# Term Structure of Interest Rates

Term Structure of Interest Rates. The term structure of interest rate refers to the relationship between time to maturity and yields for a particular **category of bonds** at a particular point in time. Particular theories are developed to explain the nature of bond yields over time.

### The following theories are vital in this regard.

*Expectations Theory*

*Expectations Theory*

**Expectations theory of term structure of interest rates** states that market participants and the market force as well will determine the return from holding a security where the return from holding an n-period bond equals the average return expected from holding a series of one-year bonds over the same n-periods. The long-term rate of interest would be equal to an average of the present yield on short-term bond plus the expected future yields on short-term bonds which are expected to prevail over the long-term period. For each period, the total rate of return is expected to be the same on all securities regardless of the time to maturity. The term structure of interest rate consists of a set of forward rates and spot rate. Spot rate is the rate which is applicable today and the forward rates are expected to prevail in the future.

The rate of return an investor requires to invest is a function of three factors as:

- risk-free real rate of return,
- inflation and iii. risk premium.

More specifically,

E(r) = risk-free rate + inflation + risk premium |

The rate of return for the n th year bond can be estimated by using the following formula:

(1 + _{t}r_{n}) = [(1 + _{t }r _{1} ) (1 + _{t + 1 }r _{1}) – – – – – (1 + _{t + n-1 }r _{1})] ^{1/n }-1

where,

(1 + _{t}r_{n}) = the rate of return on a n-year bond,

(1 + _{t }r _{1} ) = current rate on a one-year bond,

(1 + _{t + 1 }r _{1}) = the expected rate on a bond with one-year to maturity beginning one-year from now,

(1 + _{t + n-1 }r _{1}) = the expected rate on a bond with one-year to maturity beginning n -1-year from now,

n = bond maturity period.

**The formula is applicable for any number of periods. Any long-term rate is geometric average of consecutive one-period rate.**

### Let us consider an example.

Suppose one-year bond rate is 10 per cent and two forward rates are 11 per cent and 12 per cent. Find out the three-year bond.

This problem can be solved by using geometric average of the current one-year rate and the expected forward rates for the subsequent two years as follows:

(1 + _{t}r_{3}) = [(1 + _{t }r_{1} ) (1 + _{t + 1 }r _{1}) (1 + _{t + 2 }r _{1})] ^{1/3 }-1.0

= [(1.10)(1.11)(1.12)]^{1/3 }– 1.0

= [1.3675] ^{1/3} – 1.0 = 1.11- 1.0 = 0.11 = 11%

*Liquidity Preference Theory*

*Liquidity Preference Theory*

**Liquidity preference theory** assets that as in the expectations theory, interest rates reflect the sum of current and expected short rates plus liquidity premium. Because of the uncertainty in future, investors prefer to invest in short-term bonds. On the other hand, borrowers prefer long-term to invest in capital assets. Under these circumstances, investors are supposed to receive a liquidity premium to invest in long-term bonds. Therefore, this theory implies that long-term bonds should offer higher yields.

*Preferred Ha**bital Theory*

*Preferred Ha*

*bital Theory*

The difference between the expectations theory and liquidity preference theory is the recognition that future interest rate expectations are uncertain. To compensate this uncertainty, risk-averse investors would like a higher rate for long-term bonds. Being third theory of term structure of interest rates, preferred habital theory assets that investors usually prefer maturity sectors or habits. A financial institution with many five-year maturity deposits to pay off will not wish to take the reinvestment rate risk that would result from investing in one-year **Treasury bills**. This theory means that the borrowers and lenders can be induced to shift maturities if they are adequately compensated by an appropriate risk premium.

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