Convexity of a bond

 

Modified duration comes from first order derivative of present value of future cash flows from a bond. For small changes in yield the first order derivative is adequate, however, for large changes in yield, it gives inaccurate results.

It’s evident from the formula of duration that it’s not a static property of a bond but depends on time and change in yield. Since the price-yield relationship is not linear but convex (slope or duration is different at different points in the curve), larger the change in yield larger the error in estimating the impact on price of a bond using modified duration.

So we should consider second order derivative. Convexity is the second order derivative of the price equation w.r.t. yield. It’s a measure of the degree to which a bond’s price-yield curve departs from a straight line.

 

dP/dr = -1/(1+r) [CF1/(1+r) + 2CF2/(1+r)2 + ….+ nCFn/(1+r)n ]

d2P/dr2 = -2*CF1/(1+r)3 - 6*CF2/(1+r)4 - ….- n(n+1)CFn/(1+r)n+2

The above equation can be written as

d2P/dr2 = -1/(1+r)2 [ 1 * 2 CF1/(1+r) + 2 * 3 * CF2/(1+r)2 + … n (n+1)*CFn/(1+r)n]

d2P/dr2 = -1/(1+r)2 [ ∑  n * (n+1) * CFn/(1+r)n ]

Divide both sides of the equation by P

 

d2P/dr2 (1/P)= -1/(1+r)2 [ ∑ n n * (n+1) * CFn/(1+r)n ] (1/P)

This is called convexity of a bond which represents price change due to large changes in yield and is the second order derivative of change of price equation

d2P = Convexity * dr2 * P

Change in price can be expressed as

Considering linear and convex equations, change in price can be expressed as

dP = - Duration mod * dr * P + 0.5 * Convexity * dr2 * P

dP = - Duration mod * dr *P + 0.5 * Convexity * dr2 * P

 

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June 21, 2008 | Filed Under Bonds 

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