Problem

Based on economists' forecasts and analysis, 1-year Treasury bill rates and liquidity premiums for the next four years are expected to be as follows:
\[
\begin{array}{cll}
R_{1}=0.908 & \\
E\left(2^{\left.x_{1}\right)}=2.058\right. & L_{2}=0.098 \\
E\left(3^{\left.x_{1}\right)}=2.158\right. & L_{3}=0.128 \\
E\left({ }_{4}{ }_{1}\right)=2.458 & L_{4}=0.148
\end{array}
\]

Using the liquidity premium theory, determine the current (long-term) rates. (Do not round intermediate calculations. Round your answers to 2 decimal places.)

Answer

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Answer

Final Answer: The current long-term rates for 2, 3, and 4 years are \(\boxed{1.53\%}\), \(\boxed{1.78\%}\), and \(\boxed{1.99\%}\) respectively.

Steps

Step 1 :Given the following values:

Step 2 :\(R_{1} = 0.908\)

Step 3 :\(E(R_{2}) = 2.058\)

Step 4 :\(L_{2} = 0.098\)

Step 5 :\(E(R_{3}) = 2.158\)

Step 6 :\(L_{3} = 0.128\)

Step 7 :\(E(R_{4}) = 2.458\)

Step 8 :\(L_{4} = 0.148\)

Step 9 :We can calculate the long-term rates for 2, 3, and 4 years using the liquidity premium theory formula:

Step 10 :\(R_{n} = \frac{1}{n} \left( R_{1} + E(R_{2}) + L_{2} + E(R_{3}) + L_{3} + ... + E(R_{n}) + L_{n} \right)\)

Step 11 :Substituting the given values into the formula, we get:

Step 12 :\(R_{2} = \frac{1}{2} \left( 0.908 + 2.058 + 0.098 \right) = 1.53\)

Step 13 :\(R_{3} = \frac{1}{3} \left( 0.908 + 2.058 + 0.098 + 2.158 + 0.128 \right) = 1.78\)

Step 14 :\(R_{4} = \frac{1}{4} \left( 0.908 + 2.058 + 0.098 + 2.158 + 0.128 + 2.458 + 0.148 \right) = 1.99\)

Step 15 :Final Answer: The current long-term rates for 2, 3, and 4 years are \(\boxed{1.53\%}\), \(\boxed{1.78\%}\), and \(\boxed{1.99\%}\) respectively.

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