# Yield curves and Bonds

Just struggling with putting the two together and google seems to be failing me.

(1) Are the yields on yield curves annual continuously compounded rates, effective rates or something else like a "bond yield" type of calculation. What I mean by this last one, is, for example, if a current bonds were priced \$100 (same as face value) and paid \$5 each year then the yield would be \$5 but it doesn't seem that this 5% is a guaranteed "effective rate" since the first coupon of \$5 in a years time may not be able to be reinvested at a "5%" rate...

(2) Assuming the yield curves yield rates are effective rates, for example, for a yield curve with points (1,4%) and (2,5%) where its (x,y)=(maturity,yield) does this suggest that, one could currently invest in a zero coupon bond and receive a 4% return at the end of the year, and that one could also invest in a 2 year zero coupon bond and receive (1.05)^2-1= 10.25% 2-year-return at the end of the two years. Is this one way of thinking about it?

You can skip what's below (a bit of musing) but I would really appreciate some clarification, hopefully along these simplistic lines, to the above.

Things seem to get complicated for me when I try think of non-zero coupon bonds because, for example, for two years, if a coupon bond was priced \$1000 now and payed \$51.25 at the end of each year, wouldn't this technically also be a 10.25% for the two years, except that the first coupon could be reinvested at a risk free rate making it more attractive then a zero coupon bond paying 10.25% at the end of two years.

I'm going to paraphrase your questions.

(1) Please explain what the yield curve tells us and how it is quoted

There are a lot of people out there constructing yield curves. I guess any curve that is made of up any set of numbers that could be called "yields" is a yield curve in the loose sense. I can't say everyone does what they should when making such a curve either.

However, when we use the term "yield curve" we are normally referring to the set of annualized yields for zero coupon treasury securities. The computation of zero coupon yields doesn't have any reinvestment concerns.

Now, treasury bonds and notes do pay coupons, so we can't just observe yields of zero coupon bonds at these maturities. Therefore we must use the prices of existing bonds to infer the yields for theoretical (non existent) securities of the desired maturity. Basically we use short term yields to figure out the present value of each of the coupons and subtract that amount from the price before computing the bond's yield. This process is called bootstrapping. (By the way we can also look at the prices of strips, but they sell at a slight discount and this is not the usual way to compute the yield curve.)

Anyway, the resulting curve is inferred from outstanding bond prices, but it may not represent the actual yield of any security you can buy.

(2) Does 4% and 5% for the 1 and 2 year bonds mean I can get 4% for a one-year zero coupon bond or 5% per year for a two year zero?

Yes. Though as I mentioned, those securities may not actually exist, so it is a bit theoretical. If they did exist and the markets are working well, then those would be the yields you would get.

(3) Are coupon bonds better because you can reinvest the dividends?

When we compute the "yield" of a coupon bond, usually we mean yield to maturity. The mathematics of YTM implicitly assume that you will reinvest dividends at the same rate as you got on this bond (i.e., its YTM). Is having reinvestable coupons a good thing? Depends on whether bond yields on competing alternatives have gone up or down by the time the coupon comes out. If they have fallen, then the zero coupon bond will be better because coupons will end up reinvested at a lower rate.

One way to think about this is that zero coupon bonds have the highest duration of all bonds with their maturity. Coupons get paid out earlier, reducing the risk of losses if bond yields rise. But they also reduce the gains if bond yields fall over the life of the bond. In short they trade off interest rate risk for reinvestment risk.

Bottom line: zero coupon bonds are no better nor worse than coupon bonds, but they do have a little more interest rate risk. Whether you will be better off investing in zeros or coupon bonds depends on what happens to competing interest rates over the life of the bond.

• Absolutely OUTSTANDING explanation here. Thank you for the great lesson in how coupon bonds work! Mar 20, 2021 at 21:27