good question. When the shock is symmetrical, the two expressions are the same: the denominator you are showing 2 *Δy * P = yield(higher) - yield(lower) * P. In the problem above, it's difficult (impossible?) to use your formula. But imagine if the middle yield was changed to be the midpoint, from 4.230% to 4.235%, then we would have:

• $106.30 @ 4.250%
• $106.10 @ 4.225% (tweaked to lie in the middle so there is one shock value)
• $105.80 @ 4.200%

And we can apply yours, by observing that the Δy = 0.0250%. That is, our P(y @ 4.225%) = 106.10 and then we have two prices 4.225% +/- 0.0250% such that:

• ($105.80 - 106.30) / (2 * 0.0250%) * -1/$106.10 = -9.43 years, which would be the same as:
• ($106.30 - 105.80) / (4.250% - 4.200%) * -1/106.10 = -9.43 years; as this is rise/run * -1/P, the order could be switched

For me at least, the key insight is that either ($105.80 - 106.30) / (2 * 0.0250%) or ($106.30 - 105.80) / (4.250% - 4.200%) is just retrieving a slope of a line ("rise over run" as they say) so there needs to be consistency in numerator and denominator. I hope that's helpful!

 

[pint0]

thank you so much! This is really helpful!