# Estimate yield of coupon bond given yield of zero coupon bond

Suppose that now is August 2006 and we have the following zero-coupon bonds:

Maturity: August 2007, Price: 95,53

Maturity: August 2008, Price: 91,07

Maturity: August 2009, Price: 86,2

Maturity: August 2010, Price: 81,08

Would you expect the yield on a non-zero coupon bond maturing in August 2010 to be higher or lower than the yield on the 2010 zero-coupon bond?

My attempt:

I think we should somehow use the fact that if we calculate the yields on these zero-coupon bonds then the term structure will be upward-sloping. By the expectations hypothesis, an upward sloping yield curve implies that the market is expecting higher spot rates in the future. But I don't know what conclusion we can draw about non-zero coupon bond yield from it.

• Hint: calculate the yield of the Aug 2010 zero coupon bond. Then build a Aug 10 coupon bond whose coupon equals that yield and see if it’s price is higher or lower than par.
– dm63
Oct 15, 2021 at 10:46
• @dm63 but what will be the yield to maturity of this coupon bond? Oct 15, 2021 at 12:02
• Well, if the price of the bond you have created is >100, the yield is less than the yield of the Aug 10 zero coupon bond.
– dm63
Oct 15, 2021 at 15:59

The yield on a discount (zero-coupon) bond maturing in 2010 should be higher than that of a coupon bond maturing in 2010 under the stated circumstances.

This is because some of the cash flow of the coupon bond will be realized earlier than that of the discount bond, and as shown in the table below, the yield curve, as far as these two bonds are concerned, is upward sloping.

Maturity Price Annualized Yield (%)
Aug 2007 95.53 4.7
Aug 2008 91.07 4.8
Aug 2009 86.20 5.1
Aug 2010 81.08 5.4

Let’s say the fair coupon on the 2010 coupon paying bond is C. Then this bond is worth 100. Its cash flows in 2007,2008,2009,2010 are respectively C,C,C, (100+C) and we can use the discount factors in the OP to calculate the present value as follows: $$PV = 0.9553C +0.9107 C+ 0.8620C+ 0.8108(100+C)$$. Equating this to 100 and solving, we find C to be 5.34, less than the yield of the 2010 zero coupon bond.

• I agree that a coupon bond would always have a lower yield(-to-maturity) than a discount bond with the same maturity given these discount bond prices or an (strictly) upward sloping yield curve. However, though informative, I think your answer is incomplete given the OP. The yield-to-maturity of a coupon bond, given these discount bond prices, depends on the coupon rate but the OP does not state the bond’s coupon rate. Oct 17, 2021 at 17:38

It depends on the value of the bonds that you are comparing. The yield of a bond it's related to its market value. Note that the value of a bond can be given in term of its yield as

$$V (y) = \sum^N_{i = 1} \dfrac{C_i}{\left(1 + y \right)^{t_i}},$$ where $$C_i$$ are the coupons. Note that I'm implicitly including the principal in the last coupon, $$C_N$$.

Then, determining the yield of a bond its just a matter of solving $$0 = V_{\rm quoted} - \sum^N_{i = 1} \dfrac{C_i}{\left(1 + y \right)^{t_i}}.$$

If both bonds, the one with zero coupons and the one with coupons trade at the same price $$V$$, then the one with coupons will have a higher yield. Note that the yield is a way of measuring cost-effectiveness. Therefore, the quote-price of the bond it's an important feature.