<p>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.</p>

<h2>Yield curve premiums</h2>

<p>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.</p>

<p>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.</p>

<p>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.)</p>

<h2>Contracts</h2>

<p>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.</p>

<p>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, $F_{0,T} = S_0(1+r)^T$. </p>

<p>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  $f_{0,T} = S_0(1+r)^T +c_T$</p>

<p>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:
$$ S_0 = \frac{E(S_T) - c_T - k}{(1+r)^T}. $$</p>

<p>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.</p>

<h2>Yield curve</h2>

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

<p>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
$$ r_T = ((1+r_1) \Pi_{i=1}^{T-1} (1+f_{i,1}^i))^{1/T} - 1.$$</p>

<p>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).</p>

CC BY-SA 3.0
2016-08-13T11:53:12.630Z
2016-08-13T11:53:12.630Z
Gspia
<p>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 <strong>the forward rate is greater than the future spot rate</strong>.  However, I am confused on what exactly the forward rate and future spot rate are.  My understanding is that:</p>

<p>The <strong>forward rate (<span class="math-container">$f_2$</span>)</strong> is the hypothetical rate you would need to eliminate an arbitrage opportunity in the example below:</p>

<p><span class="math-container">$(1 + r_1)(1 + f_2)$</span> = <span class="math-container">$(1 + r_2)^2$</span></p>

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

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

<p>Thanks for your help.</p>

CC BY-SA 3.0
2015-04-06T20:21:35.827Z
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...
dismalscience
<interest-rate><liquidity>
Relationship between forward rate, future spot rate, and liquidity preference
sir_thursday
# Relationship between forward rate, future spot rate, and liquidity preference

(Asked by: username_0 on 2015-04-06)

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)?

Thanks for your help.

**Tags:** interest-rate, liquidity
> **Comments:**
> **username_1**: 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...  

---

## 1 answers

### Answer by username_2 on 2016-08-13. Score: 2

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.

## Yield curve premiums

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).


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