# Convexity of a Perpetual Bond

#### Alexei

Would it be correct to calculate the convexity of a perpetual bond paying \$1/year in a market with x as a yield for all maturities as 2/x^2?

#### dazzwater

I read somewhere that the duration of a perpetual bond is (1+yield)/yield.

If this is true, maybe it would help in getting your answer, if you can figure out the P and dP/dy and subsequently the 2nd derivative.

But I wonder what the yield curve is supposed to look like, and whether it can be illustrated from the info you provided above.

#### Alexei

Can you prove it, (1+yield)/yield?

#### financeguy

so general disclaimer: this is not what i do
but once upon a time i learned bond/annuity math
the PV/price of this perpetuity is 1/x, where 1 is the coupon payment and x is the interest rate
convexity = (1/price)*[d^2(price)/d(yield)^2] = 1/[(1/x)]*[d^2(1/x)/dx^2] = x*[d(-1/x^2)/dx] = x*[2/x^3] = 2/x^2
ta da

#### financeguy

no love?
(modified) duration = -(1/price)*d(price)/d(yield) = -x*d(1/x)/dx = -x*(-1/x^2) = 1/x
the duration of a perpetuity due, which starts payment today rather than a perpetuity immediate which starts payment next year would be future valued 1 year from that, i.e. (1/x)*(1+x) = (1+x)/x, but i believe the problem you are trying to solve for a perpetual bond assumes first payment next year rather than today.

#### Alexei

I got 1 + 1/x for duration and then I calculated 2/x^2 for convexity. Thanks.

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