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.)
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.
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.