Bootstrapping a Three-Year Zero-Coupon Bond from Coupon Bond Prices

Summary

The question asks how to infer the price and yield of a three-year zero-coupon bond from prices for one-year and two-year zero-coupon bonds and a three-year annual-coupon bond. It proposes valuing the coupon payments using the shorter bonds’ prices, subtracting those values from the coupon bond’s price, and scaling the remaining value to a face value of 100 to estimate the zero’s price. It then attempts to convert that price into a yield to maturity.

The example raises a useful bootstrapping concept: discount each cash flow using the spot rate for its own maturity, then infer the discount factor for the remaining cash flow. However, the proposed coupon valuations and yield calculation are not established as correct in the document, and the quoted price used in the yield expression differs from the preceding estimate. The text contains no answer resolving these issues, so it serves as a problem statement rather than a worked solution.

Key ideas

• Bond cash flows at different maturities require their corresponding discount factors.
• A coupon bond’s price can be decomposed into the present values of its coupon and principal payments.
• The example’s proposed bootstrapping steps and yield calculation are left unverified.
• The price stated in the yield formula differs from the price estimated immediately before it.

Tags

# How to compute the yield to maturity on a zero-coupon 3 year bond in this case?


# How to compute the yield to maturity on a zero-coupon 3 year bond in this case?












Suppose I have the following problem:

> We have data on three bonds: a one-year zero-coupon bond (bond A), a twoyear zero-coupon bond (bond B), and a three-year bond with an annual coupon equal to 5% of its face value (bond C). All three bonds have face values of 100 USD. The price of bond A is $90, the price of bond B is 85 USD, and the price of bond C is 105 USD.

Suppose I want to solve the following question:

> What are the price and yield of a three-year zero-coupon bond with face value of 100 USD?

I approached this problem in the following way:

By non-arbitrage, the first coupon must be worth $4.5$ since the 1 period bond with $100$ USD as face value costs 90. The second coupon must cost $4.25$ by the same reason. Then, the price of a zero coupon bond with face value 105 should be the price of bond C less the present value of coupons: $105 - 4.5 - 4.25 = 96.25$.

Now, by non-arbitrage again, the price of a zero coupon bond with 3 years to maturity with 100 face value should be:

$$P = \frac{100}{105}\cdot 96.25 \approx 91.6667$$

The yield should be

$$ytm = \left(\frac{100}{91.97}\right)^{1/3} - 1 \approx 2.9\%$$

Is this correct? Does this make sense? Thanks!

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