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Pricing Zero-Coupon Bonds from Spot and Forward Rates

Article Quant Q&A · Author: bng
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Summary

The document explains how to price a zero-coupon bond when yields differ across maturities. Its key distinction is between a bond’s maturity-specific zero-coupon spot rate and the sequence of one-year forward rates. Under annual compounding, discounting the face value by the product of the yearly forward-rate factors gives the same price as discounting by the corresponding maturity spot rate raised to the number of years.

The answer cautions that the required spot curve is not usually observed directly for every maturity. It must be bootstrapped from traded bonds, and long-maturity zero-coupon bonds may not trade in practice. Therefore, the calculation is a useful pricing framework, but its inputs may be theoretical or estimated rather than directly quoted market rates.

Key ideas

  • A zero-coupon bond is priced using the spot rate for its maturity.
  • A sequence of annual forward rates can be combined to discount the bond’s face value.
  • The compounded forward-rate factors correspond to the maturity spot-rate discount factor.
  • Spot rates for maturities without traded zero-coupon bonds may need to be bootstrapped.

Tags

Fixed incomeDerivatives pricingStatistics
Full text
# Pricing zero coupon bonds on a yield curve


# Pricing zero coupon bonds on a yield curve












I'm getting confused about how I should price the current price of a zero coupon bond when there are several yields to choose from. For instance, lets say that there is an upward sloping yield curve. The rates are $r_1 < r_2 < \cdots < r_{10}$.

Typically, when we price the current price of zero coupon bond that matures in 1 year, the calculation is simply $$ P = \frac{100}{1 + r_1} $$ However, when we price the current price of a 10 year zero coupon bond, I feel that it is overly simplistic to calculate the price as $$ P = \frac{100}{(1+r_{10})^{10}} $$ Rather, I feel that the correct way to price this is to think the rates as forward rates for each year so that $$ P = \frac{100}{(1+r_1)(1+r_2) \cdots (1+r_{10})} $$

Any input will be much appreciated. Thanks in advance.

## Answer by cykor21 (score 1, accepted)

https://quant.stackexchange.com/a/33500

Your approach is correct, but the practical difficulty is that you do not see zero-coupon spot rates for maturities longer then 1 year; and zero-coupon spot rates are the relevant rates for pricing zero-coupon bonds.

So 10 year zero-coupon spot rate curve needs to be bootstrapped from other bonds but in practice there are no 10 year zero-coupon bonds traded so this exercise has only theoretical meaning.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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