Dear Financial Adviser,

My spouse and ae each 62 and hope to ref In tea years. After retment we wil receive $9100 per month after taxes fom our
‘employers’ pension plans and $3100 per month after

Question

Dear Financial Adviser,

My spouse and ae each 62 and hope to ref In tea years. After retment we wil receive $9100 per month after taxes fom our
‘employers’ pension plans and $3100 per month after taxes from Social Securty, Unfortunately our monthiy ving expenses are
$1650, Our social cbigatons peciud further economics

We have $820,000 investcd in a high-orac, tox ree municpalbond mutual fund. The return on the fund is 50% per year. We plano
make annual withdrawals fom the mutual fund o cover the ference between our pension and Social Securly income and our ing
expenses.

Sincerely,

Lusury Challenged

Morbiohooc, MA

ou can assim thatthe withdrawal (one per year wilt I a checking account (o Interest) unti spent. The couple wi

account cover the moniny shor 2 § chai ple Latte
How many yeors before Luxury Challenged rns out of money?

Note: Do not ound intermediate calculations. Round your answer to 2 decimal places.

VIPSolutions · Accepted Answer

Let’s break down the problem:

### **Step 1: Calculate the annual income and expenses**
- **Monthly income (pension + Social Security):** $9,100 + $3,100 = $12,200  
- **Annual income:** $12,200 × 12 = $146,400  
- **Annual expenses:** $16,600 × 12 = $199,200  
- **Annual shortfall:** $199,200 (expenses) - $146,400 (income) = **$52,800**

### **Step 2: Initial investment and interest**
- **Initial mutual fund investment:** $820,000  
- **Annual interest rate:** 5%

### **Step 3: Find the number of years until the fund is depleted**
This is a standard annuity withdrawal problem:

The formula for the number of years ($ n $) until the fund runs out is:
$$
n = \frac{\ln\left(\frac{W}{W - rP}\right)}{\ln(1+r)}
$$
Where:
- $ W $ = initial investment = $820,000
- $ P $ = annual withdrawal = $52,800
- $ r $ = annual interest rate = 0.05

Let's compute step by step:

#### Calculate $ rP $:
$ rP = 0.05 \times 820,000 = 41,000 $

#### Calculate $ W - rP $:
$ W - rP = 820,000 - 41,000 = 779,000 $

#### Calculate $ \frac{W}{W - rP} $:
$ \frac{820,000}{779,000} \approx 1.0526 $

#### Take the natural log of both the numerator and denominator:
$ \ln(1.0526) \approx 0.0513 $
$ \ln(1.05) \approx 0.04879 $

#### Plug into the formula:
$$
n = \frac{0.0513}{0.04879} \approx 1.0516
$$

#### Wait! This value seems off. Let's use the correct annuity depletion formula:

$$
n = \frac{\ln \left( \frac{P}{P - rW} \right)}{\ln(1+r)}
$$

Where:
- $ W $ = $820,000
- $ P $ = $52,800
- $ r $ = 0.05

#### Compute $ P - rW $:
$ P - rW = 52,800 - 0.05 \times 820,000 = 52,800 - 41,000 = 11,800 $

#### Compute $ \frac{P}{P - rW} $:
$ \frac{52,800}{11,800} \approx 4.4746 $

#### Take the natural log of numerator and denominator:
$ \ln(4.4746) \approx 1.4986 $
$ \ln(1.05) \approx 0.04879 $

#### Plug into the formula:
$$
n = \frac{1.4986}{0.04879} \approx 30.72
$$

### **Final Answer:**
**Luxury Challenged will run out of money in approximately 30.72 years.**

**(Round to two decimal places as requested.)**

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