Chapter6: DISCOUNTED CASH FLOW VALUATION
6.2 Valuing Level Cash Flows: Annuities and Perpetuities
We will frequently encounter situations in which we have multiple cash flows that are all the same amount. For example, a common type of loan repayment plan calls for the borrower to repay the loan by making a series of equal payments over some length of time. Almost all consumer loans (such as car loans) and home mortgages feature equal payments, usually made each month.
More generally, a series of constant or level cash flows that occur at the end of each period for some fixed number of periods is called an ordinary annuityA level stream of cash
flows for a fixed period of time.; more correctly, the cash flows are said to be in ordinary
annuity form. Annuities appear frequently in financial arrangements, and there are some useful shortcuts for determining their values. We consider these next. p. 153
PRESENT VALUE FOR ANNUITY CASH FLOWS
Suppose we were examining an asset that promised to pay $500 at the end of each of the next three years. The cash flows from this asset are in the form of a three-year, $500 annuity. If we wanted to earn 10 percent on our money, how much would we offer for this annuity?
From the previous section, we know that we can discount each of these $500 payments back to the present at 10 percent to determine the total present value:
This approach works just fine. However, we will often encounter situations in which the number of cash flows is quite large. For example, a typical home mortgage calls for monthly payments over 30 years, for a total of 360 payments. If we were trying to determine the present value of those payments, it would be useful to have a shortcut. Because the cash flows of an annuity are all the same, we can come up with a handy variation on the basic present value equation. The present value of an annuity of C dollars
per period for t periods when the rate of return or interest rate is r is given by:
The term in parentheses on the first line is sometimes called the present value interest
factor for annuities and abbreviated PVIFA(r, t).
The expression for the annuity present value may look a little complicated, but it isn't difficult to use. Notice that the term in square brackets on the second line, 1/(1 + r)t, is the
same present value factor we've been calculating. In our example from the beginning of this section, the interest rate is 10 percent and there are three years involved. The usual present value factor is thus:
To calculate the annuity present value factor, we just plug this in:
Just as we calculated before, the present value of our $500 annuity is then:
After carefully going over your budget, you have determined you can afford to pay
$632 per month toward a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow?
To determine how much you can borrow, we need to calculate the present value of
$632 per month for 48 months at 1 percent per month. The loan payments are in ordinary annuity form, so the annuity present value factor is:
With this factor, we can calculate the present value of the 48 payments of $632 each
Therefore, $24,000 is what you can afford to borrow and repay.
Just as there are tables for ordinary present value factors, there are tables for annuity factors as well. Table 6.1 contains a few such factors; Table A.3 in the appendix to the book contains a larger set. To find the annuity present value factor we calculated just before Example 6.5, look for the row corresponding to three periods and then find the column for 10 percent. The number you see at that intersection should be 2.4869 (rounded to four decimal places), as we calculated. Once again, try calculating a few of these factors yourself and compare your answers to the ones in the table to make sure you know how to do it. If you are using a financial calculator, just enter $1 as the payment and calculate the present value; the result should be the annuity present value factor. TABLE 6.1 Annuity Present Value Interest Factors
Annuity Present Values
To find annuity present values with a financial calculator, we need to use
the key (you were probably wondering what it was for). Compared to
finding the present value of a single amount, there are two important
differences. First, we enter the annuity cash flow using the key.
Second, we don't enter anything for the future value, . So, for
example, the problem we have been examining is a three-year, $500
annuity. If the discount rate is 10 percent, we need to do the following
(after clearing out the calculator!):
As usual, we get a negative sign on the PV.
Annuity Present Values
Using a spreadsheet to find annuity present values goes like this:
Finding the Payment
Suppose you wish to start up a new business that specializes in the latest of health food trends, frozen yak milk. To produce and market your product, the Yakkee Doodle Dandy, you need to borrow $100,000. Because it strikes you as unlikely that this particular fad will be long-lived, you propose to pay off the loan quickly by making five equal annual payments. If the interest rate is 18 percent, what will the payment be? In this case, we know the present value is $100,000. The interest rate is 18 percent, and there are five years. The payments are all equal, so we need to find the relevant annuity factor and solve for the unknown cash flow:
Therefore, you'll make five payments of just under $32,000 each.
Finding annuity payments is easy with a financial calculator. In our yak
example, the PV is $100,000, the interest rate is 18 percent, and there are five years. We find the payment as follows:
Here, we get a negative sign on the payment because the payment is an
outflow for us.
Using a spreadsheet to work the same problem goes like this:
You ran a little short on your spring break vacation, so you put $1,000 on your credit
card. You can afford only the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000?
What we have here is an annuity of $20 per month at 1.5 percent per month for some
unknown length of time. The present value is $1,000 (the amount you owe today). We need to do a little algebra (or use a financial calculator):
At this point, the problem boils down to asking, How long does it take for your money to quadruple at 1.5 percent per month? Based on our previous chapter, the answer is about 93 months:
It will take you about 93/12 = 7.75 years to pay off the $1,000 at this rate. If you use a
financial calculator for problems like this, you should be aware that some automatically round up to the next whole period.
Finding the Number of Payments
To solve this one on a financial calculator, do the following:
Notice that we put a negative sign on the payment you must make, and we have solved
for the number of months. You still have to divide by 12 to get our answer. Also, some financial calculators won't report a fractional value for N; they automatically (without telling you) round up to the next whole period (not to the nearest value). With a
spreadsheet, use the function =NPER(rate,pmt,pv,fv); be sure to put in a zero for fv and to enter ?20 as the payment.
Finding the Rate
The last question we might want to ask concerns the interest rate implicit in an annuity. For example, an insurance company offers to pay you $1,000 per year for 10 years if you will pay $6,710 up front. What rate is implicit in this 10-year annuity? In this case, we know the present value ($6,710), we know the cash flows ($1,000 per year), and we know the life of the investment (10 years). What we don't know is the discount rate:
So, the annuity factor for 10 periods is equal to 6.71, and we need to solve this equation for the unknown value of r. Unfortunately, this is mathematically impossible to do directly. The only way to do it is to use a table or trial and error to find a value for r.
If you look across the row corresponding to 10 periods in Table A.3, you will see a factor of 6.7101 for 8 percent, so we see right away that the insurance company is offering just about 8 percent. Alternatively, we could just start trying different values until we got very close to the answer. Using this trial-and-error approach can be a little tedious, but
1fortunately machines are good at that sort of thing.
To illustrate how to find the answer by trial and error, suppose a relative of yours wants to borrow $3,000. She offers to repay you $1,000 every year for four years. What interest rate are you being offered?
The cash flows here have the form of a four-year, $1,000 annuity. The present value is $3,000. We need to find the discount rate, r. Our goal in doing so is primarily to give you a
feel for the relationship between annuity values and discount rates.
We need to start somewhere, and 10 percent is probably as good a place as any to begin. At 10 percent, the annuity factor is:
The present value of the cash flows at 10 percent is thus:
You can see that we're already in the right ballpark.
Is 10 percent too high or too low? Recall that present values and discount rates move in opposite directions: Increasing the discount rate lowers the PV and vice versa. Our present value here is too high, so the discount rate is too low. If we try 12 percent, we're almost there:
We are still a little low on the discount rate (because the PV is a little high), so we'll try 13 percent:
This is less than $3,000, so we now know that the answer is between 12 percent and 13 percent, and it looks to be about 12.5 percent. For practice, work at it for a while longer and see if you find that the answer is about 12.59 percent.
To illustrate a situation in which finding the unknown rate can be useful, let us consider that the Tri-State Megabucks lottery in Maine, Vermont, and New Hampshire offers you a choice of how to take your winnings (most lotteries do this). In a recent drawing, participants were offered the option of receiving a lump sum payment of $250,000 or an annuity of $500,000 to be received in equal installments over a 25-year period. (At the time, the lump sum payment was always half the annuity option.) Which option was better?
To answer, suppose you were to compare $250,000 today to an annuity of $500,000/25 = $20,000 per year for 25 years. At what rate do these have the same value? This is the same type of problem we've been looking at; we need to find the unknown rate, r, for a
present value of $250,000, a $20,000 payment, and a 25-year period. If you grind through the calculations (or get a little machine assistance), you should find that the unknown rate is about 6.24 percent. You should take the annuity option if that rate is attractive relative to other investments available to you. Notice that we have ignored taxes in this example, and taxes can significantly affect our conclusion. Be sure to consult your tax adviser anytime you win the lottery.
Finding the Rate
Alternatively, you could use a financial calculator to do the following:
Notice that we put a negative sign on the present value (why?). With a
spreadsheet, use the function =RATE(nper,pmt,pv,fv); be sure to put in a
zero for fv and to enter 1,000 as the payment and ?3,000 as the pv.
FUTURE VALUE FOR ANNUITIES
On occasion, it's also handy to know a shortcut for calculating the future value of an annuity. As you might guess, there are future value factors for annuities as well as present value factors. In general, here is the future value factor for an annuity:
To see how we use annuity future value factors, suppose you plan to contribute $2,000 every year to a retirement account paying 8 percent. If you retire in 30 years, how much will you have?
The number of years here, t, is 30, and the interest rate, r, is 8 percent; so we can
calculate the annuity future value factor as:
The future value of this 30-year, $2,000 annuity is thus:
Future Values of Annuities
Of course, you could solve this problem using a financial calculator by doing the following:
Notice that we put a negative sign on the payment (why?). With a
spreadsheet, use the function =FV(rate,nper, pmt,pv); be sure to put in a zero for pv and to enter ?2,000 as the payment.
Sometimes we need to find the unknown rate, r, in the context of an annuity future value.
For example, if you had invested $100 per month in stocks over the 25-year period ended December 1978, your investment would have grown to $76,374. This period had the worst
stretch of stock returns of any 25-year period between 1925 and 2005. How bad was it?