Risk Prep – Practice exam 1 – Q 115

[Anonymous]

Q-115: A ‘consol’ is a perpetual bond issued by the UK government. Its running yield is 5%. What is its duration?

I disagred with the answer. It should be 21 yeaars.

Running yield = Coupon / Value, if it’s perpetual than it gives (FV*i)/(FV*i/i) = i Where FV = Face value and i = interest rate

So 1/(Running yield) = 1/i

Duration = ( Coupon * Sum(n*v^n) + n*FV*(v^n) )/( Coupon * Sum(v^n) + FV*(v^n)) so if n = infinite than
Duration = ( Coupon * Sum(n*v^n) )/( Coupon * Sum(v^n) ) = ( Sum(n*v^n) )/( Sum(v^n) ) which gives us
Duration = ( (1/i^2) + (1/i) ) / ( (1/i) ) = 1 + 1/i

The anwser that we got is 1/(Running yield) = 1/i but it should be 1/(Running yield) + 1 = 1/i + 1

Anyone on that ???

[Jens Kroeske]

The Macaulay duration of a consol is (1+y)/y = 1+1/y (where y is the yield) as you rightly derived. I think the questions asks for the Modified duration which is (Macaulay duration)/(1+y) = 1/y (which is actually easier to derive using the fact that PV = coupon/yield* FV, taking the derivative with respect to the yield y and dividing by -PV).

Hope this helps

[Anonymous]

I haven’t saw that it could be the modified duration that was ask

Thanks!

PS Its my actuaial backgroud that tells me that duration = Macaulay duration

[Author]

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