Yield curve and sensitivity

 If you assume there is no default risk, fixed coupon bonds pay fixed ( predictable) cashflows throughout its lifetime. That doesn’t mean it is risk-free. You can make capital gain or loss on fixed coupon bonds depending on the prevailing market rate. Quantifying it, can help in risk mitigation. For example in US rate is expected to go up in future and in India it is widely expected that rate needs to come down. This will have impact on all bond prices. In such cases, we need to know how the yield changes of a bond affects its prices.

 

Assume a coupon with 100 face value , with 10% annual coupon rate for 10 years. For simplicity, On day one , let’s assume interest rate is so volatile and it can vary between 0-24% rather than staying at 10%. What will be the bond price at each yield? We can use yield to maturity equation below to calculate bond prices at each yield and we can plot it in a diagram( java program attached. Raname priceyield.txt to jar and double click to launch if you have java setup)

 

P = c1/(1+y) + c2/(1+y)^2 +....+cn/(1+y)^n + f/(1+y)^n

 

This is called yield price curve. At yield zero, the bond price(200) is just sum of coupon values plus the face value we get at the end. As yield increases, prices of the bond falls.

 

The shape of the curve is hyperbola. The slope or the price sensitivity to the yield is high at the beginning and the slope is low at the end. The change of yield from 0% to 1% causes major price change compared to yield change of 23% to  24%.Slope/Sensitivity is the first order derivative at any point on the curve. First order derivatives will be useful in predicting nearby values on the curve approximately. Say you are driving your motorbike at 5km per hour near a signal. Velocity is first order derivative of displacement w.r.t time. I can approximately calculate (using first two terms of Taylor series) the distance covered by you for the next few seconds using this velocity of 5km/hour. The calculation gives approximate result as calculation is done along the tangent line rather than on the curve. If I use acceleration information , I can obtain better approximation (using first 3 terms of Taylor series). Similarly in the yield price curve, we need the second order derivative also , which is called convexity to calculate price changes more accurately.

 

Now consider the same bond’s yield price curve with different maturity scenarios ( 9 years to maturity, 5 years to maturity , 1 year to maturity ) . Curve shift happens as year passes. It flattens out as maturity nears. This is because most of the cash flows has already occurred and less discounting effect(Price needs to adjust  high interest rate for less number of years).