06-Liquidity Preference Theory
Expectations Theory Review

Given that
YTM 1t  10%

YTM 1et 1  12%
Expectations Theory:
(1.10)(1.12)  (1  YTM 2t ) 2
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–
Given that we want to invest for two years, we
should be indifferent between either strategy.
On average, either strategy gives the same return.
Expectations Theory Review

The yield curve is usually upward sloping.

According to Expectations Theory: The market
usually expects interest rates to increase.

But interest rates are stationary: they decrease
from one period to the next about as often as
they increase.
Should You Be Indifferent?

Both bonds are default-free

Does one strategy expose you other kinds of risk?

If so, then the return from this strategy should be
higher to entice investors to buy these bonds.
–
The price of this strategy should be lower.
Should you Be Indifferent? View#1

You’re locked in to get the return with the two
year bond

There is uncertainty regarding the actual return
you’ll get by buying the one year bond and
rolling it over.

Perhaps buying the one-year bond is perceived
as more risky than just locking in and buying
the two-year bond.
Should you Be Indifferent? View#2

Suppose that in 1 year, there is a chance you
may need to liquidate and get cash to pay off
some financial obligation.

In the example, initially, YTM 1et 1  12%.
Should you Be Indifferent? View#2

Suppose at time t+1, 1-year rates jump to 14%
YTM 1t 1  14%

What do you get back from each strategy?
–
Strategy of rolling over short-term bonds:


Face value back from the bond (1000)
Return = 1000/909.09 -1 = 10%
Should you Be Indifferent? View#2

What is price of 2-year bond?
–
–
–
–
It only has 1-year left until it matures
1-yr rates are 14%
Price = 1000/1.14 = 877.19
Return = 877.19/826.45 -1 = 6.14%
Should you Be Indifferent? View#2

With the two year bond, you are exposed to
greater risk if you need to cash out of the
investment strategy before the end of the 2nd
year, that is, if you need liquidity.

Perhaps buying the two-year bond is
perceived as more risky than buying the oneyear bond and rolling over the proceeds.
Liquidity Preference Theory

Liquidity Preference Theory
–
–
View #2 dominates View #1
Long term default-free bonds are considered to be
more risky that short-term bonds, since in the short
term, if liquidity is needed, the return from long term
bonds is more uncertain.
(1  YTM 1t )(1  YTM 1e,t 1 )  (1  YTM 2t ) 2
Liquidity Preference Theory



What about forward rates?
Forward rates by definition always satisfy
(1  YTM 1t )(1  f 2 )  (1  YTM 2t ) 2
Hence, if
(1  YTM 1t )(1  YTM 1e,t 1 )  (1  YTM 2t ) 2
then
YTM 1e,t 1  f 2
Liquidity Preference Theory

According to the LPT:
f 2  YTM
e
1,t 1
 2
where
2  two - year liquidity premium
Example

Suppose at time t, the market expects:
YTM 1t 1  12% with 50% probability
YTM 1t 1  8% with 50% probability

It follows that on average, the market expects
YTM
e
1t 1
 10%
Example

Suppose as of time t:
–
–
–
YTM on a 1-year zero is 10% (YTM 1t  10%)
YTM on a 2-year zero is 12% (YTM 2t  12%)
What are f 2 and 2 ?
(1.10)(1  f 2 )  (1.12) 2
f 2  14.36%  YTM 1et 1  2
2  14.36%  10%  4.36%
Longer Term Bonds

For a three-year bond, it is always true that
(1  YTM 1t )(1  f 2 )(1  f 3 )  (1  YTM 3t ) 3
which to an approximation implies
YTM 1t  f 2  f 3
 YTM 3t
3
by the definition of forward rates.
Longer Term Bonds

Liquidity Preference Theory says that
f 2  YTM 1et 1 ,
f 3  YTM 1et  2
forward rates are greater than expected future
short-term rates since forward rates include the
liquidity premium.
Longer Term Bonds

This implies that
YTM 1t  f 2  f 3
YTM 1t  YTM 1et 1  YTM 1et  2
 YTM 3t 
3
3

The liquidity preference theory says that the
n-period spot rate is greater than the average
of the one period rates expected to occur
over the n-period life of the bond.
Example

Expected one-period spot rates
YTM 1t  4%, YTM 1et 1  4%, YTM 1et  2  4%, YTM 1et 3  4%
2  0.5%, 3  1%, 4  1.3%

Then
f 2  4%  .5%  4.5%
f 3  4%  1%  5%
f 4  4%  1.3%  5.3%
YTM 2t  (4%  4.5%) / 2  4.25%
YTM 3t  (4%  4.5%  5%) / 3  4.5%
YTM 4t  (4%  4.5%  5%  5.3%) / 4  4.7%
Example
YTM 2t  4.25%
YTM 3t  4.5%
YTM 4t  4.7%


A flat trend in expected short-term rates
produces an upward sloping yield curve,
because of the liquidity premium.
In general, n increases with n.
Example

You work for the bond trading desk of a
large investment bank.
YTM 1t  9%
YTM 2t  10.5%
State 1: YTM 1t 1  17% (50% probability)
State 2 : YTM 1t 1  3%
(50% probability)
What is YTM 1et 1 ?
What is 2 - yr forward rate?
What is 2t ?
Example
YTM

 0.50(0.03)  0.50(0.17)  10%
2-year forward rate:
–
–

e
1t 1
(1.09)(1+f2)=(1.105)2
f2= 12.02%
2-year risk premium: 2  12.02%  10%  2.02%
Example

A client, who is concerned about interest rate risk, has
asked for your help in constructing a forward loan. She
wants to enter into a contract to
–
–


borrow $50 thousand from your firm a year from now
to be repaid one year after.
What is the lowest interest rate you could charge the
client and make a profit on the transaction for your
services?
Show how you would structure your holdings of zerocoupon bonds so that your firm can exactly match the
cash flows required by the loan.
Example
YTM1t  9%


Assume face value of bonds = 1000
Buy 50 1-yr zero bonds.
–


YTM 2t  10.5%
Price: 50,000/1.09 = 45,872
Fund purchase by borrowing 45,872 at
10.5%
In one year,
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bonds pay you 50,000
Give cash to client
Example

In two years,
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liability has grown to
45,872(1.105)2 = 56,011
Client owes you 50,000+ interest
As long as interest > 6,011, you have made a
profit
6,011/50,000 = 12.02%
Example




But 12.02% is the 2-year forward rate
Not a coincidence.
You can always lock in future loans at the
forward rate.
As long as your client is willing to pay more
than 12.02%, you have made a profit.
Example

Why would your client be willing to lock in at
any rate above 12.02%?
–
–
–
–
The client could lock in her own rate of 12.02%
May not be able to do so as efficiently as the
bank.
The bank makes a business of buying and selling
bonds. The client does not.
The client is paying a fee for the services of the
bank.
Example

e
YTM
But if,
1t 1  10% the client expects to pay
a higher rate on average. Why is she willing
to do this?

Because she is hedged against the state that
rates jump to 17%.
Yield Curve and Recessions
Yield Curve and Recessions

Why does a flat yield curve predict recessions?

Assuming risk-premia are constant, a flatter yield curve
indicates the market expects short-term rates to be
lower in the future than they are now.

Why do forecasts of low short-term rates also indicate
recessions?
Yield Curve and Recessions

Why do forecasts of low short-term rates also
indicate recessions?
1)
during recessions, supply curve shifts left – firms
don’t need as much debt
2)
during recessions, inflationary pressures are lower
1)
2)
Demand curve shifts right
Supply curve shifts left