Investment

Fangyue Lu

2015/11/02

Duration is a measure of the effective maturity of a bond, which is the slope of the price-yield curve expressed as a fraction of the bond price. It can be used to measure the sensitivity of bond price changes to yield changes. It means the weighted average of the times until each payment is received, with the weights proportional to the present value of the payment. The Duration’s formula is as follow：

[pic 1]

CFt is cash flow on year t. And Y is the discount rate

There are some rules for Duration. The first rule is that the duration of a zero-coupon bond equals its time to maturity. If you’re holding maturity constant, a bond’s duration is higher when the coupon rate is lower. If you’re holding the coupon rate constant, a bond’s duration generally increases with its time to maturity. Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower. Price change is proportional to duration:

D* = modified duration

D* = D / (1+y)

[pic 2]P/P = - D* x [pic 3]y

The actual relationship between bond price and yields is not linear, duration rule is a good approximation for small changes in bond yield. When yield change is big, need to consider convexity. Convexity measures the curvature of the price-yield curve, it is the second derivative (the rate of change of the slope) of the price-yield curve divided by the bond price

[pic 4]

Correction:  [pic 5]

The rules for convexity is that holding maturity constant, a bond’s convexity is higher when the coupon rate is lower. If you are holding the coupon rate constant, a bond’s convexity generally increases with its time to maturity. If you are holding other factors constant, the convexity of a coupon bond is higher when the bond’s yield to maturity is lower.