# doc

```Hye Jung Kim
Hyuk-Min Choung
BONUS 3
Starting on his 25th birthday and continuing through his 60th birthday, Fred deposits
\$7,500 each year on his birthday into a retirement fund earning an annual effective
rate of 5%. Immediately after the last deposit, the accumulated value of the fund is
transferred to a fund earning an annual effective rate of j. Five years later, a twentyfive year annuity-due paying \$5,800 each month is purchased with the funds. The
purchase price of the annuity was determined using an annual effective rate of
interest of 4%. Find j.

In this question, we are trying to find the variable j, which is the annual
effective rate that was used to calculate the fund earning after calculating the
future annuity (S). It is given that Fred deposits \$ 7,500 each year on his
birthday, specifying that the payment ( R) is only made once a month. Since
he started his payment on his 25th birthday and ended on his 60th birthday,
he made 36 payments (n) because 60-25=35 and 1 has to be added to 35 to
include the payment made on his 25th birthday. The problem states that the
annual effective rate is 5%, making i as .05.
Now we have all three variables (R, n, i) to calculate the future annuity (S).
The future annuity formula is:
Then, plug in all the known variables.
With the value of future annuity (\$718,772.42), we will be calculating the
variable j. Since the money (S) accumulated from his 25th birthday to 60th
birthday is immediately transferred to a fund, compound interest formula
[P(1+i)n] will be used, and the problem states “five years later” giving the
number of n is 5.
When known variables are all plugged in, we still have j left as a variable.
Therefore, we have to equal to equation to another number to be able to
solve for j.
Then, it is given that twenty-five year annuity-due paying of \$5,800 each
month is purchased with the funds while the annual effective rate of interest
of 4%. However, in this case the rate would have to be divided into 12
because the payment is due each month.
Hye Jung Kim
Hyuk-Min Choung
With this information, another annuity equation will be set up. However,
present annuity will be calculated because the compound interest that we set
up 718,72.42(1+j)^5 will give the present annuity value. Since n is 300 (25
has to times by 25 because the payment is due monthly), interest rate is
.0033, and R is 5800, the present annuity would look like this,
Therefore, these two equations that we have derived from previous steps
will be set up equal to each other.
Then, we will be solving for j.
Hye Jung Kim
Hyuk-Min Choung
BONUS 4
When computed using an effective interest rate of i, is known that the present value
of \$2,000 at the end of each year for 2n years plus an additional \$1,000 at the end of
each of the first n years is \$52,800. Using this same interest rate, the present value
of an n year deferred annuity-immediate paying \$4,000 per year for n years is
\$27,400. Find n.

In this question, we are trying to solve for unknown variable n. When
thinking in a term of timeline. There is \$2000 payment at the end of each
year for 2n years with I effective rate. This will be called the first cash flow
because in the year of n, another cash flow is introduced. Starting the year of
n, additional \$1000 with the same effective rate i. This \$1,000 monthly
payment will be called second cash flow.
Since the problem states that addition of first and second cash flows equal to
the present value of \$52,800, present annuity will be used to calculate each
cash flow.
First cash flow equation will be set up as:
Second cash flow equation will be:
Then, each cash flow will be added to equal to \$ 52,800.
Second part of the problem states that the same interest rate, which will be
still i, used to calculate the present value of an n year deferred annuityimmediately paying \$4,000 per year for n years when the present value is
\$27,400.
Therefore, using the known variables present tense annuity equation will be
set up. However, since the problem specifically states that it is a deferred
annuity, the entire present annuity will be divided by (1+i)n, giving the
equation of:
Hye Jung Kim
Hyuk-Min Choung
Now, there are two equations with two variables, i and n. Therefore, system
of equation will be set up with those two equations to solve for both i and n
with Wolfram Alpha Website. After plugging all the information in the
Wolfram Alpha Website, it will state that the value of i is .0364 (3.64%) and n
is 17.9 years.
*When using wolfram alpha website, type the first equation then make sure
to put comma before typing the second equation.
```
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