Deriving a Forward Rate from Zero-Coupon Bond Prices

Summary

The document explores how a forward interest rate follows from the prices of zero-coupon bonds with different maturities. It proposes holding a long position in the longer-maturity bond and a short position in a scaled amount of the shorter-maturity bond, then asks how this portfolio demonstrates the rate locked in between the two dates.

The author relates the forward rate to the ratio of compounded spot rates and considers the portfolio’s value at the intermediate date and at final maturity. The central issue is how to make the no-arbitrage argument rigorous: compare the cash flows from the bond positions with reinvestment at the implied forward rate. The post presents an attempted line of reasoning and asks for clarification, but gives no completed proof or market application. Its setup also leaves details of financing and cash-flow timing to be resolved.

Key ideas

• Zero-coupon bond prices at two maturities imply a forward rate over the interval between them.
• A long and short bond portfolio can be used to examine the cash flows implied by that rate.
• A no-arbitrage proof should compare the portfolio’s dated cash flows with reinvestment at the forward rate.
• The document presents an incomplete proof attempt and does not resolve its financing and timing details.

Tags

# Simple Forward Interest Rate Proof


# 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.

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