# Relationship between forward rate, future spot rate, and liquidity preference

I'm getting confused over the relationship between forward rates, spot rates, and liquidity preference. I know that liquidity preference theory (i.e. that investors prefer shorter term investments because they are more liquid) states that the forward rate is greater than the future spot rate. However, I am confused on what exactly the forward rate and future spot rate are. My understanding is that:

The forward rate ($$f_2$$) is the hypothetical rate you would need to eliminate an arbitrage opportunity in the example below:

$$(1 + r_1)(1 + f_2)$$ = $$(1 + r_2)^2$$

The spot rate is the the expected yield on a zero-coupon bond. But how does this play into the example above?

Moreover- just to make sure- am I correct in that longer maturity -> greater interest rate risk (because of inflation / liquidity pref)?

• It might be helpful to think about this in more intuitive terms. The future spot rate is the rate that you'd pay to buy something at a particular point in the future, while the forward rate is the rate you'd pay today to buy something to be received in the future. In the first case, you hold on to cash, and wait to buy the thing; in the latter case, you pay for the thing now, and you wait and receive it later. If you have a preference for liquidity, that means that you'd rather hold on to the cash, all else being equal. So that means that you'd need to be compensated for buying a forward... Apr 6, 2015 at 20:21

In addition to comment given by @dismalscience, here you may find partial answer (hope I got everything right below). Since many similar terms refer to concepts that are close to each other, I'm also regularly fighting to get these somehow in order. For example, there are forward and futures contracts that use similar terms and are related to the yield curve concepts.

On liquidity: lenders like to prefer short periods, because they are more easy to convert to cash, while borrowers prefer long periods, since roll-over terms involve risk. So lender wants compensation for which the borrower is ready to pay. Longer you go, the premium is likely to increase but at decreasing rate. This is different to the liquidity concept when the market for a bond is very thin.

It is healthy to consider also other aspects with respect to risk-free bonds. E.g. how good vs. bad times affect premiums. And why term premium? There is some evidence of negative correlation between short bond return and economic growth, and especially this correlation has larger magnitude than the correlation the long term bonds have with economic growth. This in turn, would mean that short term bonds work out as better hedges against economic downturn leading to larger premium requirement for the longer bonds.

It is possible, that we have positive liquidity premium at the same time when other premiums are not, even so that the yield curve inverts, because the spot rate is geometric mean of forward rates. In this situation, the forward rate curve would be below the spot yield curve. (This is not shown in the equations below.)

## Contracts

Forward rate (i.e. forward price) is the price agreed on the contract initiation date. Pricing means a method to find out the forward rate "today" for the contract.

Let us denote the value of the underlying asset $S_T$. Then, the value of contract is $S_T - F_{0,T}$ at the "end" (time $T$) and we could write it as $V_{0,T}(T) = S_T - F_{0,T}$. To get the value today, we discount with risk-free curve, and is common to set the forward price $F_{0,T}$ such that neither party does not have to pay to the other: $V_{0,T}(0) = S_0 - F_{0,T}/(1+r)^T = 0$, where $r$ is the risk-free interest rate. So, $F_{0,T} = S_0(1+r)^T$.

Futures rate (i.e. futures price) is priced similarly, $f_{0,T} = S_0(1+r)^T$, and it is said that "the futures price is the spot price compounded at the risk-free rate". For the value, the marking-to-market makes the value zero in each day, but just before that, the value is $f_{0,T}(t) - f_{0,T}(t-1)$. If there is a carrying cost $c$, the price is $f_{0,T} = S_0(1+r)^T +c_T$

Future rate and expected spot rate: consider carrying costs $c$ and risk $k$, for which you don't want to pay, and that we don't know, what the spot price will be at $T$, thus using expectation: $$S_0 = \frac{E(S_T) - c_T - k}{(1+r)^T}.$$

Put the future price into that and you will obtain $f_{0,T} = E(S_T) - k$, that is, the futures price equals expected spot price less risk premium.

## Yield curve

Forward rate $f$ is an interest rate today for a loan to be made in future.

The spot rate $r_T$ is the geometric mean of the forward rates, that is, $$(1+r_T)^T = (1+r_1) \Pi_{i=1}^{T-1} (1+f_{i,1}^i)$$ or $$r_T = ((1+r_1) \Pi_{i=1}^{T-1} (1+f_{i,1}^i))^{1/T} - 1.$$

Do forward rates imply future spot rate movements? Not necessarely, but find out more about the trade called "rolling down the yield curve". However, the forward rates do provide an estimate of the expected spot rate that is biased by liquidity premium (premium is not shown in the above equation).