# Simple bond mathematics

Dean Baker doesn’t seem to quite understand how the inverse relation between price and yield works in practice:

Back in the summer of 2003, the interest rate on the 10-year treasury bond bottomed out at 3.05 percent. Today, it stands at around 4.6 percent. This means that the bonds China held back then have lost approximately one-third of their value. (The price of the bond is inversely proportional to the yield. The actual calculation of the bond price is a bit more complicated, since it does matter when they reach maturity.) If the yield on 10-year treasury bonds rises back to its avearge rate for the decade of the 90s (6.8 percent), then the value of the bonds would drop by another 30 percent.

Let’s go to the SmartMoney bond calculator, and look at the price of a 10-year, 5% bond. (Feel free to use any other maturities or coupon rates: the results won’t be all that different.)

If the yield is 3.05%, the price is 116.6.

If the yield is 4.6%, the price is 103.5. That’s a drop of 11.2%, which is nowhere near “approximately one-third”.

If the yield is 6.8%, the price is 87.2. That’s a drop of a further 15.7%, which is nowhere near “30 percent”.

But of course those drops overstate reality quite a lot. Because between summer 2003 and now, there have been 7 coupon payments, totalling 17.5 cents. So if you bought that bond at 116.6 in 2003, it might be worth only 103.5 today, but you will have received 17.5 cents in coupon payments along the way — which, added to the value of the bond, brings you to 121, or a 3.7% net gain.

Of course, there are lots of extra variables involved, including the fact that a bond’s maturity goes down over time. But I really don’t think it’s possible for a US Treasury bond to lose two-thirds of its value, as Baker implies that it can — especially if you take coupon payments into account.

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### 2 Responses to Simple bond mathematics

1. Dean Baker says:

Felix,

Felix,

you’re right on this one — my next sentence was “The price of the bond is inversely proportional to the yield. The actual calculation of the bond price is a bit more complicated, since it does matter when they reach maturity,” but I should have taken the time to do the calculations. I had intended to use the 30-year bond where the yield/price relationship is more nearly proportional, but i was too lazy to look up the yields on the 30-year.

2. Felix says:

OK, let’s use a 5% 30-year bond.

At 3.05%, the price is 137.98.

At 4.6%, the price is 106.44, a drop of 22.8%. If you make 17.5 cents in coupons along the way, you’ve ended up with a total loss of 10%.

At 6.8%, the price is 77.21, a drop of a further 27.4% — for a total drop of 44%, at the very riskiest end of the yield curve, with yields more than doubling, and no coupon payments at all being taken into account.

And I did quote that next sentence of yours!