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Term structures of interest rate in breifly explain pure expectations theory, liquid premium theory and projecting future bond prices.
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Recall that interest rates are the price of money (borrowing
or lending) and, in equilibrium, interest rates equate the amount of borrowing to the amount of saving.
The term structure of interest rates refers to different interest rates that exist over different term-to-maturity loans.
In the most basic sense, theories to explain the term structure are still based on interest rates equating the supply and demand for loanable funds.
Different rates may exist over different terms because of
expectations of changing inflation and differing preferences regarding longer-term vs. shorter-term saving.
Two main theories exist to enrich this explanation and help explain different rates over different maturity terms.
Sample Term Structure
Term to Maturity (Years)
Spot Rate
The curve plotted through the above points is also called the “yield curve”
n
n
n
n-
n-
where fn is the one period forward rate for a loan repaid in period n (i.e., borrowed in period n-1 and repaid in period n)
n-t
n
n
n
n-t
n-t
1/t
Given the data presented before, determine 1 f 3
and 2 f 5
Results: 1 f 3 =10.2577945%; 2 f 5 =10.4622321%
Current spot rates are observable today and can be contracted today.
A future spot rate will be the rate for a loan obtained in the future and repaid in a later period. Unlike forward rates, future spot rates will not be fixed (or contracted) until the future time period when the loan begins (forward rates can be locked in today).
Thus we do not currently know what will happen to future spot rates of interest. However, if we understand the theories of the term structure, we can make informed predictions or expectations about future spot rates.
We denote our current expectation of the future spot rate as follows: E[n-trn] is the expected future spot rate of interest for a loan repaid in period n and borrowed in period n-t.
You are considering locking in your mortgage rate for one year or for five years. You would use the longer term if you thought interest rates would be much higher in one year. I.e., if you expect future spot rates in one year to be much higher, you will choose the longer term mortgage right now … so, yes, it matters for your personal life.
As a financial manager, you must decide whether your firm should borrow long term or short term. You would prefer to borrow short term as these rates are currently lower, however, you are concerned about what rates you will face when you refinance your loan. I.e., you are concerned about future spot rates at the time you refinance and your expectations will affect your current decision to finance long or short term… so, yes, it matters for the corporation.
Consider that given expectations for inflation over the
next year, investors require 4% for a one year loan.
Suppose investors currently expect inflation for the next year (the second year) to be higher so that they expect to require 6% for a one year loan (starting one year from now).
Then, the Pure-Expectations Hypothesis, is consistent with the current 2-year spot rate defined as follows: (1+r 2 )^2 =(1+r 1 )(1+E[ 1 r 2 ]) = (1.04)x(1.06) so r 2 =4.995238% Restated, if we observe r 1 =4% and r 2 =4.995238%, then, under the Pure-Expectations Hypothesis, we would have E[ 1 r 2 ] to be 6% (which is equal to f 2 ).
Empirical evidence seems to suggest that
investors have relatively short time horizons for bond investments. Thus, since they are risk averse, they will require a premium to invest in longer term bonds.
The Liquidity-Preference Hypothesis states
that longer term loans have a liquidity premium built into their interest rates and thus calculated forward rates will incorporate the liquidity premium and will overstate the expected future one-period spot rates.
Reconsider investors’ expectations for inflation and
future spot rates. Suppose over the next year, investors require 4% for a one year loan and expect to require 6% for a one year loan (starting one year from now).
Under the Liquidity-Preference Hypothesis, the
current 2-year spot rate will be defined as follows: (1+r 2 )^2 =(1+r 1 )(1+E[ 1 r 2 ]) + LP 2 (LP 2 = liquidity premium: assumed to be 0.25% for a 2 year loan) (1+r 2 )^2 = (1.04)x(1.06) + 0.0025 so r 2 =5.11422%
Consider a three-year bond with annual coupons (paid annually) of $100 and a face value of $1,000 paid at maturity. Spot rates are observed as follows: r 1 =9%, r 2 =10%, r 3 =11%
What is the current price of the bond?
What is its yield to maturity (as an effective annual rate)?
What is the expected price of the bond in 2 years?
under the Pure-Expectations Hypothesis
under the Liquidity-Preference Hypothesis assume f 3 overstates E[ 2 r 3 ] by 0.5%
The Term Structure of Interest Rates shows the relation between interest rates for different term-to- maturity loans.
Two theories to explain the Term Structure are the Pure-Expectations Hypothesis and the Liquidity- Preference Hypothesis.
Empirical evidence is most consistent with the Liquidity-Preference Hypothesis.
Knowledge of the Term Structure and the theories is useful for predicting future interest rates and future bond prices.
This is useful for individuals and financial managers when deciding whether long- or short-term loans should be used for financing.
