Let's carefully analyze the problem and proceed step-by-step. --- ### **Problem Restatement:** - **Original annuity:** - Pays **$1,000** at the **end of each month**. - Duration: same as the new annuity (say **n** months). - **Desired new annuity:** - Has the **same term** (duration). - Pays **once per year**, **at the beginning of each year**. - Payment amount: **X** (to be determined). - **Interest rate:** - Nominal discount rate of **3% per annum**, payable quarterly. --- ### **Step 1: Understand the original annuity's cash flows** - **Monthly payments:** $1,000 at the end of each month. - **Number of payments:** Let's denote as **N** months total. > **Note:** The problem states "has the same term," so the total length of both annuities is the same. - Since the new annuity has **annual payments**, the total durations should match. ### **Step 2: Determine the present value (PV) of the original annuity** - Payments are **monthly**, at the end of each month. - Discount rate: nominal **3% compounded quarterly**. --- ### **Step 3: Convert the nominal rate to an effective monthly rate** - Nominal rate \( i^{(quarter)} = 3\% = 0.03 \) per quarter. - Quarterly discount factor: \( d_q = 1 + 0.03/4 = 1 + 0.0075 = 1.0075 \). - **Effective monthly discount rate:** Since the rate is compounded quarterly, the effective monthly rate \( i_m \) can be found as: \[ (1 + i_{\text{quarter}})^{1/3} - 1 \] but it's more precise to note: \[ (1 + i_q)^{1/3} - 1 \] where \( i_q = 0.03 \). Calculating: \[ i_m = (1 + 0.03)^{1/3} - 1 \] \[ i_m = 1.03^{1/3} - 1 \] Using logarithms or a calculator: \[ 1.03^{1/3} \approx e^{\frac{1}{3} \ln 1.03} \] \[ \ln 1.03 \approx 0.029558 \] \[ \frac{1}{3} \times 0.029558 \approx 0.0098527 \] \[ e^{0.0098527} \approx 1.0099 \] So, \[ i_m \approx 1.0099 - 1 = 0.0099 \text{ or } 0.99\% \] --- ### **Step 4: Compute the present value of the original monthly payments** - Payments: \( PMT = \$1,000 \) at the end of each month, - Number of months: \( N \), - Monthly discount rate: \( i_m \approx 0.0099 \). The PV of an **ordinary annuity** of \( N \) payments: \[ PV_{\text{original}} = 1000 \times \frac{1 - (1 + i_m)^{-N}}{i_m} \] --- ### **Step 5: Determine the total term \( N \)** - The **term** is the same for both annuities. - The new annuity pays once per year at the **beginning** of each year. - Since the original pays monthly, the total months \( N \). Suppose the total term is \( T \) years, then: \[ N = 12 \times T \] The problem doesn't specify \( T \), but it's implied the total duration is the same, so **we'll keep it in terms of \( N \)**. --- ### **Step 6: Convert the PV of the original annuity to an equivalent annual amount** - The **present value** of the original annuity is: \[ PV_{\text{original}} = 1000 \times \frac{1 - (1 + i_m)^{-N}}{i_m} \] - The new annuity is **annual**, pays at **beginning of each year** (annuity-due with annual payments). - To find the equivalent annual payment \( X \), we set: \[ PV_{\text{new}} = PV_{\text{original}} \] - The PV of the **annual payments at the beginning of each year** (annuity-due): \[ PV_{\text{annuity due}} = X \times \frac{1 - (1 + i_{annual})^{-T}}{i_{annual}} \times (1 + i_{annual}) \] where: - \( T \) is the number of years, - \( i_{annual} \) is the effective annual discount rate. --- ### **Step 7: Find the effective annual discount rate** - Given the nominal rate of 3% compounded quarterly: \[ i_{quarter} = 0.0075 \] - Effective annual rate: \[ i_{annual} = (1 + i_q)^4 - 1 \] \[ i_{annual} = (1.0075)^4 - 1 \] Calculate: \[ (1.0075)^4 = e^{4 \times \ln 1.0075} \] \[ \ln 1.0075 \approx 0.007472 \] \[ 4 \times 0.007472 = 0.02989 \] \[ e^{0.02989} \approx 1.0303 \] Thus, \[ i_{annual} \approx 1.0303 - 1 = 0.0303 \text{ or } 3.03\% \] --- ### **Step 8: Final calculation of the annual payment \( X \)** The PV of the original monthly payments: \[ PV_{\text{original}} = 1000 \times \frac{1 - (1 + 0.0099)^{-N}}{0.0099} \] The PV of the new annual payments: \[ PV_{\text{new}} = X \times \frac{1 - (1 + 0.0303)^{-T}}{0.0303} \times (1 + 0.0303) \] Since the total term is \( N \) months, the number of years: \[ T = \frac{N}{12} \] Therefore: \[ PV_{\text{new}} = X \times \left( \frac{1 - (1 + 0.0303)^{-\frac{N}{12}}}{0.0303} \right) \times 1.0303 \] --- ### **Step 9: Equate the present values** \[ PV_{\text{original}} = PV_{\text{new}} \] \[ 1000 \times \frac{1 - (1 + 0.0099)^{-N}}{0.0099} = X \times \left( \frac{1 - (1 + 0.0303)^{-\frac{N}{12}}}{0.0303} \right) \times 1.0303 \] Solve for \( X \): \[ X = \frac{1000 \times \frac{1 - (1 + 0.0099)^{-N}}{0.0099}}{\left( \frac{1 - (1 + 0.0303)^{-\frac{N}{12}}}{0.0303} \right) \times 1.0303} \] --- ### **Final step: Numerical example** Suppose the total term is **10 years**, so: \[ N = 12 \times 10 = 120 \text{ months} \] Calculate: \[ (1 + 0.0099)^{-120} \approx (1.0099)^{-120} \] \[ \ln 1.0099 \approx 0.00985 \] \[ -120 \times 0.00985 = -1.182 \] \[ e^{-1.182} \approx 0.306 \] Now: \[ PV_{\text{original}} = 1000 \times \frac{1 - 0.306}{0.0099} = 1000 \times \frac{0.694}{0.0099} \approx 1000 \times 70.05 \approx \$70,050 \] Calculate the denominator: \[ \frac{N}{12} = 10 \text{ years} \] \[ (1 + 0.0303)^{-10} = (1.0303)^{-10} \] \[ \ln 1.0303 \approx 0.0300 \] \[ -10 \times 0.0300 = -0.300 \] \[ e^{-0.300} \approx 0.741 \] Then: \[ 1 - 0.741 = 0.259 \] \[ \frac{0.259}{0.0303} \approx 8.55 \] Multiply by 1.0303: \[ 8.55 \times 1.0303 \approx 8.81 \] Finally, compute \( X \): \[ X = \frac{70,050}{8.81} \approx \$7,956 \] --- ### **### **Final Answer:** **The annual payment \( X \) at the beginning of each year should be approximately \$7,956.** --- ### **Summary:** - Convert the monthly discount rate to effective monthly. - Calculate the PV of the original monthly annuity. - Find the effective annual discount rate. - Equate PVs to solve for the annual payment. - For a 10-year term, the approximate annual payment is **\$7,956**. **Note:** The actual value depends on the total term \( N \). If the duration differs, adjust \( N \) accordingly.