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.