Question

Today, you turn 23. your birthday wish is that you will be a millionaire by your 40th birthday. in an attempt to reach this goal, you decide to save $75 a day, every day, until you turn 40. you open an investment account and deposit your first $75 today. what rate of return must you earn to achieve your goal? 7.67 percent 8.09 percent 9.90 percent 10.06 percent 10.54 percent

Answer 1

Answer: Since 13335.18058 is closest to 13333.33333, 8.09% is the interest rate I must earn to achieve my goal of becoming a millionaire.

The interest rates for Future Values and Present Values of annuities are always calculated by iteration.

Since we have four options we can substitute the value of r for each of the interest rates for each period.

In this question we have

Payment = $75

Frequency of saving : Daily

No. of years :   $40 - 23 =17$

So, number of periods = $17 * 365 = 6205$

The interest rate quoted usually an annual interest rate. For daily compounding we need to divide the interest rate by 365. So, $r = \frac{i}{365}$

The Future Value of an annuity formula is:

$FV_{annuity}=PMT \left (\frac{(1+r)^{n}-1}{r} \right )$

Substituting the values in the equation above we get,

$1000000 =75 \left (\frac{(1+r)^{n}-1}{r} \right )$

$\frac{1000000}{75} = \left (\frac{(1+r)^{n}-1}{r} \right )$

$13333.33333 = \left (\frac{(1+r)^{n}-1}{r} \right )$ ------ (1)

The only value that comes closest to the value on the Left hand side of the equation is when the interest rate is 8.09%.

If annual interest rate is 8.09%, the daily interest rate is $\frac{0.0809}{365} = 0.000221644$

Substituting this in the equation (1) we get,

$\left (\frac{(1+0.000221644)^{6205}-1}{0.000221644} \right )$

$\left (\frac{3.955660571-1}{0.000221644} \right )$

$13335.18058$

Since 13335.18058 is closest to 13333.33333, 8.09% is the interest rate I must earn to achieve my goal of becoming a millionaire.