Bond Pricing, Yield and Interest Rates: The Mathematics Behind Bond Investing

SYNOPSIS

A bond's face value is fixed, but its market price keeps moving above or below that value as interest rates and demand change. This piece breaks down how pricing, yield, duration and convexity work together, so you can see the simple logic behind every bond's price.

Two people hold the exact same bond, yet one is sitting on a profit and the other on a loss. The only thing that changed between them is the interest rate.

When you buy a bond, the first number you notice is its face value. But the moment you go to the market to actually buy or sell one, you realise the price on the screen is rarely that same number. It keeps moving. On some days a bond sells for more than its face value and on some days for less. What decides this is how good the bond looks next to the other bonds available at that time. A bond selling above its face value is said to be at a premium. One selling below it is at a discount.

Now why does this premium or discount happen. It comes from the way return is measured. The coupon rate is the fixed interest on the face value. This never changes. The current yield takes the same coupon but divides it by the market price, so it goes up when the price goes down. Then there is yield to maturity or YTM. This one shows the full return if the bond is held till the end, after counting both the interest and the price gain or loss. Out of the three, YTM is the most complete number.

There is also one small thing that comes up while buying. Between two interest dates, interest keeps adding up daily. This is the accrued interest. The price you see on the screen without it is the clean price. The price you actually pay, after adding this interest, is the dirty price.

The pricing part sounds heavy but the base idea is simple. A bond's price is just the present value of all its future cash flows. Every coupon and the final face value is brought back to today's value using the market rate. This is exactly why bond prices move opposite to interest rates. If rates go up, the discounting is stronger, so the price falls. If rates come down, the price goes up.

The next question is how much the price moves. For this we use duration. Duration tells how sensitive a bond is to a rate change. Modified duration puts a number to it. If the modified duration is 5, the price falls by around 5% for a % rise in rates. But the actual movement is not a perfect straight line. There is a small curve in it. This curve is what convexity measures. Because of it the price falls a bit less and rises a bit more than duration alone tells you.

One example makes all this clear. Take a bond of face value 1,000 paying a 7% coupon. Say market rates now rise to 8%. New bonds are paying more, so nobody wants this old 7% bond at the same price. To make it worth buying, the price has to fall below 1,000, till the return on it matches the 8 % going around. The coupon does not change. Only the price adjusts. This one small adjustment is the whole idea behind bond mathematics.

Manav Trivedi