Deriving FRA Forward Rates from Zero-Coupon Bond Prices
Summary
The document explains why a forward rate agreement cannot be priced by treating an interest rate like a tradable stock. For a forward on an asset, the current asset price and the discount bond price can determine the forward value through replication. An interest rate, however, is not itself an asset that can be bought and held, so its quoted market rate cannot serve as the underlying price in that formula.
Instead, the FRA’s forward rate is derived from observable zero-coupon bond prices. A position in bonds maturing at the loan’s start and end dates replicates the agreement’s cash flows, yielding the implied forward rate from the ratio of their prices. The response emphasizes observability and replication as the reason for the different calculation. It is a concise conceptual answer rather than a broader treatment of FRA conventions, collateral, day-count rules, or valuation under different interest-rate frameworks.
Key ideas
- An interest rate is not a tradable asset that can be bought and held like a stock.
- The asset-forward pricing formula cannot be applied by substituting an unobservable future interest rate for the asset price.
- Zero-coupon bonds at the FRA dates provide observable prices for deriving its implied forward rate.
- The bond portfolio replicates the FRA cash flows and grounds the rate in tradable instruments.
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Full text
# FRA vs "forward on interest rate"?
# FRA vs "forward on interest rate"?
Consider a forward on a general underlying security in which the holder pays $K$ and receives $S_T$ at $T$. We can show that the forward price at $t\leq T$ is $$F=S_t/Z(t,T),$$ where $Z(t,T)$ denotes the price of a maturity-$T$ ZCB at $t$. This is because the forward is replicated by a portfolio which is short $K$ ZCBs and long the stock.
Now consider an FRA in which the holder pays $(T-T')K$ and receives $(T-T')L_{T'}[T',T]$ at $T$, where $L_{T'}[T',T]$ is the market's realized interest rate for borrowing/lending from $T'$ to $T$. If an FRA is just a forward on an interest rate, why can't we directly apply the above formula to the FRA --- treating $(T-T')L_t[T',T]$, the market's interest rate at $t$ for borrowing/lending from $T'$ to $T$, as an analogue of $S_t$ --- to obtain a forward price of $$F=\frac{(T-T')L_t[T',T]}{Z(t,T)}?$$
Instead, my book derives the FRA's forward price by constructing a replicating portfolio which is long a maturity-$T'$ ZCB and short $1+K(T-T')$ maturity-$T$ ZCBs to derive a forward price of $$F=\left(\frac{Z(t,T')}{Z(t,T)}-1\right)\frac{1}{T-T'}.$$
Why don't the results agree? Is there something different about FRAs which make them not equivalent to a forward on an interest rate?
## Answer by dm63 (score 1)
https://quant.stackexchange.com/a/81443
The idea of the exercise is to calculate the forward rate from things that are observable. The ZCBs are observable . You cannot directly observe $L_t[T’,T]$. Indeed, that is what you are trying to calculate- the correct price now for the interest rate at a future time. Does that clarify ?
One further comment : interest rates are fundamentally different from stocks, because a stock is an asset which you can buy and hold. An interest rate is not an asset that you can buy. The assets in the interest rate world are the ZCBs, so you need to construct the forward rate from the prices of those.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.