4. Pat plans to retire at 57 , and will receive a cash payment of $$ \$ 50000 $$ from a profit sharing plan. The total sum received will be used to set up an annuity with an insurance company at $$ 8 \% / $$ a compounded semi-annually until age 65 . How large is each payment, if Pat is to receive two equal payments per year? ( 4 marks)

Question

UpStudy AI · Accepted Answer

To determine the size of each payment Pat will receive from the annuity, we can use the present value of an annuity formula. Here's a step-by-step breakdown:

### **Given:**
- **Present Value (PV):** \$50,000
- **Annual Interest Rate:** 8% compounded semi-annually
- **Time Period:** 8 years (from age 57 to 65)
- **Number of Payments per Year:** 2

### **Steps:**

1. **Determine the Effective Interest Rate per Period:**
   - Since the interest is compounded semi-annually, the rate per period $$ r $$ is:
     $$
     r = \frac{8\%}{2} = 4\% \text{ or } 0.04 \text{ per half-year}
     $$

2. **Calculate the Total Number of Periods ($$ n $$):**
   - Over 8 years with 2 payments per year:
     $$
     n = 8 \times 2 = 16 \text{ periods}
     $$

3. **Use the Present Value of an Annuity Formula:**
   $$
   PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
   $$
   - Plugging in the known values:
     $$
     50,000 = PMT \times \left( \frac{1 - (1.04)^{-16}}{0.04} \right)
     $$

4. **Calculate the Annuity Factor:**
   - First, compute $$ (1.04)^{-16} \approx 0.5366 $$
   - Then:
     $$
     \frac{1 - 0.5366}{0.04} = \frac{0.4634}{0.04} \approx 11.585
     $$

5. **Solve for $$ PMT $$:**
   $$
   PMT = \frac{50,000}{11.585} \approx 4,312
   $$

### **Conclusion:**
Each semi-annual payment Pat will receive is approximately **\$4,312**.
Each semi-annual payment Pat will receive is approximately \$4,312.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Calculus Questions