# 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? • Um, rotate picture first please? – BCLC Jul 25 '14 at 9:59

## 1 Answer

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.