第一段为甚么是End？不是应该BGM吗 因为是先拿钱？

问题如下：

Question An individual wants to be able to spend €80,000 per year for an anticipated 25 years in retirement. To fund this retirement account, he will make annual deposits of €6,608 at the end of each of his working years. He can earn 6% compounded annually on all investments. The minimum number of deposits that are needed to reach his retirement goal is closest to:

选项：

A.51 B.40 C.28

解释：

Solution

B is correct. The following figure represents the timeline for the problem:

Using a financial calculator, the funds needed at retirement (R on the timeline) are calculated: N = 25; I/Y = 6%; PMT = €80,000; Future value (FV) = €0; Mode = End. The calculated present value (PV) is €1,022,668.$\begin{array}{c}\text{PV}=A\left[\frac{1?\frac{1}{{\left(1+r\right)}^N}}{r}\right]\\ =80,000\left[\frac{1?\frac{1}{{\left(1.06\right)}^{25}}}{0.06}\right]\\ =1,022,688\end{array}$ Then, €1,022,668 is used as the FV (at R on the timeline) for the accumulation phase annuity as per: I/Y = 6%; PV = €0; PMT = -€6,608; FV = €1,022,668; Mode = End. The computed N is 40.

$1,022,668=6,608\left[\frac{{\left(1+0.06\right)}^N?1}{0.06}\right]$

A is incorrect. 80,000 is multiplied by 25 years (2,000,000) and the result is used as the FV of the 6,608 annuity at 6% and PV = 0. The result is N = 50.67.

C is incorrect. 80,000 is multiplied by 25 years and then discounted at 6% for 25 years (465,997). The result is used as the PV of the 6,608 annuity at 6% as follows: I/Y = 6%; PMT = 6,608; PV =465,997; FV = 0; Calculate N: N = -28.40 and the minus sign is ignored.