Questions: The Wall Street Journal reports that the rate on 4-year Treasury securities is 5.55 percent and the rate on 5-year Treasury securities is 5.80 percent. The 1-year interest rate expected in four years, E(5 r1), is 6.40 percent. According to the liquidity premium hypotheses, what is the liquidity premium on the 5-year Treasury security, L5? Note: Do not round intermediate calculations. Round your percentage answer to 2 decimal places (i.e., 0.1234 should be entered as 12.34).

The Wall Street Journal reports that the rate on 4-year Treasury securities is 5.55 percent and the rate on 5-year Treasury securities is 5.80 percent. The 1-year interest rate expected in four years, E(5 r1), is 6.40 percent. According to the liquidity premium hypotheses, what is the liquidity premium on the 5-year Treasury security, L5?
Note: Do not round intermediate calculations. Round your percentage answer to 2 decimal places (i.e., 0.1234 should be entered as 12.34).

Transcript text: The Wall Street Journal reports that the rate on 4-year Treasury securities is 5.55 percent and the rate on 5-year Treasury securities is 5.80 percent. The 1 -year interest rate expected in four years, $E\left({ }_{5} r_{1}\right)$, is 6.40 percent. According to the liquidity premium hypotheses, what is the liquidity premium on the 5 -year Treasury security, $L_{5}$ ? Note: Do not round intermediate calculations. Round your percentage answer to 2 decimal places (i.e., 0.1234 should be entered as 12.34).

Solution

To determine the liquidity premium on the 5-year Treasury security ($L_5$) according to the liquidity premium hypothesis, we need to use the given interest rates and the expected 1-year interest rate in four years. The liquidity premium hypothesis suggests that the yield on a long-term bond is equal to the average of the short-term interest rates expected over the life of the long-term bond plus a liquidity premium.

Given:

4-year Treasury rate ($r_4$) = 5.55%
5-year Treasury rate ($r_5$) = 5.80%
Expected 1-year interest rate in four years ($E(5r_1)$) = 6.40%

The formula for the 5-year Treasury rate under the liquidity premium hypothesis is: \[ r_5 = \frac{r_1 + r_2 + r_3 + r_4 + E(5r_1)}{5} + L_5 \]

We need to isolate $L_5$ in this equation. First, we need to find the average of the short-term interest rates expected over the life of the 5-year bond. Since we are not given the individual short-term rates for the first four years, we will assume that the 4-year rate is the average of the first four years' rates.

Thus, we can write: \[ r_4 = \frac{r_1 + r_2 + r_3 + r_4}{4} \]

Given $r_4 = 5.55\%$, we assume: \[ r_1 = r_2 = r_3 = r_4 = 5.55\% \]

Now, we can calculate the average of the short-term rates over the 5-year period: \[ \text{Average short-term rates} = \frac{5.55\% + 5.55\% + 5.55\% + 5.55\% + 6.40\%}{5} \] \[ = \frac{4 \times 5.55\% + 6.40\%}{5} \] \[ = \frac{22.20\% + 6.40\%}{5} \] \[ = \frac{28.60\%}{5} \] \[ = 5.72\% \]

Now, we know that: \[ r_5 = 5.80\% \] \[ r_5 = \text{Average short-term rates} + L_5 \] \[ 5.80\% = 5.72\% + L_5 \]

Solving for $L_5$: \[ L_5 = 5.80\% - 5.72\% \] \[ L_5 = 0.08\% \]

Therefore, the liquidity premium on the 5-year Treasury security ($L_5$) is 0.08%.

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