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# Annuities are explicated.

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Part I. Basic Computations

1. Mike's Sport Shop deposits \$3,600 at the end of each year for 12 years at 7% annual interest.

a. How much will this ordinary annuity be worth at the end of the 12 years? (5 points)

b. How much more will this annuity be worth (annuity due) if Mike deposits the money at the beginning of each year instead of at the end of each year? (5 points)

2. Barb and John Reed want to know how much they must deposit in a retirement savings account today to have payments of \$1,750 every six months for 15 years. The retirement account is paying 8% annual interest, compounded semiannually. (5 points)

3. Lena Dimock is saving for her college expenses. She sets aside \$200 at the beginning of each three months in an account paying 8% annual interest, compounded quarterly. How much will Lena have accumulated in the account at the end of four years? (5 points)

Part II. Case Study
Julie has just completed the rigorous process of becoming a Certified Financial Planner (CFP). She is looking forward to working with individuals on saving for retirement. She would like to show her clients the value of an annuity program as one of the best options for investing current earnings in a tax-deferred account.
1. If a client puts the equivalent of \$55 per month, or \$660 per year, into an ordinary annuity, how much money would accumulate in 20 years at 3% compounded annually? (5 points)

2. Jackie, a 25 year old client, want to retire by age 65 with \$2,000,000. How much would she have to invest annually, assuming a 6% rate of return? (5 points)

3. Another client, Wynona, decides that she will invest \$5,000 per year in a 6% annuity for the first ten years, then \$6,000 for the next ten years, and then \$4,000 per year for the last ten years, how much will she accumulate? [Hint: Treat each ten-year period as as separate annuity and compute the Future Value. After the ten years, assume that the value will continue to grow at compound interest for the remaining years of the 30 years. Use tables from Unit 6 to compute compound interest.] (5 points)

4. Research the benefits and risks associated with annuities. Based on your research, select one particular type of annuity in which you might consider investing. Describe why you have selected this annuity and how it fits into your personal financial picture. Please be specific about how this annuity fits into your plan. Please cite sources. (150 words or so)

https://brainmass.com/math/basic-algebra/annuities-explicated-419831

#### Solution Preview

Let's first define annuity. An annuity is a series of cashflow for a fixed number of periods. An ordinary annuity is when the payments (or cashflows) occur at the end of the period (for example, I pay you \$100/month for 12 month and the payment is made on the 31st of every month). An annuity due is when the payment is made on the first day of the period.

For ordinary annuities,

PV = (C/r)[1 - (1 + r)^(-n)], where r is interest rate, n is number of periods and C is per period payment.

FV = (C/r)[(1+r)^n - 1]

For annuity dues,

FV = (C/r)[(1 + r)^(n+1) - 1] - C.

Now we start answering the questions.

1.

a) This is an ordinary annuity. So FV = (C/r)[(1+r)^n - 1] = (3600/0.07)[(1+0.07)^12 - 1] = 64398.42.

b) This is an annuity due. FV = (C/r)[(1 + r)^(n+1) - 1] - C = (3600/0.07)[(1 + 0.07)^13 - 1] - 3600 = 68906.31.

There is no surprise that an annuity due is worth more than an ordinary annuity, because with annuity dues, you get the payments 1 period before the ordinary annuities, so you can start investin your payments earlier.

2. We assume that this is an ordinary annuity, when the payment time is not specified. This question is asking for the present value, there are ...

#### Solution Summary

Annuities are explicated. The benefits and risks associated with annuities are provided.

\$2.19