Amadea wants to retire in
1
7
years when she turns
6
7

.
Amadea wants to have enough money to replace
8
5
%
of her current income excluding what she expects to receive from Social Security

Question

Amadea wants to retire in 
1
7
 ﻿years when she turns 
6
7
 
.
 ﻿Amadea wants to have enough money to replace 
8
5
%
 ﻿of her current income excluding what she expects to receive from Social Security at the beginning of each year. She expects to receive $
2
2
,
0
0
0
 ﻿per year from Social Security in today's dollars. Amadea has high risk tolerance and wants to assume an 
9
%
 ﻿annual investment rate of return and that inflation will be 
3
%
 ﻿per year. Based on her family history, she expects that she will live to be 
9
5
 ﻿years old. If Amadea currently earns $
1
1
0
,
0
0
0
 ﻿per year and she expects her raises to equal the inflation rate, how much does she need at retirement to fulfill her retirement goals?
◻
 ﻿$
1
,
7
0
7
,
0
9
0
.
$
9
1
0
,
2
5
0
$
1
,
6
1
3
,
1
2
5
$
2
,
0
1
2
,
4
5
8

VIPSolutions · Accepted Answer

Let's break down the problem step by step:

### **Step 1: Years to Retirement**
- **Current age:** 50
- **Retirement age:** 67
- **Years to retirement:** $ 67 - 50 = 17 $ years

### **Step 2: Years in Retirement**
- **Expected lifespan:** 95
- **Retirement duration:** $ 95 - 67 = 28 $ years

### **Step 3: Income Replacement Goal**
- **Current income:** $110,000 per year
- **Replacement rate:** 85%
- **Income needed:** $ 110,000 \times 0.85 = \$93,500 $ per year

### **Step 4: Subtract Social Security**
- **Social Security per year:** $22,000 (in today's dollars)
- **Amount needed from portfolio:** $ 93,500 - 22,000 = \$71,500 $ per year

### **Step 5: Adjust for Inflation**
- **Inflation rate:** 3%
- **Years until retirement:** 17
- **Amount needed in first year of retirement (future dollars):**

$$
\text{Future value} = \text{Present value} \times (1 + \text{inflation})^{n}
$$
$$
= 71,500 \times (1.03)^{17}
$$

First, calculate $(1.03)^{17}$:

$$
(1.03)^{17} \approx 1.6895
$$

So,

$$
\text{Future value} = 71,500 \times 1.6895 \approx \$120,857
$$

### **Step 6: Calculate Required Nest Egg at Retirement**

This is a **present value of an annuity due** (withdrawals at the beginning of year), with:
- **First payment:** $120,857 (in future dollars)
- **Number of years:** 28
- **Portfolio return:** 9% per year
- **Inflation:** 3% per year
- **Real return:** $ (1 + 0.09) / (1 + 0.03) - 1 \approx 0.0583 $ or **5.83% real**

#### **Present Value of an Annuity Due Formula:**

$$
PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \times (1+r)
$$

- $ PMT = \$120,857 $
- $ r = 0.0583 $
- $ n = 28 $

Compute the annuity factor:

$$
\frac{1 - (1 + 0.0583)^{-28}}{0.0583}
$$

Calculate $(1 + 0.0583)^{-28}$:

$$
(1.0583)^{28} \approx e^{28 \times \ln(1.0583)} \approx e^{28 \times 0.0567} \approx e^{1.5876} \approx 4.893
$$
$$
(1.0583)^{-28} \approx 1 / 4.893 \approx 0.2043
$$

So,

$$
\frac{1 - 0.2043}{0.0583} = \frac{0.7957}{0.0583} \approx 13.64
$$

Multiply by $ (1 + r) $:

$$
13.64 \times 1.0583 \approx 14.44
$$

Now, multiply by the payment:

$$
PV = \$120,857 \times 14.44 \approx \$1,744,178
$$

### **Step 7: Closest Answer**
- The closest answer to $1,744,178 is **$1,707,090**.

---

## **Final Answer:**
$$
\boxed{\$1,707,090}
$$

Amadea needs **$1,707,090** at retirement to fulfill her retirement goals.

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