Los principios del modelo

Unequal size or spacing

Sometimes an investment has payments of unequal size or spacing. The present and future value formulas can be used to accommodate these irregularities. For instance, suppose that you deposit $100 in an investment at the end of Year One, $350 at the end of Year Three, $250 at the end of Year Four, and $200 at the end of Year Five. If the investment earns 6% compounded annually, what is the value of this investment at the end of Year Six? Let's build a table to illustrate the cash flows and their future values.

Year Deposit x FVIF = Payments 1 $100 x (1.06)5 = 133.82 2 $000 x (1.06)4 = 0.00 3 $350 x (1.06)3 = 416.86 4 $250 x (1.06)2 = 280.90 5 $200 x (1.06)1 = 212.00 6 $000 x (1.06)0 = 0.00 FV = $1,043.58

Figure 3.6: Future Value of Annuity with Unequal Payments

You can see that the first deposit is in the account for five years, earning compounded interest of $33.82 during that time; no deposit is made in the second year; the third deposit earns compound interest for three years in the amount of $66.86; and so on. The total interest the account will earn for the six years is $1,043.58 – $900.00 (the total amount of all the deposits) = $143.58.

Formula for present value of unequal payments

The formula for calculating the present value of unequal payments is a modification of the formula for the present value for an annuity.

PV = CF₁[1/(1+R)]1 _{+ CF}

2[1/(1+R)]2 + ... + CFT[1/(1+R)]T

Where:

PV = Present value of the stream of cash flows CF_i = Cash flow (payment) in period i

R = Discount rate

T = Number of periods in the stream of cash flows

The idea is to discount each payment or cash flow from the time of receipt back to the present time. The present value is the sum of all of the discounted cash flows.

Example To illustrate, we will calculate the present value of an investment with expected payments of $250 at the end of Year One, $500 at the end of Year Two, $400 at the end of Year Four, and $100 at the end of Year Five. The investor's required rate of return is 8%.

Year Payment x PVIF = Present Value

5 $100 x 1/(1.08)5 = 68.06 4 $400 x 1/(1.08)4 = 294.01 3 $000 x 1/(1.08)3 _{= 0.00} 2 $500 x 1/(1.08)2 = 428.67 1 $250 x 1/(1.08)1 = 231.48 PV = $1,022.22

Figure 3.7: Present Value of Unequal Discounted Cash Flows

Using a financial calculator

The present value of this investment is $1,022.22. Another approach to solving this problem is to use a financial calculator that allows the input of individual cash flows. The process is to input each cash flow and the appropriate discount rate. Then, push the present value key to derive the present value of the investment as $1,022.22. Many calculators can handle this type of calculation. The logic is to input each payment for the period and the interest rate the investment is earning. Then push the future value key to derive the future value.

You have completed the section on calculating the present value and future value of annuities, perpetuities, and uneven payment streams. Please check your understanding of these

calculations by completing the Practice Exercise that begins on page 3-37. Check your answers with the Answer Key.

Following the Practice Exercise is a summary of the concepts we have presented in this unit. Please read the summary, then answer the questions in Progress Check 3. It is important that you master these basic concepts, as they form the foundation for understanding the material in succeeding units of this course. If you answer any of the questions incorrectly, we

PRACTICE EXERCISE 3.4

Directions: Calculate the answer to each question and write the correct answer in the blank. Check your solution with the Answer Key on the next page.

18. A security is structured to pay cash flows of $100 per year for ten years. If an investor's required rate of return is 5%, what price would the investor be willing to pay for the security?

$_______________________

19. You have agreed to pay five installments of $100 for a television, the first payment being due immediately. If the discount rate is 10%, what is the equivalent cash price for the television?

$_______________________

20. PQR Inc. sets up a fund into which it will make annual payments that will

accumulate at 5% per annum. The fund is designed to repay a bond issue at the end of 25 years. If the annual payment into the fund is fixed at $5,000,000, how large is the bond issue?

ANSWER KEY

18. A security is structured to pay cash flows of $100 per year for ten years. If an investor's required rate of return is 5%, what price would the investor be willing to pay for the security?

$772

Building the table and adding the present Using a financial calculator: value interest factors for the ten cash Number of payments = 10

flows. Interest rate = 5%

PV_{a = $100 (7.7217)} Payment = $100

PV_a = $772.17 Press the present value key and you should get $772.17

19. You have agreed to pay five installments of $100 for a television, the first payment being due immediately. If the discount rate is 10%, what is the equivalent cash price for the television?

$417

With the formula, calculate the present Using a financial calculator: value of the last four payments and add Number of payments = 4 the total to the deposit. Interest rate = 10%

PV_{a = $100 (3.1699)} Payment = $100

PV_a = $317 Press the present value key and Cash Price = $317 + $100 = $417 you should get $316.99. Then

add the deposit of $100 to get a cash price of $417.

20. PQR Inc. sets up a fund into which it will make annual payments that will

accumulate at 5% per annum. The fund is designed to repay a bond issue at the end of 25 years. If the annual payment into the fund is fixed at $5,000,000, how large is the bond issue?

$238,635,494

Using a financial calculator: Number of payments = 25 Interest rate = 5%

PRACTICE EXERCISE 3.4

(Continued)

21. You are offered a contract that will pay $50.00 per year indefinitely. If your discount rate is 10%, what would you be willing to pay for the offered contract?

$_______________________

22. If your required return is 10%, what would you be willing to pay for a perpetuity cash flow of $5.00 per year?

ANSWER KEY

21. You are offered a contract that will pay $50.00 per year indefinitely. If your discount rate is 10%, what would you be willing to pay for the offered contract?

$500.00

PV_p = A x (1/R)

PV_p = $50.00 (1 / 0.10) PV_p = $500.00

22. If your required return is 10%, what would you be willing to pay for a perpetuity cash flow of $5.00 per year?

$50.00

PVp = A x (1/R)

PV_p = $5.00 (1 / 0.10) PV_p = $50.00

UNIT SUMMARY

In this unit, we focused on the changes in the value of money over time.

Future value — simple /

compound interest

The future value of money at the end of a defined period of time results from adding the amount of interest that may be earned on an investment to the principal amount.

• Simple interest is paid on the principal at the end of the payment period at a defined rate.

• Interest also may be compounded, which means that interest is reinvested after each payment period and earns interest in each subsequent period. Interest may be compounded at the end of discrete annual or non-annual periods, or continuously throughout the life of the investment.

Present value – discount

Sometimes we want to know the current value of a cash flow that will occur at some future date. Discrete or continuous discounting is the process for equating a future cash flow with its present value. The discount rate may be referred to as the opportunity cost of an investment or the investor's required rate of return.

Annuity _{An annuity is a security that involves a series of equal payments}

made at regular time intervals. To derive the future value of an annuity, each payment is compounded for the remaining life of the security. A payment made at the end of the first year of a ten-year annuity is compounded for nine years, the second payment is compounded for eight years, etc. All compounded payments added together equal the future value.

Calculating the present value of an annuity is the reverse process of compounding. The payment due at the end of the tenth year is

discounted over the ten year period, the ninth payment is discounted over nine years, etc. The present values of all payments added together equal the present value of the annuity.

Perpetuity A perpetuity is a special type of annuity with an infinite number of payment periods. The future value of a perpetuity is infinite — and of little consequence. The present value is easy to calculate and provides the investor with information to facilitate a comparison of investment prices.

The present and future values of investments with payments of uneven size or spacing also may be calculated by totaling the compounded or discounted values of the cash flows.

You can see that the present / future value of an investment depends on the discount / interest rate, the number of payments, the length of time until maturity, and the amount of each cash flow. By

manipulating these factors, we can discover the future value of an interest-bearing investment or the discounted present value of future cash flows.

You have completed Unit Three: Time Value of Money. Before proceeding to Unit Four:

Valuing Financial Assets, please answer the following questions. They will check your

comprehension of the concepts presented in this unit. If you answer any questions incorrectly, we suggest that you review the appropriate text.