Question 7.1: Sheila is investing €10,000 a year in a savings scheme that ......
Sheila is investing €10,000 a year in a savings scheme that pays 10% interest every year. What will the value of her investment be after 5 years?
Sheila invests €10,000 at the start of year 1 and so this earns 5 years of compound interest of 10% and so its future value in 5 years is given by 10000 ∗ 1.1^{5} = €16,105. The future value of the payments that she makes is presented in Table 7.2.
Therefore, the value of her investment at the end of 5 years is the sum of the future values of each payment at the end of 5 years = 16,105 + 14,641 + 13,310 + 12,100 + 11,000 = €67,156.
We note that this is the sum of a geometric series and so in general if an investor makes a payment of A at the start of each year for n years at a rate r of interest then the investment value at the end of n years is
A\,(1\,+\,r)^{n}\,+\,A\,(1\,+\,r)^{n-1}+\,…\,+A\,(1\,+\,r)= A\left(1\,+\,r\right)\left[1\,+\,A\left(1\,+\,r\right)+\,…\,+A\left(1\,+\,r\right)^{n-1}\right]
= A\left(1\,+\,r\right)\frac{(1+r)^{n}-1}{(1+r)-1}
= A\left(1\,+\,r\right)\frac{\left(1+r\right)^{n}-1}{r}
We apply the formula to check our calculation.
10000\;(1\;+\;0.1)\;{\frac{(1+0.1)^{5}-1}{0.1}}= 11000{\frac{(1.1)^{5}-1}{0.1}}
= 11000{\frac{(1.61051)-1}{0.1}}
= €67,156
| TABLE 7.2 | |||
| Calculation of Future Value of Annuity | |||
| Year | Amount | Future Value (r = 0.1) | |
| 1 | 10,000 | 10,000\ast1.1^{5}=€16,105 | |
| 2 | 10,000 | 10,000\ast1.1^{4}=€14,641 | |
| 3 | 10,000 | 10,000\ast1.1^{3}=€13,310 | |
| 4 | 10,000 | 10,000\ast1.1^{2}=€12,100 | |
| 5 | 10,000 | 10,000\ast1.1^{1}=€11,000 | |
| Total | €67,156 | ||