Let's break down the problem step by step. **Given:** - Principal: \( P = 20,000 \) - Loan term: \( n = 20 \) years - **Method a:** Amortization, **effective rate:** \( i_a = 6.5\% \) - **Method b:** Sinking fund, lender receives **effective rate:** \( i_l = 8\% \), sinking fund earns **rate:** \( j \) - Both methods require **equal annual payment:** \( X \) We are to **find \( j \)** so that the annual payment \( X \) is the same under both methods. --- ### 1. **Amortization Method** Annual payment for a fully amortized loan: \[ X = P \cdot \frac{i_a}{1 - (1+i_a)^{-n}} \] Plug in values: \[ X = 20,000 \cdot \frac{0.065}{1 - (1.065)^{-20}} \] First, calculate \( (1.065)^{-20} \): \[ (1.065)^{20} \approx 3.5075 \implies (1.065)^{-20} \approx 0.2852 \] So, \[ X = 20,000 \cdot \frac{0.065}{1 - 0.2852} = 20,000 \cdot \frac{0.065}{0.7148} = 20,000 \cdot 0.09093 = \$1,818.60 \] --- ### 2. **Sinking Fund Method** For each year: - **Interest payment to lender:** \( P \cdot i_l = 20,000 \times 0.08 = \$1,600 \) - **Sinking fund deposit:** \( S \) (to accumulate to $20,000 in 20 years at rate \( j \)) Sinking fund deposit: \[ S = \frac{P}{s_{\overline{n}|j}} \] Where: \[ s_{\overline{n}|j} = \frac{(1+j)^n - 1}{j} \] So, \[ S = 20,000 \cdot \frac{j}{(1+j)^{20} - 1} \] **Total annual payment:** \( X = \text{Interest} + \text{Sinking Fund Deposit} \) \[ X = 1,600 + 20,000 \cdot \frac{j}{(1+j)^{20} - 1} \] --- ### 3. **Equate the Payments** Set the two expressions for \( X \) equal: \[ 1,818.60 = 1,600 + 20,000 \cdot \frac{j}{(1+j)^{20} - 1} \] \[ 1,818.60 - 1,600 = 20,000 \cdot \frac{j}{(1+j)^{20} - 1} \] \[ 218.60 = 20,000 \cdot \frac{j}{(1+j)^{20} - 1} \] \[ \frac{218.60}{20,000} = \frac{j}{(1+j)^{20} - 1} \] \[ 0.01093 = \frac{j}{(1+j)^{20} - 1} \] \[ 0.01093 \times ((1+j)^{20} - 1) = j \] \[ 0.01093 (1+j)^{20} - 0.01093 = j \] \[ 0.01093 (1+j)^{20} - j = 0.01093 \] \[ 0.01093 (1+j)^{20} - j = 0.01093 \] --- ### 4. **Solve for \( j \):** Let \( x = 1+j \): \[ 0.01093 x^{20} - (x-1) = 0.01093 \] \[ 0.01093 x^{20} - x + 1 = 0.01093 \] \[ 0.01093 x^{20} - x + 1 - 0.01093 = 0 \] \[ 0.01093 x^{20} - x + 0.98907 = 0 \] Now solve for \( x \) (and thus \( j \)). This is a transcendental equation and must be solved numerically. --- #### **Numerical solution:** Start with an estimate. The amortization rate is 6.5%, but in the sinking fund method, the fund must earn less than this since the lender is being paid 8%. Try \( j \approx 4\% \) to \( 6\% \). Let’s try \( j = 5\% \): \[ x = 1.05 \] \[ x^{20} = 1.05^{20} \approx 2.6533 \] \[ 0.01093 \times 2.6533 = 0.02901 \] \[ 0.02901 - 1.05 + 0.98907 = 0.02901 - 0.06093 = -0.03192 \] That’s negative. Try \( j = 6\% \): \[ x = 1.06, \quad x^{20} = 3.2071 \] \[ 0.01093 \times 3.2071 = 0.03504 \] \[ 0.03504 - 1.06 + 0.98907 = 0.03504 - 0.07093 = -0.03589 \] Still negative, but slightly more negative. Try \( j = 4\% \): \[ x = 1.04, \quad x^{20} = 2.1911 \] \[ 0.01093 \times 2.1911 = 0.02394 \] \[ 0.02394 - 1.04 + 0.98907 = 0.02394 - 0.05093 = -0.02699 \] Still negative, but less so. Let's try \( j = 7\% \): \[ x = 1.07, \quad x^{20} = 3.8697 \] \[ 0.01093 \times 3.8697 = 0.04232 \] \[ 0.04232 - 1.07 + 0.98907 = 0.04232 - 0.08093 = -0.03861 \] All values are negative. Let's check our setup. Alternatively, let's use the version from earlier: \[ 0.01093 (1+j)^{20} - j = 0.01093 \] \[ 0.01093 (1+j)^{20} = 0.01093 + j \] \[ (1+j)^{20} = 1 + \frac{j}{0.01093} \] Try \( j = 0.05 \): \[ 1 + \frac{0.05}{0.01093} = 1 + 4.574 = 5.574 \] \[ (1.05)^{20} = 2.6533 \] 2.65 is much less than 5.57, so \( j \) needs to be higher. Try \( j = 0.15 \): \[ 1 + \frac{0.15}{0.01093} = 1 + 13.73 = 14.73 \] \[ (1.15)^{20} = 16.37 \] Now, 16.37 is more than 14.73, so the correct value is between 5% and 15%. Try \( j = 0.12 \): \[ 1 + \frac{0.12}{0.01093} = 1 + 10.98 = 11.98 \] \[ (1.12)^{20} = 9.646 \] Still less. Try \( j = 0.135 \): \[ 1 + \frac{0.135}{0.01093} = 1 + 12.35 = 13.35 \] \[ (1.135)^{20} = 13.36 \] Now, \( (1.135)^{20} \approx 13.36 \) and the right hand side is 13.35. This is **very close**! --- ### **Final Answer** \[ \boxed{j \approx 13.5\%} \] **So, the sinking fund must earn an annual effective rate of approximately 13.5% for the payments to be equal under both methods.**