According to the Liquidity Premium Theory, a long-term rate of interest is an average of short-term rates plus a liquidity premium. In other words, investors expect to be compensated for holding long-term bonds instead of short-term bonds as long-term bonds are perceived to be riskier. This causes the yield curve to be steeper than it would be under the Expectations Theory. This is again a function of the state of the economy and is actually an improvement that is built on the Expectations Theory.

As an example, suppose that the one-year rates over the next five years are expected to be 5%, 6%, 7%, 8% and 9%, respectively. The liquidity premium for holding bonds of these maturities equals 0% for a one-year bond, 0.25% for a two-year bond, 0.5% for a three-year bond, 0.75% for a four-year bond and 1.00% for a five-year bond. What are the implied two-year and five-year rates under the:

§ Expectations Theory

§ Liquidity Premium Theory

Under the Expectations Theory, the implied two-year rate is computed as:

(1 + X)^{2 }= (1 + 0.05)(1 + 0.06)

(1 + X)^{2} = 1.113

1 + X = 1.113^{1/2}

X = 1.113^{1/2} – 1

X = 0.055 = 5.5%

Under the Liquidity Premium Theory, the implied two-year rate is computed as:

(1 + X)^{2 }= (1 + 0.05 + 0)(1 + 0.06 + 0.0025)

(1 + X)^{2} = (1.05)(1.0625)

(1 + X)^{2} = (1.115625)

1 + X = 1.115625^{1/2}

X = 1.115625^{1/2} – 1

X = 0.05623 = 5.623%

Under the Expectations Theory, the implied five-year rate is computed as:

The basic rule that you need to understand here is that the as the time frame widens, the gap between the yield calculated by Expectations theory and the Liquidity premium theory also differs. That is because as the term to maturity goes up, investors tend to demand greater liquidity premium to park their money.

According to the Liquidity Premium Theory, a long-term rate of interest is an average of short-term rates plus a liquidity premium. In other words, investors expect to be compensated for holding long-term bonds instead of short-term bonds as long-term bonds are perceived to be riskier. This causes the yield curve to be steeper than it would be under the Expectations Theory. This is again a function of the state of the economy and is actually an improvement that is built on the Expectations Theory.

As an example, suppose that the one-year rates over the next five years are expected to be 5%, 6%, 7%, 8% and 9%, respectively. The liquidity premium for holding bonds of these maturities equals 0% for a one-year bond, 0.25% for a two-year bond, 0.5% for a three-year bond, 0.75% for a four-year bond and 1.00% for a five-year bond. What are the implied two-year and five-year rates under the:

§ Expectations Theory

§ Liquidity Premium Theory

Under the Expectations Theory, the implied two-year rate is computed as:

(1 + X)

^{2 }= (1 + 0.05)(1 + 0.06)(1 + X)

^{2}= 1.1131 + X = 1.113

^{1/2}X = 1.113

^{1/2}– 1X = 0.055 = 5.5%

Under the Liquidity Premium Theory, the implied two-year rate is computed as:

(1 + X)

^{2 }= (1 + 0.05 + 0)(1 + 0.06 + 0.0025)(1 + X)

^{2}= (1.05)(1.0625)(1 + X)

^{2}= (1.115625)1 + X = 1.115625

^{1/2}X = 1.115625

^{1/2}– 1X = 0.05623 = 5.623%

Under the Expectations Theory, the implied five-year rate is computed as:

(1 + X)

^{5 }= (1 + 0.05)(1+ 0.06)(1 + 0.07)(1 + 0.08)(1 + 0.09)(1 + X)

^{5}= 1.4019392521 + X = 1.401939252

^{1/5}X = 1.401939252

^{1/5}– 1X = 0.0699 = 6.99%

Under the Liquidity Premium Theory, the implied five-year rate is computed as:

(1 + X)

^{5 }= (1 + 0.05)(1+ 0.0625)(1 + 0.0750)(1 + 0.0875)(1 + 0.10)(1 + X)

^{5}= 1.434658891 + X = 1.43465889

^{1/5}X = 1.43465889

^{1/5}– 1X = 0.0749 = 7.49%

## The basic rule that you need to understand here is that the as the time frame widens, the gap between the yield calculated by Expectations theory and the Liquidity premium theory also differs. That is because as the term to maturity goes up, investors tend to demand greater liquidity premium to park their money.