# Present value of annuities and annuity payments

Present value of annuities and annuity payments

The present value of an annuity is the sum of the discounted value of all future cash flows.

An annuity that pays \$1,000 at the end of each year

An annuity that pays \$500 at the beginning of every six months

An annuity that pays \$500 at the end of every six months

An annuity that pays \$1,000 at the beginning of each year

An ordinary annuity selling at \$11,930.70 today promises to make equal payments at the end of each year for the next twelve years (N). If the annuity’s appropriate interest rate (I) remains at 9.50% during this time, the annual annuity payment (PMT) will be _

You just won the lottery. Congratulations! The jackpot is \$85,000,000, paid in twelve equal annual payments. The first payment on the lottery jackpot will be made today. In present value terms, you really won ___________assuming annual interest rate of 9.50%.

1. Implied interest rate and period

Consider the case of the following annuities, and the need to compute either their expected rate of return or duration.

Jacob inherited an annuity worth \$4,862.06 from his uncle. The annuity will pay him five equal payments of \$1,250 at the end of each year. The annuity fund is offering a return of __

Jacob’s friend, Devan, wants to go to business school. While his father will share some of the expenses, Devan still needs to put in the rest on his own. But Devan has no money saved for it yet. According to his calculations, it will cost him \$28,589 to complete the business program, including tuition, cost of living, and other expenses. He has decided to deposit \$3,800 at the end of every year in a mutual fund, from which he expects to earn a fixed 9% rate of return. It will take approximately ____________for Devan to save enough money to go to business school.

1. Perpetuities

Perpetuities are also called annuities with an extended or unlimited life. Based on your understanding of perpetuities, answer the following questions.

Which of the following are characteristics of a perpetuity? Check all that apply.

The current value of a perpetuity is based more on the discounted value of its nearer (in time) cash flows and less by the discounted value of its more distant (in the future) cash flows.

The value of a perpetuity is equal to the sum of the present value of its expected future cash flows.

A perpetuity is a stream of unequal cash flows.

The value of a perpetuity cannot be determined.

A local bank’s advertising reads: “Give us \$50,000 today, and we’ll pay you \$800 every year forever.” If you plan to live forever, what annual interest rate will you earn on your deposit?

1.92%

2.24%

1.60%

1.44%

Oops! When you went in to make your deposit, the bank representative said the amount of required deposit reported in the advertisement was incorrect and should have read \$75,000. This revision, which will ——- the interest rate earned on your deposited funds, will adjust your earned interest rate to ———–

1. Uneven cash flows
A series of cash flows may not always necessarily be an annuity. Cash flows can also be uneven and variable in amount, but the concept of the time value of money will continue to apply.

Consider the following case:

The Purple Lion Beverage Company expects the following cash flows from its manufacturing plant in Palau over the next four years:

Annual Cash Flows
Year 1 Year 2 Year 3 Year 4
\$250,000 \$37,500 \$180,000 \$300,000
The CFO of the company believes that an appropriate annual interest rate on this investment is 6.5%. What is the present value of this uneven cash flow stream, rounded to the nearest whole dollar?

\$1,475,000

\$1,692,500

\$467,500

\$650,014

Identify whether the situations described in the following table are examples of uneven cash flows or annuity payments:

Description Uneven Cash Flows Annuity Payments
You recently moved to a new apartment and signed a contract to pay monthly rent to your landlord for a year.
SOE Corp. hires an average of 10 people every year and matches the contribution of each employee toward his or her retirement fund.
Franklinia Venture Capital (FVC) invested in a budding entrepreneur’s restaurant. The restaurant owner promises to pay FVC 10% of the profit each month for the next 10 years.
You have committed to deposit \$600 in a fixed interest–bearing account every quarter for four years.
.

1. Nonannual compounding period

The number of compounding periods in one year is called compounding frequency. The compounding frequency affects both the present and future values of cash flows.

An investor can invest money with a particular bank and earn a stated interest rate of 6.60%; however, interest will be compounded quarterly. What are the nominal, periodic, and effective interest rates for this investment opportunity?

Interest Rates
Nominal rate
Periodic rate
Effective annual rate
You want to invest \$24,000 and are looking for safe investment options. Your bank is offering you a certificate of deposit that pays a nominal rate of 6% that is compounded quarterly. What is the effective rate of return that you will earn from this investment?

6.221%

6.450%

6.319%

6.136%

Suppose you decide to deposit \$24,000 in a savings account that pays a nominal rate of 12%, but interest is compounded daily. Based on a 365-day year, how much would you have in the account after four months? (Hint: To calculate the number of days, divide the number of months by 12 and multiply by 365.)

\$25,479.16

\$26,478.34

\$25,978.75

\$24,979.57

1. Mortgage payments

Mortgages, loans taken to purchase a property, involve regular payments at fixed intervals and are treated as reverse annuities. Mortgages are the reverse of annuities, because you get a lump-sum amount as a loan in the beginning, and then you make monthly payments to the lender.

You’ve decided to buy a house that is valued at \$1 million. You have \$200,000 to use as a down payment on the house, and want to take out a mortgage for the remainder of the purchase price. Your bank has approved your \$800,000 mortgage, and is offering a standard 30-year mortgage at a 9% fixed nominal interest rate (called the loan’s annual percentage rate or APR). Under this loan proposal, your mortgage payment will be ———– per month. (Note: Round the final value of any interest rate used to four decimal places.)

Your friends suggest that you take a 15-year mortgage, because a 30-year mortgage is too long and you will pay a lot of money on interest. If your bank approves a 15-year, \$800,000 loan at a fixed nominal interest rate of 9% (APR), then the difference in the monthly payment of the 15-year mortgage and 30-year mortgage will be ———–?(Note: Round the final value of any interest rate used to four decimal places. )

It is likely that you won’t like the prospect of paying more money each month, but if you do take out a 15-year mortgage, you will make far fewer payments and will pay a lot less in interest. How much more total interest will you pay over the life of the loan if you take out a 30-year mortgage instead of a 15-year mortgage?

\$856,769.40

\$1,010,987.89

\$1,182,341.77

\$1,096,664.83

Which of the following statements is not true about mortgages?

If the payment is less than the interest due, the ending balance of the loan will decrease.

Mortgages are examples of amortized loans.

Every payment made toward an amortized loan consists of two parts—interest and repayment of principal.

The ending balance of an amortized loan contract will be zero.

1. Loan amortization and capital recovery

Ian loaned his friend \$30,000 to start a new business. He considers this loan to be an investment, and therefore requires his friend to pay him an interest rate of 8% on the loan. He also expects his friend to pay back the loan over the next four years by making annual payments at the end of each year. Ian texted and asked that you help him calculate the annual payments that he should expect to receive so that he can recover his initial investment and earn the agreed-upon 8% on his investment.

Calculate the annual payment and complete the following capital recovery schedule:

Year Beginning Amount Payment Interest Paid Principal Paid Ending Balance
1 \$30,000.00
2
3
4 \$0.02

1. More on the time value of money

Lloyd is a divorce attorney who practices law in Florida. He wants to join the American Divorce Lawyers Association (ADLA), a professional organization for divorce attorneys. The membership dues for the ADLA are \$800 per year and must be paid at the beginning of each year. For instance, membership dues for the first year are paid today, and dues for the second year are payable one year from today. However, the ADLA also has an option for members to buy a lifetime membership today for \$8,500 and never have to pay annual membership dues.

Obviously, the lifetime membership isn’t a good deal if you only remain a member for a couple of years, but if you remain a member for 40 years, it’s a great deal. Suppose that the appropriate annual interest rate is 7.6%. What is the minimum number of years that Lloyd must remain a member of the ADLA so that the lifetime membership is cheaper (on a present value basis) than paying \$800 in annual membership dues? (Note: Round your answer up to the nearest year.)

13 years

19 years

16 years

23 years

In 1626, Dutchman Peter Minuit purchased Manhattan Island from a local Native American tribe. Historians estimate that the price he paid for the island was about \$24 worth of goods, including beads, trinkets, cloth, kettles, and axe heads. Many people find it laughable that Manhattan Island would be sold for \$24, but you need to consider the future value (FV) of that price in more current times. If the \$24 purchase price could have been invested at a 4.5% annual interest rate, what is its value as of 2018 (392 years later)?

\$635,647,030.80

\$987,122,447.84

\$859,993,041.68

\$747,820,036.24

1. More on the time value of money

Lloyd is a divorce attorney who practices law in Florida. He wants to join the American Divorce Lawyers Association (ADLA), a professional organization for divorce attorneys. The membership dues for the ADLA are \$800 per year and must be paid at the beginning of each year. For instance, membership dues for the first year are paid today, and dues for the second year are payable one year from today. However, the ADLA also has an option for members to buy a lifetime membership today for \$8,500 and never have to pay annual membership dues.

Obviously, the lifetime membership isn’t a good deal if you only remain a member for a couple of years, but if you remain a member for 40 years, it’s a great deal. Suppose that the appropriate annual interest rate is 7.6%. What is the minimum number of years that Lloyd must remain a member of the ADLA so that the lifetime membership is cheaper (on a present value basis) than paying \$800 in annual membership dues? (Note: Round your answer up to the nearest year.)

13 years

19 years

16 years

23 years

In 1626, Dutchman Peter Minuit purchased Manhattan Island from a local Native American tribe. Historians estimate that the price he paid for the island was about \$24 worth of goods, including beads, trinkets, cloth, kettles, and axe heads. Many people find it laughable that Manhattan Island would be sold for \$24, but you need to consider the future value (FV) of that price in more current times. If the \$24 purchase price could have been invested at a 4.5% annual interest rate, what is its value as of 2018 (392 years later)?

\$635,647,030.80

\$987,122,447.84

\$859,993,041.68

\$747,820,036.24

1. Present value

Finding a present value is the reverse of finding a future value.

is the process of calculating the present value of a cash flow or a series of cash flows to be received in the future.

Which of the following investments that pay will \$18,500 in 5 years will have a higher price today?

The security that earns an interest rate of 5.50%.

The security that earns an interest rate of 8.25%.

Eric wants to invest in government securities that promise to pay \$1,000 at maturity. The opportunity cost (interest rate) of holding the security is 8.20%. Assuming that both investments have equal risk and Eric’s investment time horizon is flexible, which of the following investment options will exhibit the lower price?

An investment that matures in five years

An investment that matures in six years

Which of the following is true about present value calculations?

Other things remaining equal, the present value of a future cash flow increases if the investment time period increases.

Other things remaining equal, the present value of a future cash flow decreases if the investment time period increases.

1. Finding the interest rate and the number of years

The future value and present value equations also help in finding the interest rate and the number of years that correspond to present and future value calculations.

If a security currently worth \$2,000 will be worth \$2,249.73 three years in the future, what is the implied interest rate the investor will earn on the security—assuming that no additional deposits or withdrawals are made?

4.00%

3.20%

0.37%

8.89%

If an investment of \$45,000 is earning an interest rate of 12.00%, compounded annually, then it will take ——— for this investment to reach a value of \$74,936.54—assuming that no additional deposits or withdrawals are made during this time.

Which of the following statements is true—assuming that no additional deposits or withdrawals are made?

It takes 10.50 years for \$500 to double if invested at an annual rate of 5%.

It takes 14.21 years for \$500 to double if invested at an annual rate of 5%.

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