Bootstrapping Spot Rates from Semiannual Coupon Bonds

Summary

The document shows how to infer spot rates from the prices and cash flows of semiannual coupon bonds. For a bond whose only payment occurs at maturity, the price is equated to the discounted coupon plus principal, which gives the rate for that maturity period. A second bond with an earlier coupon and a later principal payment can then be used to solve for the next spot rate.

The worked setup discounts each cash flow using the rate for its payment date. The first-period coupon uses the already inferred six-month rate, while the later coupon and principal are discounted over two six-month periods using the one-year rate. This is a basic bootstrapping argument: solve sequentially for discount rates from bond prices and known cash flows. The document offers only a short numerical setup and does not discuss day-count conventions, compounding choices beyond the stated equations, credit risk, or how to handle a larger bond set.

Key ideas

• A bond price equals the present value of its contractual cash flows under the chosen discount rates.
• A single-payment bond can be used to infer the spot rate for its maturity period.
• Later-maturity spot rates can be solved using bonds with multiple cash flows once earlier rates are known.
• Each payment should be discounted over the period corresponding to its maturity.

Tags

# Valuation of Fixed-Income Securities


# Valuation of Fixed-Income Securities












Could one of you please assist with question 4 shown in the image above?

Thank you in advance! JR

## Answer by AlexM (score 1, accepted)

https://quant.stackexchange.com/a/36357

Bonds X and Y pay semiannual coupons. Then the cash flow for X is a single payment at the maturity of 3\$ (half of 6\$ since it's semiannual) plus the par value. The discounted cash flow has to equal the price of the bond. Then you have :$$100,98 = \frac{103}{1+R_{0.5}}$$ Solving for $R_{0.5}$ gives you 2%

Then you can perform the same calculation for Y, knowing that you have a first cash flow of 4\$ (that has to be discounted with $R_{0.5}$) for the first 6-month period and a second for the second period for the coupon and the principal payment (that should be discounted with $R_1$).

## Answer by AlexM (score 1)

https://quant.stackexchange.com/a/36359

I wish I could "add a comment" to your answer but I can't for the moment. Of course, the coupon for Y is 4\$ I corrected my answer. However, you are missing to square the denominator (because you are discounting the cash flow of the second period). You should have : $$103.59 = \frac{4}{1+R_{0.5}} + \frac{104}{(1+R_1)^2}$$ Since you know $R_{0.5}$ you can solve for $R_{1}$. Hope this helped!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.