Question

Assume that you are 30 years old today, and that you are planning on retirement at age 65. You expect your salary to be $42,000 one year from now and you also expect your salary to increase at a rate of 5% per year as long as you work. To save for your retirement, you plan on making annual contributions to a retirement account. Your first contribution will be made on your 31st birthday and will be 8% of this year's salary. Likewise, you expect to deposit 8% of your salary each year until you reach age 65. Assume that the rate of interest is 9%.
Required:
a. The present value (PV) (at age 30) of your retirement savings is ________.
b. A rich donor gives a hospital $1,040,000 one year from today. Each year after that, the hospital will receive a payment 6% larger than the previous payment, with the last payment occurring in ten years' time. What is the present value (PV) of this donation, given that the interest rate is
11%?
c. If the current rate of interest is 8%, then the present value (PV) of an investment that pays $1200 per year and lasts 24 years is closest to ________.

Answer 1

Answer:

The answer to this question  can be defined as follows:

In point a, answer is "$61,303".  

In point b, answer is " $7,681,257.74".

In point c, answer is "$12,635".

Explanation:

Given value:

In point a:

Year 1 = 0.08(42,000)

          = $3,360

Time = 30 years

Rate Of  Growth  = 5%

Rate  Of Interest = 9%

Formula:

Present Value $= \frac{P}{(r - g)}[1 - (\frac{(1 + g)}{(1 + r)})^n]$

                         $=\frac{3,360}{(0.09 - 0.05)}[1 - (\frac{1.05}{1.09})^{35}]$

                         $=\frac{3,360}{(0.04)}[1 - (0.270207895)]\\\\=\frac{3,360}{(0.04)}[ 0.729792105]\\\\=\frac{2452.10147}{(0.04)}\\\\= 61,302.5368 \\\\ = \bold{61,303}$

In point b:

$PV= [ \frac{P}{(r-g)}] \times [1-[\frac{(1+g)}{(1+r)}]^{n}]$

      $= [ \frac{1,040,000}{(11 \%-6\% )}] \times [1-[\frac{(1+6 \% )}{(1+11 \%)}]^{10}] \\\\= [ \frac{1,040,000}{(5 \%)}] \times [1-[\frac{1.06}{(1.11)}]^{10}] \\\\= [ \frac{1,040,000}{(5 \%)}] \times [1-[(0.954954955)]^{10}] \\\\= [ \frac{1,040,000}{(5 \%)}] \times [1- 0.630708763] \\\\= [ \frac{1,040,000}{(5 \%)}] \times 0.369291237\\\\= [ \frac{1,040,000}{(5 \%)}] \times 0.369291237\\\\= 20800000 \times 0.369291237 \\\\= 7,681,257.74$

In point c:

$PV= \frac{PMT \times (1- \frac{1}{1+r^n})}{r}$

      $= \frac{1200 \times 1- (\frac{1}{1.08^{24}})}{0.08}\\\\= \frac{1200 \times 1- (0.157699337)}{0.08}\\\\= \frac{1200 \times 0.842300663}{0.08}\\\\= \frac{1010.7608}{0.08}\\\\=12634.51\\\\= \bold{12635}$