The Secret To A Comfortable Retirement: Level Consumption

 Problem:

Solution:

If the inflation rate equals the interest rate, the real value of money does not change over time. This means the retiree can spend the same amount each year in real terms (adjusted for inflation) without reducing the purchasing power of her savings.

Given:

• Total savings: $450,000
• Retirement duration: 30 years
• Interest rate = Inflation rate

The problem essentially asks for the annuity payment $C$ that the retiree can withdraw each year for 30 years, such that the entire $450,000 is used up by the end of 30 years.

Since the interest rate equals the inflation rate, the retiree can withdraw an equal amount each year. This can be calculated using the formula for the present value of an annuity:

$$PV = C \times \left(\frac{1 - (1 + r)^{-t}}{r}\right)$$

However, when $r = 0$ (since the real interest rate is effectively zero when the interest rate equals the inflation rate), the formula simplifies to:

$$C = \frac{PV}{t}$$

Where:

• $PV = 450,000$ dollars (total savings)
• $t = 30$ years (retirement duration)

This simplifies to:

$$C = \frac{450,000}{30}$$

Let's calculate the amount she can spend each year in real terms.

The retiree can spend $15,000 each year in real terms over the 30-year retirement period.