Deriving Zero-Coupon Bond Prices from Forward Rates

Summary

The answer explains how a zero-coupon bond discount factor relates to a sequence of forward rates. Its replication argument describes rolling an investment through consecutive periods, using each period's forward rate as the rate available today for that future interval. Under exponential compounding, the accumulated growth factor is built from the sum of the period rates in the exponent, rather than their average.

This gives an intuitive link between forward rates and the price of a bond maturing at the end of the sequence. The document is brief and provides no numerical example or discussion of alternative compounding conventions. Its displayed expression describes accumulated growth, while a discount factor is conventionally the reciprocal growth factor, so the sign convention in the quoted formula should be checked against how rates and discounting are defined.

Key ideas

• A zero-coupon bond price is a discount factor for its maturity.
• Rolling an investment across successive periods uses the forward rate for each interval.
• With exponential compounding, accumulated rates enter through a sum in the exponent rather than an average.
• The displayed growth expression and discount-factor convention require careful attention to the sign.

Tags

# Formula for the forward rates?


# Formula for the forward rates?












I'm reading a book about interest rate modelling. It states the following formula

P(0,T) = exp(-sum of the forward rates)

But I thought it's the average of the forward rates?

## Answer by Richi Wa (score 4, accepted)

https://quant.stackexchange.com/a/14165

The price of the zero-coupon bond is the discount factor for this maturity. In the world of exponential compounding formulas are of the form $\exp(\sum \cdots)$. With a replication argument if we want to invest money for $n$ years what can we do. We invest for one year $r_0 = F(0,1)$ then after this year we invest for another year, the rate for this today is $F(1,2)$, after another year we invest again for one year, the rate for this today is $F(2,3)$. After all the discount factor is simply $$ \exp(\sum_{i=1}^n F(0,n-1,n)), $$ where the $0$ indicates that the forward rates are traded/observed at time $0$ and $n-1$,$n$ means that it is the forward rate for the respective year. So it is the sum, not the average.

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