Why Forward Rates Differ from Future Spot Rates
Summary
The question compares a simply compounded forward rate observed at an earlier date with a spot rate that will be set over the same future period. It defines zero-coupon bond prices, derives a forward rate from the ratio of current bond prices, and appears to obtain the same expression for the future spot rate by applying a zero-coupon price relation. This leads to confusion about why the rates should not be equal.
The response identifies the timing error: the future spot rate is unknown at the earlier date. Current prices of bonds maturing at the two dates determine the forward rate, while the rate available when the first bond matures will be set later. A strategy that buys the nearer-maturity bond and reinvests at that future rate therefore does not have a known return today equivalent to buying the longer-maturity bond. The exchange explains the distinction conceptually, but does not discuss risk premia or whether forward rates predict realized future spot rates.
Key ideas
- A forward rate is derived from zero-coupon prices observed at the current time.
- A future spot rate is determined later and is not known when the forward rate is calculated.
- The no-arbitrage bond-price relation does not make a future reinvestment rate certain.
- A forward rate and the later realized spot rate need not be identical.
Tags
Full text
# forward vs spot simply-compounded spot interest rate
# forward vs spot simply-compounded spot interest rate
Question about forward vs spot simply-compounded spot interest rate.Some definitions
- $P(a,b)$ a zero coupond price at time $a$ and maturity $b$
- $L(a,b)$ simply compounded spot interest rate set at time $a$ and payement at time $b$
- $F(t,a,b)$ simply compounded forward interest rate at time $t$, set at time $a$ and payement at time $b$
For a timeline $t<T<S$, we have formula for spot rate \begin{equation} L(t,T) = \frac{1}{T-t}[\frac{1}{P(t,T)}-1] \hspace{1cm} L(T,S) = \frac{1}{S-T}[\frac{1}{P(T,S)}-1] \end{equation} By arbitrage free hypothesis, we have the zero coupond relation $P(t,S) = P(t,T)P(T,S)$, hence \begin{equation} P(t,S)(S-T)L(T,S) = P(t,T) - P(t,S) \end{equation} On the other hand, we have the formula for forward interest rate \begin{equation} F(t,T,S)=\frac{1}{S-T}[\frac{P(t,T)}{P(t,S)}-1] \leftrightarrow P(t,S)(S-T)F(t,T,S) = P(t,T) - P(t,S) \end{equation} The idea is $F(t,T,S)$ has to be a forward rate at time $t$ of the spot rate $L(T,S)$, but what i've proved, they are equal. I see that is not logic, but i can not see what was wrong in my formula.
Can someone help me please?
Thanks
## Answer by RRL (score 4)
https://quant.stackexchange.com/a/12765
The flaw is $L(T,S)$ is a future spot rate that is determined at time $T>t$ and unknown at present.
It is correct that
$$F(t,T,S)=\frac{1}{S-T}\left[\frac{P(t,T)}{P(t,S)}-1\right] \iff P(t,S)(S-T)F(t,T,S) = P(t,T) - P(t,S), $$
as this is just the definition of the forward rate.
However, you are saying that
$$\frac1{P(t,T)}\frac1{P(T,S)}=\frac1{P(t,S)},$$
where
$$P(T,S)=\frac1{1+L(T,S)(S-T)}$$
The zero-coupon bonds maturing at $T$ and $S$ have known prices at present. You don't know that buying the $T$-maturity bond today and rolling into a new $S-T$-maturity zero-coupon bond at future time $T$ will produce the same return as buying an $S$-maturity bond today.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.