This is another study series looking into the bond price. This time, I’d like to take a look at the Duration, in a plain word, the derivative of bond price with respect to the interest rate. It’s actually simple than you think, and the conclusion can be very insightful.
Price change due to interest rate
For a bond with no coupon, we have introduced last time as the price today discounted by a factor:
Price today = 100 * Discount FactorThe discount factor is always less than 1, since otherwise it’ll be harder to ask you lead out $100 today. It needs to be cheap enough thus the discount factor. Say in an ideal case, which is not too different than the real case, the discount factor has the following relationship with the interest rate:
Discount Factor = exp(- Interest Rate * Time)Here, the time refers to the number of years to wait till matured. Plugging this into the price, and taking a derivation of price with respect to the interest rate:
dPrice/dInterestRate = - Time * PricePlease don’t be bothered by the math, it’s easier to look at the result for now.
When interest rate increases
This is the definition of the Duration, a quantity tells us how fast the price changes if the interest rate changes. Most of time we only care about the percentage change of the price instead of the amount of the price change, thus dividing the Price from both side:
dPrice/dInterestRate/Price = - TimeThe relative price change decreases when the interest rate increases, thanks to the negative sign. Moreover the magnitude of the decrease depends on a simple variable, the time.
Since the unit of this volatility term is time, the term is referred as Duration.
If we are comparing a 30 years bond and 1 year bond, we can draw the conclusion the price volatility due to interest rate can be drastically larger for the 30 years bond than the 1 year bond.
What happens to the coupon bond?
The above discussion applies to the zero coupon bond, but does it also hold for the regular bond that pays you coupon every six month? Sort of. Let’s take a look together.
First, the good news for the coupon bond, from last article, is that it can be made of series of zero-coupon bonds: the one-time principle payment in the end and series of coupons along the way. So in order to calculate the volatility, we can figure out volatility of each part and then add them together weighted by their compositions.
Volatility = Sum of each part weighted VolatilitySo, the volatility from the principle payment is same as the zero-coupon calculation, -30 for 30 years. Similarly the volatility from the first coupon is -0.5 (six month), and -1 (one year) for the second one etc. Adding all of them together considering their weighted percentage:
Volatility = -0.5wp - 1wp - 1.5wp - ... -30wpYou might have 60 payments (two payments a year) plus the last payment altogether. wp refers to the weighted percentage of each payment. Thus all the wp are same except the last one. Essentially it asks us to calculate the average volatility of the above 61 parts. Let me skip the math and just tell you the answer is -19.6.
Weight from the last future
Although the answer isn’t -30, it’s not difficult to find out the weight of the last principle payment is relatively large, in this case, it’s 29.4% like 1/3 of the composition. Thus -30 gets its chance to play more important part in determine the average value -19.6 .
What does this all means? It means, the future isn’t clear, the farther we go, the more volatile it gets, especially in the context of the price with respect to the interest rate. If in the future the interest rate increase by 1%, you should expect the price drop as large as 20%. Insane, but not un-common.
Wow, now you don’t think it’s funny, do you? What is the way to decrease the volatility, you might ask. Based on what we know, a six month zero-couple bond gives us the volatility of -0.5. For the same 1% of interest rate change, the price only drops by 0.5%. The tomorrow of six month is much clearer than the 30 years future. This is too obvious, isn’t it? Yes, but what we kinda of quantified it.
Note: we only used 1% interest rate change. Go figure when the interest rate changes more than that.
Summary
For environment with potential large interest rate changes, the price of the bond can be quite volatile. Thus you can’t merely bet on the return of 30 years bond blindly unless you can really sit through the next 30 years without doing anything.
