A savings balance with a random yearly growth factor

Question

You deposit \1000. Each year, independently, the balance is multiplied by a random growth factor spread evenly between 1.0 and 1.1 (uniform on [1.0, 1.1]). The account runs for three years. What balance should you expect after three years?

Accepted Answer

Each year multiplies the balance by a random factor, and the years are independent. Condition on the balance before each year with the tower rule. Step 1, the average yearly factor. A uniform draw on [1.0, 1.1] has mean at the midpoint: E[	ext{factor}] = \frac{1.0 + 1.1}{2} = 1.05. Step 2, the recursion. Let Bt be the balance after t years. Given B{t-1}, the next year multiplies it by an independent factor averaging 1.05, so E[Bt] = 1.05 \, E[B{t-1}]. Step 3, iterate from B0 = 1000 over three years: E[B3] = 1000 	imes 1.05^3 = 1000 	imes 1.157625 = 1157.625. So you expect a balance of abou…

A savings balance with a random yearly growth factor

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