TVM Discounted Cash Flow Problem Sets

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Chapter 5

5-1 Future Value Compute the future value in year 9 of a $2,000 deposit in year 1 and another $1,500 deposit at the end of year 3 using a 10 percent interest rate. (LG5-1) LEARNING GOALS

LG5-1 Compound multiple cash flows to the future.

Finding the Future Value of Several Cash Flows LG5-1

Consider the following contributions to a savings account over time. You make a $100 deposit today, followed by a $125 deposit next year, and a $150 deposit at the end of the second year. If interest rates are 7 percent, what’s the future value of your deposits at the end of the third year? The time line for this problem is illustrated as:

Note that the first deposit will compound for three years. That is, the future value in year 3 of a cash flow in year 0 will compound 3 (= 3 − 0) times. The deposit at the end of the first year will compound twice (= 3 − 1). In general, a deposit in year m will compound N− m times for a future value in year N. We can find the total amount at the end of three years by computing the future value of each deposit and then adding them together. Using the future value equation from Chapter 4, the future value of today’s deposit is $100 × (1 + 0.07)3 = $122.50. Similarly, the future value of the next two deposits are $125 × (1 + 0.07)2 = $143.11 and $150 × (1 + 0.07)1 = $160.50, respectively.

Putting these three individual future value equations together would yield:

FV3 = $100 × (1 + 0.07)3 + $125 × (1 + 0.07)2 + $150 × (1 + 0.07)1 = $426.11

The general equation for computing the future value of multiple and varying cash flows (or payments) is:

In this equation, the letters m, n, and p denote when the cash flows occur in time. Each deposit can be different from the others.

5-3 Future Value of an Annuity what is the future value of a $900 annuity payment over five years if interest rates are 8 percent? (LG5-2) Compute the future value of frequent, level cash flows.

Future Value of Level Cash Flows LG5-2

Now suppose that each cash flow is the same and occurs every year. Level sets of frequent cash flows are common in finance—we call them annuities. The first cash flow of an annuity occurs at the end of the first year (or other time period) and continues every year to the last year. We derive the equation for the future value of an annuity from the general equation for future value of multiple cash flows, equation 5-1. Since each cash flow is the same, and the cash flows are every period, the equation appears as:

FVAN = Future value of first payment

× Future value of second payment + ··· + Last payment

= PMT × (1 + i)N−1 + PMT

× (1 + i)N−2 + PMT × (1 + i)N−3 + ··· + PMT(1 + i)0

The term FVA is used to denote that this is the future value of an annuity. Factoring out the common level cash flow, PMT, we can summarize and reduce the equation as:

annuity A stream of level and frequent cash flows paid at the end of each time period—often referred to as an ordinary annuity.

Suppose that $100 deposits are made at the end of each year for five years. If interest rates are 8 percent per year, the future value of this annuity stream is computed using equation 5-2 as:

We can show these deposits and future value on a time line as:

Five deposits of $100 each were made. So, the $586.66 future value represents $86.66 of interest paid. As with almost any TVM problem, the length of time of the annuity and the interest rate for compounding are very important factors in accumulating wealth within the annuity. Consider the examples in Table 5.1. A $50 deposit made every year for 20 years will grow to $1,839.28 with a 6 percent interest rate. Doubling the annual deposits to $100 also doubles the future value to $3,678.56. However, making $100 deposits for twice the amount of time, 40 years, more than quadruples the future value to $15,476.20! Longer time periods lead to more total compounding and much more wealth. Interest rates also have this effect. Doubling the interest rate from 6 to 12 percent on the 40-year annuity results in nearly a five-fold increase in the future value to $76,709.14.

Finding the Present Value of Several Cash Flows LG5-3

Consider the cash flows that we showed at the very beginning of the chapter: You deposit $100 today, followed by a $125 deposit next year and a $150 deposit at the end of the second year. In the previous situation, we sought the future value when interest rates are 7 percent. Instead of future value, we compute the present value of these three cash flows. The time line for this problem appears as:

The first cash flow is already in year zero, so its value will not change. We will discount the second cash flow one year and the third cash flow two years. Using the present value equation from the previous chapter, the present value of today’s payment is simply $100 ÷ (1 + 0.07)0 = $100. Similarly, the present value of the next two cash flows are $125 ÷ (1 + 0.07)1 = $116.82 and $150 ÷ (1 + 0.07)2 = $131.02, respectively. Therefore, the present value of these cash flows is $347.84 (= $100 + $116.82 + $131.02).

Putting these three individual present value equations together would yield:

PV = [$100 ÷ (1 + 0.07)0] + [$125 ÷ (1 + 0.07)1] + [$150 ÷ (1 + 0.07)2] = $347.84

The general equation for discounting multiple and varying cash flows is:

In this equation, the letters m, n, and p denote when the cash flows occur in time. Each deposit can differ from the others in terms of size and timing.

The five TVM buttons/functions in financial calculators have been fine, so far, for the types of TVM problems we’ve been solving. Sometimes we had to use them two or three times for a single problem, but that was usually because we needed an intermediate calculation to input into another TVM equation.

Luckily, most financial calculators also have built-in worksheets specifically designed for computing TVM in problems with multiple no constant cash flows.

To make calculator worksheets as flexible as possible, they are usually divided into two parts: one for input, which we’ll refer to as the CF (cash flow) worksheet, and one or more for showing the calculator solutions. We’ll go over the conventions concerning the CF worksheet here, and we’ll discuss the output solutions in Chapter 13.

The CF worksheet is usually designed to handle inputting sets of multiple cash flows as quickly as possible. As a result, it normally consists of two sets of variables or cells—one for the cash flows and one to hold a set of frequency counts for the cash flows, so that we can tell it we have seven $1,500 cash flows in a row instead of having to enter $1,500 seven times.

Using the frequency counts to reduce the number of inputs is handy, but you must take care. Frequency counts are only good for embedded annuities of identical cash flows. You have to ensure that you don’t mistake another kind of cash flow for an annuity.

Also, using frequency counts will usually affect the way that the calculator counts time periods. As an example, let’s talk about how we would put the set of cash flows shown here into a CF worksheet:

To designate which particular value we’ll place into each particular cash flow cell in this worksheet, we’ll note the value and the cell identifier, such as CF0, CF1, and so forth. We’ll do the same for the frequency cells, using F1, F2, etc., to identify which CF cell the frequency cell goes with. (Note that, in most calculators, CF0 is treated as a unique value with an unalterable frequency of 1; we’re going to make the same assumption here so you’ll never see a listing for F0.) For this sample timeline, our inputs would be:

To compute the present value of these cash flows, use the NPV calculator function. The NPV function computes the present value of all the future cash flows and then adds the year 0 cash flow. Then, on the NPV worksheet, you would simply need to enter the interest rate and solve for the NPV:

Note a few important things about this example:

1. We had to manually enter a value of $0 for CF3. If we hadn’t, the calculator wouldn’t have known about it and would have implicitly assumed that CF4 came one period after CF2.

2. Once we use a frequency cell for one cash flow, all numbering on any subsequent cash flows that we enter into the calculator is going to be messed up, at least from our point of view. For instance, the first $75 isn’t what we would call “CF5,” is it? We’d call it “CF7” because it comes at time period 7; but calculators usually treat CF5 as “the fifth set of cash flows,” so we’ll just have to try to do the same to be consistent.

3. If we really don’t need to use frequency cells, we will usually just leave them out of the guidance instructions in this chapter to save space.

5-5 Present Value Compute the present value of a $2,000 deposit in year 1 and another $1,500 deposit at the end of year 3 if interest rates are 10 percent. (LG5-3) Discount multiple cash flows to the present.

Finding the Present Value of Several Cash Flows LG5-3

Consider the cash flows that we showed at the very beginning of the chapter: You deposit $100 today, followed by a $125 deposit next year, and a $150 deposit at the end of the second year. In the previous situation, we sought the future value when interest rates are 7 percent. Instead of future value, we compute the present value of these three cash flows. The time line for this problem appears as:

Putting these three individual present value equations together would yield:

PV = [$100 ÷ (1 + 0.07)0] + [$125 ÷ (1 + 0.07)1] + [$150 ÷ (1 + 0.07)2] = $347.84

The general equation for discounting multiple and varying cash flows is:

In this equation, the letters m, n, and p denote when the cash flows occur in time. Each deposit can differ from the others in terms of size and timing.

Luckily, most financial calculators also have built-in worksheets specifically designed for computing TVM in problems with multiple nonconstant cash flows.

Note a few important things about this example:

1. We had to manually enter a value of $0 for CF3. If we hadn’t, the calculator wouldn’t have known about it and would have implicitly assumed that CF4 came one period after CF2.

3. If we really don’t need to use frequency cells, we will usually just leave them out of the guidance instructions in this chapter to save space.

5-7 Present Value of an Annuity what’s the present value of a $900 annuity payment over five years if interest rates are 8 percent? (LG5-4) Compute the present value of an annuity.

Present Value of Level Cash Flows LG5-4

Page 109You will find that this present value of an annuity concept will have many business and personal applications throughout your life. Most loans are set up so that the amount borrowed (the present value) is repaid through level payments made every period (the annuity). Lenders will examine borrowers’ budgets and determine how much each borrower can afford as a payment. The maximum loan offered will be the present value of that annuity payment. The equation for the present value of an annuity can be derived from the general equation for the present value of multiple cash flows, equation 5-3. Since each cash flow is the same, and the borrower pays the cash flows every period, the present value of an annuity, PVA, can be written as:

Suppose that someone makes $100 payments at the end of each year for five years. If interest rates are 8 percent per year, the present value of this annuity stream is computed using equation 5-4 as:

TABLE 5.2 Magnitude of the Annuity, Number of Years Invested, and Interest Rate on PV

The time line for these payments and present value appears as:

Notice that although five payments of $100 each were made, $500 total, the present value is only $399.27. As we’ve noted previously, the span of time over which the borrower pays the annuity and the interest rate for discounting strongly affect present value computations. When you borrow money from the bank, the bank views the amount it lends as the present value of the annuity it receives over time from the borrower. Consider the examples in Table 5.2.

A $50 deposit made every year for 20 years is discounted to $573.50 with a 6 percent discount rate. Doubling the annual cash flow to $100 also doubles the present value to $1,146.99. But extending the time period does not impact the present value as much as you might expect. Making $100 payments for twice the amount of time—40 years—does not double the present value. As you can see in Table 5.2, the present value increases less than 50 percent to only $1,504.63! If the discount rate increases from 6 percent to 12 percent on the 40-year annuity, the present value will shrink to $824.38.

Your firm needs to buy additional physical therapy equipment that costs $20,000. The equipment manufacturer will give you the equipment now if you will pay $6,000 per year for the next four years. If your firm can borrow money at a 9 percent interest rate, should you pay the manufacturer the $20,000 now or accept the 4-year annuity offer of $6,000?

SOLUTION:

We can find the cost of the 4-year, $6,000 annuity in present value terms using equation 5-4:

The cost of paying for the equipment over time is $19,438.32. This is less, in present value terms, than paying $20,000 cash. The firm should take the annuity payment plan.

Similar to Problems 5-7, 5-8, self-test problem 2

Page 110The present value of a cash flow made far into the future is not very valuable today, as Figure 5.2 illustrates. That’s why doubling the number of years in the table from 20 to 40 only increased the present value by approximately 30 percent. Notice how the present value of $100 annuity payments declines for the cash flows made later in time, especially at higher discount rates. The $100 cash flow in year 20 is worth less than $15 today if we use a 10 percent discount rate; they’re worth more than double, at nearly $38 today, if we use a discount rate of 5 percent. The figure also shows how quickly present value declines with a higher discount rate relative to a lower rate. As we showed above, the present values of the annuities in the figure are the sums of the present values shown. Since the present values for the 10 percent discount rate are smaller, the present value of an annuity is smaller as interest rates rise.

Present Value of Multiple Annuities

Just as we can combine annuities to solve various future value problems, we can also combine annuities to solve some present value problems with changing cash flows. Consider Alex Rodriguez’s (A-Rod’s) baseball contract in 2000 with the Texas Rangers. This contract made A-Rod into the “$252 million man.” The contract was structured so that the Rangers paid A-Rod a $10 million signing bonus, $21 million per year in 2001 through 2004, $25 million per year in 2005 and 2006, and $27 million per year in 2007 through 2010.1 Notethatadding the signing bonus to the annual salary equals the $252 million figure. However, Rodriguez would receive the salary in the future. Using an 8 percent discount rate, what is the present value of A-Rod’s contract?

The reported values for many sports contracts may be misleading in present value terms.

FIGURE 5.2 Present Value of Each Annuity Cash Flow

We begin by showing the salary cash flows with the time line:

First create a $27 million, 10-year annuity. Here are the associated cash flows:

Now create a –$2 million, six-year annuity:

Notice that creating the −$2 million annuity also resulted in the third annuity of −$4 million for four years. This time line shows three annuities. If you add the cash flows in any year, the sum is A-Rod’s salary for that year. Now we can find the present value of each annuity using equation 5-4 three times.

Adding the value of the three annuities reveals that the present value of A-Rod’s salary was $158.67 million (= $181.17m − $9.25m − $13.25m). Adding in the $10 million signing bonus produces a contract value of $168.67 million. So, the present value of A-Rod’s contract turns out to be quite considerable, but you might not call him the $252 million man!

5-39 Loan Payments You wish to buy a $25,000 car. The dealer offers you a 4-year loan with a 9 percent APR. What are the monthly payments? How would the payment differ if you paid interest only? What would the consequences of such a decision be? (LG5-9) Compute payments and amortization schedules for car and mortgage loans.

Say that you have your heart set on purchasing a beautiful, old Tudor-style house for $250,000. A mortgage broker says that you can qualify for a mortgage for 80 percent (or $200,000) of the price. If you get a 15-year mortgage, the interest rate will be 6.1 percent APR. A 30-year mortgage costs 6.4 percent. One of the factors that will help you decide which mortgage to take is the magnitude of the monthly payments. What will they be? 5

SOLUTION:

To pay off the mortgage in only 15 years, the payments would have to be larger than for the 30-year mortgage. The higher payment will be eased somewhat because the interest rate is lower on the 15-year mortgage. The payment for the 15-year mortgage is:

The payment for the 30-year mortgage would be:

So, the payments on the 15-year mortgage are nearly $450 more each month than the 30-year mortgage payments. You must decide whether the cost of paying the extra $450 each month is worth it to own the house with no debt 15 years sooner. The decision would depend on your financial budget and the strength of your desire to be debt free.

Similar to Problems 5-39, 5-40, self-test problems 1 and 4

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Using the AMORT Function in TVM Calculators

Page 121imagesTVM calculators have preprogrammed functions to compute the amount of principal paid part way through a mortgage. For example, if you took out a 30-year, $200,000 mortgage at a 6 percent APR, how much principal have you paid after five years? How much do you still owe? How much interest have you paid?

To answer these questions, first enter the mortgage information to compute the monthly payments. Then use the AMORT function. This example uses the Texas Instruments BA II + as an example. The Hewlett-Packard and other TVM calculators have similar functions. The AMORT function allows you to compute the loan balance at any time during the mortgage period. It also computes the amount of principal and interest that has been paid during any time period. To answer the questions above:

1. Press 2nd AMORT and P1 = 1 appears.s (This refers to the first payment of the mortgage.)

2. Press the down arrow ↓, P2 = appears. (This refers to the last payment made.)

3. The question refers to 5 years of payments, which is 60 months. Enter 60 and press ENTER.

The calculator has now computed the loan balance after the 60th payment and the amount of principal and interest that have been paid between the first and the 60th payment.

4. Press the down arrow ↓, displayed is BAL = 186,108.71, which is the loan balance.

5. Press the down arrow ↓, displayed is PRN = 13,891.29, which is the principal paid in the first 5 years.

6. Press the down arrow ↓, displayed is INT = −58,054.78, which is the interest paid in the first 5 years.

Note that in the beginning of a mortgage, far more interest is paid than principal.