# Simple Forward Interest Rate Proof

Just trying to check my logic here:

Let $Z(t,T)$ be a Zero-Coupon Bond with maturity $T$ bought at time $t$, $S_m$ be the spot interest rate for time $m$ and $S_n$ for time $n$ respectively, where $n >m$.

I was trying to prove to myself that holding a portfolio of long $Z(t,n)$ and short $\frac{(1+S_m)^m}{(1+S_n)^n}$ units of $Z(t,m)$ "locks" or implies that the interest rate between time $m$ and $n$ is the forward interest rate $f(m,n)$ where:

$$[1+f(m,n)]^{n-m} = \frac{(1+S_n)^n}{(1+S_m)^m}$$

So far, I've shown that at time $m$, the value of the portfolio is:

$$\frac{Z(t,n) - 1}{[1+f(m,n)]^{n-m}}$$

And at time $n$, the value of the portfolio must be zero; otherwise there would exist an arbitrage portfolio.

However, I'm having trouble making the last argument (or so I think). So far, I've framed it in the following way, which I feel is rather lackluster.

• Since there cannot exist an arbitrage portfolio, thus the fair forward rate at which we re-invest our $Z(t,m)$ at time $m$ MUST be the forward rate $f(m,n)$

I suppose my real question is - what is the proper way to do a no-arbitrage/replication proof? And what is the real-world context in terms of this proof? How would this be applied: say at time $m$, our $m$-maturity ZCB matures, and we are obligated to pay the shorted ZCB out. I'm having trouble thinking of/visualizing no-arbitrage.

• Could you please use math formatting next time? – SRKX Jun 20 '14 at 4:57
• my bad! I didn't know how. Will review your edits and note duly for next time. sorry for the trouble! – Luke Chang Jun 20 '14 at 7:23