# You are planning to save for retirement over the next 15 years. To do this, you will invest...

## Question:

You are planning to save for retirement over the next {eq}15 {/eq} years. To do this, you will invest {eq}\$1,100 {/eq} a month in a stock account and {eq}\$500 {/eq} a month in a bond account. The return on the stock account is expected to be an {eq}EAR {/eq} of {eq}7 {/eq} percent, and the bond account will pay an {eq}EAR {/eq} of {eq}4 {/eq} percent. When you retire, you will combine your money into an account with an {eq}EAR {/eq} of {eq}5 {/eq} percent. How much can you withdraw each month during retirement assuming a {eq}20 {/eq} -year withdrawal period?

(a) {eq}\$3,037.36 {/eq}

(b) {eq}\$2,904.11 {/eq}

(c) {eq}\$3,008.21 {/eq}

(d) {eq}\$2,636.19 {/eq}

(e) {eq}\$3,406.97. {/eq}

## Effective Annual Interest rate :

Effective annual rate (EAR) is the actual or real rate of interest to be received or paid on the saving or loan respectively when the compounding period is more than once in a year. EAR is always greater than the stated interest rate.

## Answer and Explanation: 1

Correct Option : Option a

Workings :

Amount deposit monthly in stock account (P1) = $1,100

Amount deposit monthly in bond account (P2) = $500

Effective annual interest rate on stock = 7%

Effective monthly interest rate on stock (r1) = 7%/12 = 0.583%

Effective annual interest rate on bond = 4%

Effective monthly interest rate on bond (r2) = 4%/12 = 0.33%

Effective interest rate while withdrawing = 5%

Effective monthly rate while withdrawing (r3) = 5%/12 = 0.416%

Number of years deposit going on = 15

Number of deposit (n1) = 180

Total years of withdrawal = 20

Total number of withdrawals (n2) = 240

{eq}\begin{align*} \rm\text{Total amount of in the account after 15 years} &= P1 \times \frac {(1+r1)^{n1}-1}{r1} + P2 \times \frac {(1+r2)^{n1}-1}{r2}\\ &= \$1,100 \times \frac {(1+0.00583)^{180}-1}{0.00583} + \$500 \times \frac {(1+0.0033)^{180}-1}{0.0033}\\ &= \$348,537.32 + \$122,643.70\\ &= \$471,181.02 \end{align*} {/eq}

Now, this amount will be the present value of all the withdrawals made.

So, Let monthly withdrawal be P3

{eq}\begin{align*} \rm\text{Present value of all withdrawals } &= P3 \times \frac {1-(1+r)^{-n}}{r}\\ \$471,181.02 &= P3 \times \frac {1-(1+0.00416)^{-240}}{0.00416}\\ \$471,181.02 &= P3 \times 151.626831\\ \$3,107.50 &= P3 \end{align*} {/eq}

So, the nearest $3,037.36

Hence, option a is correct.

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Chapter 8 / Lesson 3Learn how to find present value of annuity using the formula and see its derivation. Study its examples and see a difference between Ordinary Annuity and Annuity Due.

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