# I am planning on investing for retirement. I estimate that I will need $100,000 per year for...

## Question:

I am planning on investing for retirement. I estimate that I will need $100,000 per year for twenty years. I expect to earn 6% while accumulating and 4% in retirement.. I am now 25 expecting to retire at 67 and have nothing in the plan yet, and from this year I will be contributing equal annual amount.

(a) How big must those contributions be?

(b) I started on that plan ten years ago and am now 35 and have $80,000 in the plan. If I continue to make the contributions from part (a), what retirement income can I expect?

## Present Value:

Present value is the current worth of future annual income or expenditure. Using the annual rate of return, all the future expenses can be discounted and compared with other options to decide the future course of action.

## Answer and Explanation: 1

The present value of the amount required after maturity must be known to calculate the annuity required to save:

{eq}\begin{align*} {\rm\text{Present Value}}& = {\rm\text{P}} \times \frac{{\left( {1 - {{\left( {1 + {\rm\text{i}}} \right)}^{ - {\rm\text{n}}}}} \right)}}{{\rm\text{i}}}\\ & = \$ 100,000 \times \frac{{\left( {1 - {{\left( {1 + 0.04} \right)}^{ - 20}}} \right)}}{{0.04}}\\ &= \$ 1,359,033 \end{align*} {/eq}

Using the above amount as future value, the annuity will be calculated using the FV of annuity formula:

{eq}\begin{align*} {\rm\text{FutureValue}} &= {\rm\text{P}} \times \frac{{\left( {{{\left( {1 + {\rm\text{i}}} \right)}^{\rm\text{n}}} - 1} \right)}}{{\rm\text{i}}}\\ \$ 1,359,033 &= {\rm\text{P}} \times \frac{{\left( {{{\left( {1 + 0.06} \right)}^{42}} - 1} \right)}}{{0.06}}\\ & = \$ 7,723.95 \end{align*} {/eq}

b.If the savings are $80,00 at age of 35 and annual contributions are $7,732.95 the accumulated amount will be:

{eq}\begin{align*} {\rm\text{FutureValue}} &= {\rm\text{P}} \times \frac{{\left( {{{\left( {1 + {\rm\text{i}}} \right)}^{\rm\text{n}}} - 1} \right)}}{{\rm\text{i}}} + {\rm\text{PV}}{\left( {1 + {\rm\text{i}}} \right)^{\rm\text{n}}}\\ & = \$ 7,723.95 \times \frac{{\left( {{{\left( {1 + 0.06} \right)}^{32}} - 1} \right)}}{{0.06}} + \$ 80,000{\left( {1 + 0.06} \right)^{32}}\\ &= \$ 1,218,299.04 \end{align*} {/eq}

He can utilize this amount for the next 20 years during retirement as shown below:

Particulars | Amount |

Total amount | $1,218,299.04 |

No. of years after returenment | 20 |

Annual amount available | $60,914.95 |

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Chapter 6 / Lesson 1The value of investments changes over time, and this can be applied to multiple cash flows. Identify how to calculate both the present and future values applied specifically to cash flows.

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