Question

Muhammad, a 21-year old computer engineer, is opening an individual retirement account (IRA) at a bank. His goal is to accumulate $2.5 million in the IRA by the time he retires in 46 years. Muhammad expects his IRA to receive 8% nominal annual interest, compounded semiannually, throughout the 46 years. As a computer engineer, Muhammad believes his salary will increase at a constant 4% annual rate during his career. Muhammad wishes to make annual deposits into his IRA account over the 46 years. He wishes to start his IRA with the lowest possible deposit and then increase his deposit amount at a constant 4% rate each year.
Assuming end-of-year deposits, how much should she deposit the first year?

Answer 1

Answer:

The first annual depoisit will be of 3,373.49 dollars

Explanation:

Given the formula for future growing annuity

we need to solve for the yearly payment:

grow rate:  0.04

annual effective rate: 8% compounding semiannually:

$(/1+0.08/2)^2-1 = r_e$

r= 0.0816

FV 2,500,000

n 46

Formula for future value fo an ordinary annuity:

$C_0 \times \frac{(1+r)^n-(1+g)^n}{r-g} = FV$

$C_0 \times \frac{(1+0.0816)^{46}-(1+0.04)^{46}}{0.0816-0.04} = 2,500,000\\C_0 = 3,373.4855$

The first annual depoisit will be of 3,373.49 dollars