# A saver wants $100,000 after ten years and believes that it is possible to earn an annual rate of... ## Question: A saver wants$100,000 after ten years and believes that it is possible to earn an annual rate of 8% on invested funds. What amount must be invested each year to accumulate $100,000 if: a. the payments are made at the beginning of each year? b. they are made at the end of each year? Show work and explain. Explain the difference between the two. ## Annuity accumulation: An annuity is a set of equal payments over a specified period of time. The accumulated value is the future value of the annuity. When the payments start right away it the value of annuity due and if they start at the end of first period, it is the value of ordinary annuity. ## Answer and Explanation: 1 a) If the payments are made at the beginning of periods, then it is the value of annuity due: Future value of annuity due can be expressed as: {eq}FV=(1+r) \times P[ \frac{(1+r)^{n}-1}{r} ] {/eq} Future value (FV) =$100,000

Payment (P) = ?

r (rate) = 8.00% or 0.08

n (periods) = 10

{eq}$100,000 = (1+0.08) \times P [ \frac{(1+0.08)^{10}-1}{0.08} ] {/eq} {eq}$100,000 = (1+0.08) \times P \times 14.48656247 {/eq}

{eq}P = $6,391.62 {/eq} Hence the amount that shall be invested is$6,391.62

The Future Value of annuity can be represented as:

{eq}FV=P\times \frac{(1+r)^{n}-1}{r} {/eq}

Here:

Future value (FV) = $100,000 Payment (P) = ? r (rate) = 8.00% or 0.08 n (periods) = 10 Substituting the values we have: {eq}$100,000=P \times \frac{(1+0.08)^{10}-1}{0.08} {/eq}

{eq}$100,000=P \times 14.48656247 {/eq} {eq}P =$6,902.95 {/eq}

Hence the payment shall be \$6,902.95