“In signing a 10-year, $252 million free-agent contract ...
Chapter 10: Interest Rates and Time Value of Money
Presented To:
Dr. John Banko
Dr. Craig Tapley
Written By:
Meghan Fierko
Shannon Lauer
Daniela Scarpetta
University of Florida
FIN4403 – Honors Finance
November 16, 2009
“In signing a 10-year, $252 million free-agent contract with the Texas Rangers, [Alex] Rodriguez…became the highest-paid player in the history of baseball.” [i] Even though this may have been the highest quoted professional sports contract, is it really worth $252 million today? Sports reporters and news analysts commonly misinterpret the value of a contract by totaling the annual salary payments. However, one must consider the financial effects such as inflation, fluctuating interest rates, and loss of reinvestment opportunities to calculate the real present value of his contract.
According to USA Today[ii], Alex Rodriguez’s (A-Rod’s) contract details are the following (excluding the $10 million signing bonus), as shown in Figure 10.1 below:
Figure 10.1
The fundamental principle of Time Value of Money states that a dollar today is worth more than a dollar tomorrow because of compounding interest effects. Because his salary is paid annually as opposed to a lump sum, A-Rod loses the opportunity to invest his entire signing contract in 2001.
How can we best approximate the present value of A-Rod’s quoted $252 million contract, excluding the $10 million signing bonus?
We first discount the amounts paid every year back to 2001 at a pre-determined interest rate. In this example, we will use a 6% discount rate to calculate the present value of the future payouts. Next, to calculate the present value of the deferred payments, we first need to compound the deferred payments to their payout date using the stated 3% interest rate; according to the contract, the deferred payments will accrue at 3% interest rate starting in 2001, until the date they are paid out. Then, we discount their future value to 2001 using our 6% discount rate.
Now that all payout amounts from 2001-2020 are in PV terms as of 2001, we add these values to find the total present value of the contract. The contract is worth $178,393,182.90 in 2001, a stark contrast to the reported $242 million. This means that sports analysts and reporters overvalued A-Rod’s contract by $63,606,817.10!
This example demonstrates the importance of accurately applying time value of money principles, which we will explore further in this chapter.
Why do people invest money? Since interest rates enable peoples’ money to grow, investors know that a dollar today is worth more than a dollar tomorrow. Interest rates are the cost of money that a lender pays a borrower for their investment. As stated in the previous chapter, interest rates are determined by risk, inflation, changing consumption preferences, and production opportunities. The most basic form of interest rate that exists is the real-risk free rate of interest (i*). This rate is commonly referred to as the short-term rate on a U.S Treasury Security if inflation were non-existent. The real risk-free rate sets a minimum for the interest rate that companies use as the cost of their money. Because companies face factors such as inflation, defaults, and maturity and liquidity risk, the real-risk free rate of interest is only one component of the quoted, or nominal, interest rate that is used to value an investment, as shown in the formula below.
Nominal interest rate =
i* + Inflation Premium + Default Risk Premium + Liquidity premium + Maturity Risk Premium
This nominal interest rate can also be referred to as the annual percentage rate (APR), which we will use in later sections to find the future and present value of an investment.
SECTION 10.1: FUTURE VALUE OF A SINGLE TIME DEPOSIT
Suppose that your uncle deposited $100 in your bank account the day you were born and that the bank paid you an annual simple nominal interest rate of 6% each year. You are now ten years old and would like to withdraw the money from your account. If you made no further deposits, how much money would you have at the end of 10 years? In other words, what is the future value, FV, at year 10 of the original $100 deposit?
Since your account grows by 6% each year, at the end of year 1 you will have $100 plus the additional yearly 6% interest.
(1+ interest rate)*Principal = FV1
(1+.06)*100 = $106 = FV1
As you can see in Figure 10.2, you would earn $6 of interest the first year. Your bank would give another $6 in interest the following year, which means that in two years you have earned an extra $12 just by leaving your money in the bank.
The general formula to find the FV for year n using a simple interest rate, i, is:
Principal + (i*Principal)*n = FVn
100 + (.06*100)*2 = $112 = FV2
Now, assume that this bank offers to pay you an annual compounding nominal interest rate of 6%. How would this effect your investment?
Compounding interest builds upon your accumulated interest, in other words, you are earning interest on interest. Therefore, in this example, in year 2 your account continues to grow by 6% on the already compounded $106. This exponential growth means you have now received $6.36 worth of interest in your second year, already $0.36 more than your previous simple interest accumulation!
In order to see the difference between simple vs. compounding interest rates at each year, reference Figure 10.2 below.
Figure 10.2
[pic]
We can see from the figure above that the FV for investments that have compounding interest is calculated as:
Future Value = (Present Value)(1 + Interest Rate)Number of years
which simplifies to:
FVn = (PV)(1+i)n
In Excel, the function that allows us to determine the future value of an investment using compounding interest rates is as follows:
=FV(rate,nper,pmt,pv,type*)
*For this example we will not use type.
To find the FV at year 7 ($150.36), the Excel function would read:
=FV(.06,7,0,-100) = $150.36 = FV7
Rate = Interest rate, in decimal form, .06
Nper = Number of years, 7
Pmt = Yearly cash inflows, for this example there are none, 0
PV= Initial deposit, -100*
*When solving for PV or FV, both the FV and PMT amounts must be negative, or the PV amount must be negative. In this case, we chose to make our PV negative because it is a cash outflow.
CONTINUOUS COMPOUNDING
Another type of compounding process is continuous compounding. Like compounding, continuous interest also builds on accumulated interest. However, instead of compounding at the end of the year, interest is compounding infinitely. This formula is represented by:
PVcontinuous = Pert
If we were to use the example above, P is our $100 principal, e represents the infinite compounding, and r and t correspond to our interest rate and time, respectively. The letter e stands 2.718281825. Excel represents this number by the EXP function as seen below in Figure 10.3.
Figure 10.3
[pic]
Most banks and institutions do not rely on continuous compounding and instead compound interest monthly, quarterly, semi-annually or annually, which we will explore later in this chapter.
SECTION 10.2: PRESENT VALUE OF A SINGLE TIME DEPOSIT
Suppose your uncle deposited money into your bank account ten years ago and it is now worth $179.08. The bank gave him a 6% annual interest rate but he cannot remember how much he initially deposited. Just like we were able to figure out the future value of our investment, we can discount our future value back 10 years and determine how much your uncle invested at year 0.
We will now be solving for the present value, PV, which is represented by the formula:
[pic]
Using the values from the above example, we see that the initial deposit equals:
[pic]
Present value is important because it can help us decide between two investments or it can help us determine if a bond or stock is over or undervalued.
We can demonstrate the PV concept with a new example:
If a bank offers you $2,000 today or $3,000 four years from now at 5% interest rate, which offer should you accept? Your choice depends on which offer gives you the highest PV as of today. Therefore, we need to discount the $3,000 back four years to compare it to the other offer of $2,000 received today.
Using the PV formula, we find that the PV today of the $3,000 offer equals $2,468.11, [$3,000/(1.05)4]. Consequently, it is better to take the $3,000 investment four years from now because it is worth $468.11 more than getting the $2,000 today ($2,468.11 - $2,000).
If the year in which you receive the $3,000 is paid out in a different year, we can see how the PV differs. Through this process, we can determine the time in which the $3,000 offer is no longer the better deal, as seen in Figure 10.4.
Figure 10.4
[pic]
In Excel, the function that allows us to determine the present value of an investment using the discounting interest rate can be written as:
=PV(rate,nper,pmt,fv,type*)
*For this example we will not be using type.
To find the present value of $3,000 paid out in a specified year, such as in year 3 for the current example, the Excel function would read:
=PV(0.05,3,0,3000) = -$2,591.51 = PV0
Rate= Interest rate in decimal form, 0.05
Nper= Number of years, 3
Pmt= Yearly cash inflows, for this example there are none, 0
FV= The value after last time period that you expect to receive interest for, 3000
Our present value will be negative because Excel requires either FV or PV to be negative (an outflow).
SECTION 10.3: SOLVING FOR N AND I
We are able to solve for variables such as n and i as long as we have the other variables in our present and future value equations. Note that you will be able to use this same principle to find missing variables of the equations for annuities later in the chapter.
For example, if you deposit $2,468.11 today, at what interest rate would your money need to be invested so your original deposit is worth $4,000 at the end of five years?
The Excel function that allows us to solve for this interest rate is:
=RATE(nper,pmt,pv,fv,type,guess*)
*For this example, we will not use type or guess.
Solving the Excel function, we find that your deposit must earn an interest rate of 10.138% for it to be worth $4,000 in five years, as shown in Figure 10.5
=RATE(5,0,-2468.11,4000) = 10.138%
Figure 10.5
[pic]
Now assume that our interest rate declines to 4%. How many years would it take the $2,468.11 investment to grow to $4,000?
The Excel function that allows us to solve for number of years is:
=NPER(rate,pmt,pv,fv,type*)
*For this example we will not use type.
=NPER(.04,0,-2468.11,4000) = 12.311 years
SECTION 10.4: ANNUITIES (PV and FV)
In the previous sections, we examined present and future value calculations of single time deposits. We will now explore the valuation of an investment with multiple cash payments.
An annuity provides a stream of fixed payments across equally spaced periods, allowing investors to obtain a steady flow of income. There are two types of annuities, ordinary and due, which distinguish between the timing of payments. An ordinary, or regular, annuity results when payments are made at the end of the period, while an annuity due makes payments at the beginning of the period. If you invest $100 for the next four years and your first deposit occurs one year from today, your investment would be an ordinary annuity. On the other hand, if you make your first $100 deposit today, your investment would be considered an annuity due.
We see various examples of annuities in everyday life. Mortgage and student loan payments are typically made at the end of the month and are therefore ordinary annuities. Alternatively, rental payments and insurance premiums are usually annuities due, with payments occurring at the beginning of each month.
PRESENT VALUE OF AN ANNUITY:
Finding the PV of an annuity consists of discounting each payment back to the point in time you are solving for, and then summing these present values. The general formula can be represented as follows, where PVAn stands for the present value of an annuity at period n:
[pic]
The PV of an ordinary annuity is valued one period before the first payment, while the PV of an annuity due is valued at the time of the first payment. The following image depicts the points in time in which the present values are calculated for both ordinary annuities and annuities due.
t= 0 1 2 3 4
CF= PMT PMT PMT PMT
PV Ordinary (yr0) PV Due (yr1)
As you can see, if payments start at the end of the first year (t=1), the PV of an ordinary annuity will be calculated at year 0. The PV of an annuity due will be found at year 1, when the first payment occurs. Note the difference between period 1 and year 1. Period 1 represents the time elapsed during the first year, from t= 0 to t = 1, whereas year 1 identifies a specific point in time. In addition, the end of year 0 is the same as the beginning of year 1.
Because we evaluate the present value of an annuity due at the time of the first payment, it has one extra period of compounding interest than an ordinary annuity. Therefore, you can find the PV of an annuity due by compounding the PV of an ordinary annuity forward one period. This concept can be applied when you have determined the PV of an ordinary annuity and want to consider what the PV would be if payments instead occurred at the beginning of every period (annuity due). To find the PV of an annuity due, you would simply compound the PV of the ordinary annuity forward one period.
PV Annuity Due = (PV Ordinary Annuity)(1+i)
For example, if the present value of an ordinary annuity with an interest rate of 5% were $800 at year 0, then the PV of an annuity due at year 1 with the same payment and interest rate would be $800*(1.05), or $840.
Now suppose you will receive $250 each year for the next three years, with the first cash flow beginning one year from now. If the interest rate is 5%, what is the PV of this ordinary annuity today?
First, discount each payment back to year 0 using the stated interest rate. Then sum up these values to determine the PV of the annuity at year 0.
[pic]
[pic]
Therefore, you would be willing to invest $680.81 today in order to receive $250 for the next three years, if you can earn a 5% interest rate.
Let us examine how to solve for the present value of a 10-year annuity in Excel, using the same $250 payment and 5% interest rate.
Figure 10.6
[pic]
As shown in Figure 10.6, the present value of this annuity today is equal to $1,930.43.
A quicker way to solve this problem requires the use of Excel functions. Recall from the previous section that the PV function is:
=PV(rate,nper,pmt,fv,type)
Type distinguishes between an ordinary annuity and an annuity due. For an ordinary annuity, enter 0, and for an annuity due, enter 1. For the ordinary annuity example above, we would enter:
=PV(0.05,10,250,0,0) = PV0 = $1,930.43
Note that this is an ordinary annuity because the first payment occurs at year 1 and we are solving for the present value at year 0. However, if we wanted to know the present value of this annuity at year 1, we could treat this as an annuity due and enter:
=PV(0.05,10,250,0,1) = PV1 = $2,026.96.
The $96.52 difference between the ordinary annuity and annuity due is from the effect of one additional period of interest on the initial $250 payment.
PV Annuity Due = (PV Ordinary Annuity)(1+i)
PV Annuity Due = $1,930.43(1.05) = $2,026.96.
Now that we understand how to find the present value of an annuity, let us examine how to determine the future value of annuities.
FUTURE VALUE OF AN ANNUITY:
The future value of an annuity is found by compounding the payment values to a future date in time. Making the distinction between an ordinary annuity and an annuity due is important for finding present value as well as future value calculations. The FV of an ordinary annuity is found as of the last payment while the FV of an annuity due is found one period after the last payment, as seen below.
t = 0 1 2 3 4
CF = PMT PMT PMT
FV Ordinary (yr3) FV Due (yr4)
The future value of an ordinary annuity can be found by:
[pic]
which simplifies to:
[pic]
In the equation above, n represents the total number of periods of the annuity and t represents the period in which each payment occurs. In an ordinary annuity, compounding each payment into the future at (n-t) recognizes the fact that the future value is determined as of the last payment. Since you will not receive interest after the last payment, this payment does not need to be compounded.
For example, if you receive $250 one year from today at a 5% interest rate, how much is your annuity worth at year 8?
Because we are solving for the FV at the same time as the last cash flow occurs (year 8), this is an ordinary annuity. As shown below in Figure 10.7, the FV at year 8 is $2,387.28.
Figure 10.7
[pic]
You may also use the FV Excel function to determine the future value of an annuity:
=FV(rate,nper,pmt,pv,type)
=FV(0.05,8,-250,0,0) = FV8 = $2,387.28
Note that the payment is negative $250 because it is a cash outflow.
To make this example an annuity due, we solve for the future value at year 9 instead of at year 8 since the FV of an annuity due is found one period after the last payment. We may use the Excel formula above, replacing type 0 with type 1 to indicate an annuity due:
=FV(0.05,8,250,0,1) = FV9 = $2,506.65
We may also solve for FV of an annuity due by compounding the FV of an ordinary annuity forward one period at (1+i).
FV Annuity Due = (FV Ordinary Annuity)(1+i)
FV Annuity Due = (FV8)(1+i) = ($2,387.28)(1.05) = $2,506.65 = FV9
SECTION 10.5: SOLVING FOR PMT, N AND I
As explained in Section 10.4, we can solve for pmt, n, and i if the other variables are known. This allows us to determine how much we need to deposit each period, how long we need to deposit a specified amount, and what interest rate is necessary to achieve some specified goal, respectively.
Solving for Payment:
Suppose you are planning to buy a car for $18,000 in four years. How much should you deposit each year if you can invest your money at a 6% interest rate, assuming your first deposit will occur one year from now?
Using the PMT Excel function, we see that you must deposit $4,114.65 per year to achieve your FV goal of $18,000 in four years.
=PMT(rate,nper,pv,fv,type)
=PMT(0.06,4,0,18000,0) = -$4,114.65
Note that the payment value is negative because it is a cash outflow.
We can confirm this amount by compounding the annual payments of $4,114.65 to year 4 and summing these four FV payments, as seen below in Figure 10.8.
Figure 10.8
[pic]
Continuing this example, assume that in addition to your four yearly payments, your uncle gives you $2,000 towards the purchase of a car. Now what payment is required per year to achieve your target amount of $18,000?
There is now an initial investment to consider when determining how much we should deposit each year. We will use the same Excel PMT function, but will now have a PV of -$2,000. This amount is entered as a negative because we are investing it and it is therefore treated as a cash outflow:
=PMT(0.06,4,-2000,18000,0) = -$3,537.46
You only need to invest $3,537.46 each year because of the initial deposit, compared to the $4,114.65.
SOLVING FOR NUMBER OF YEARS
How long would it take to raise $10,000 if you invested $1,000 each year at an interest rate of 6%, with the first deposit occurring today?
Using the NPER Excel function, we find that it takes approximately 7.70 years:
=NPER(rate,pmt,pv,fv,type)
=NPER(0.06,-1000,0,10000,1) =7.697908 years
Because the first deposit occurs today, enter 1 for type, as this is an annuity due. The pmt of $1,000 is entered as a negative value while the FV of $10,000 is a positive value because your payments are cash outflows and your ending value is an inflow. Your PV is 0 since you did not make an initial investment. This can also be seen in the figure below.
Figure 10.9
[pic]
SOLVING FOR INTEREST RATE:
Now suppose you are planning for a vacation and need $5,000 at the end of three years (year 4). At what interest rate must you invest if you deposit $1,500 each year, starting one year from today?
=RATE(nper,pmt,pv,fv,type,guess*)
* For this example, we will not use guess.
=RATE(3,-1500,0,5000,0) = 0.107275126
The Excel function shows that our deposits must be invested at approximately 10.728% per year in order to achieve our goal of having $5,000 by the beginning of year 4.
SECTION 10.6: UNEVEN CASH FLOWS
Up until this point in the chapter, we have assumed that the annuity payments have been the same amount each period. Realistically though, stocks and other capital investments often produce uneven cash flows. The two main classifications for uneven cash flows are:
1) Fixed series of payments and a lump sum
t = 0 1 2 3 4
CF = $100 $100 $100 $100 $ 1000 $ 1100
2) Different payments for each period
t = 0 1 2 3 4
CF = $100 $200 $300 $400
To find the present value of these types of annuities, we discount each cash flow to the desired time and then sum up the stream of payments. The formula for solving for PV of uneven cash flows is:
[pic]
Using Excel, we can find the PV of each cash flow using the PV function. We then, sum up these values to determine the PV of the uneven cash flow annuity.
Figure 10.10 shows how to calculate the PV for the two different types of uneven cash flows.
Figure 10.10
[pic]
These examples show that our PV in year 0 is equal to $1,000 (example 1) and $754.80 (example 2). When discounting each payment to year 0, we treat each cash flow as the future value for that year, therefore eliminating the need for a payment input.
Now that you understand the concept behind uneven cash flows, the most efficient way of finding the PV is to use the NPV Excel function.
=NPV(Rate,Value1,Value2...ValueN)
For the example in which we have different payments each period (example 2), we would input:
=NPV(.10,100,200,300,400) = $754.80
Just like in previous sections, we can also solve for the future value of the uneven cash flow annuity using the FV function in Excel. First, we find the FV of each cash flow and then, we sum up these values.
Reference the figure below for guidance:
Figure 10.11
[pic]
SECTION 10.7: ANNUITIES WITH PERIODIC PAYMENTS
So far, we have discussed annuities with annual payments. Let us now examine how periodic payments impact annuity calculations.
It is very common for annuities to make non-annual, or periodic, payments. Instead of receiving payments (or making deposits) every year, you can receive payments over a specified period, such as monthly, quarterly, or semi-annually. Whenever payments occur over a non-annual period, you must adjust the interest rate and number of periods over the lifespan of the annuity accordingly. Instead of compounding interest on an annual basis using the nominal interest rate, the interest is now expressed as a fractional interest rate. The resulting periodic rate is the interest rate expressed in terms of the frequency of payments per year.
[pic]
This periodic rate takes our nominal interest rate and matches it to the number of payment periods per year. For instance, an annuity with semiannual payments and a nominal interest rate of 8% would have a periodic rate of 4%. This same annuity with quarterly payments would have a periodic rate of 2%.
Furthermore, when using nper in our Excel PV and FV calculations, it was previously assumed that the number of periods was in annual terms. Now we must express nper in terms of our payment period, since payments are occurring in non-annual intervals. The number of periods can be calculated as:
Number of Periods = (Number of Years) x (Periods per Year)
Therefore, a 4-year annuity with semi-annual payments will have 8 periods. A 5-year annuity with quarterly payments will have 20 periods.
To demonstrate these concepts, assume that you receive $500 every month over the course of two years and can invest these payments at a 6% nominal interest rate. What is the present value of this monthly annuity if you receive the first payment one month from now?
First, we determine how many periods comprise the life of this 2-year annuity. Because payments occur every month and there are 12 months in a year, the total number of periods is 24. Next, find the periodic interest rate. If the annual rate is 6%, the monthly periodic rate is 0.5%, (0.06/12=.005).
Using the Excel PV function, we find that the present value of this monthly annuity at time 0 is $11,281.43, as seen below in Figure 10.12
=PV(0.005,24,500,0,0) = $11,281.43 = PV0
Figure 10.12
[pic]
Now suppose instead of monthly payments, we only receive $500 per quarter for a two-year time span. How does this change the present value of the annuity?
We now have 8 periods and a periodic interest rate of 1.5%, assuming the nominal interest rate remains at 6%. The present value is $3,742.96 at year 0.
=PV(.015,8,500,0,0) = $3,742.96 = PV0
OTHER COMPOUNDING PERIODS
In the previous two examples, we discussed annuities with periodic payments where interest was compounded annually. What happens if interest is compounded more often than once a year and payments occur annually? We will need to use an effective annual interest rate (EAR) instead of the periodic interest rate, to adjust for the effect of compounding.
The general formula for finding the effective annual rate is:
[pic][pic]
[pic]
where m is the number of compounding periods per year and (inom/m) is the periodic rate.
If we invest in a fund that pays $1,000 per year and interest is compounded quarterly, what is the effective annual interest rate? Using the formula above, we find that:
[pic]
Notice that the effective interest rate of 8.24% is higher than the nominal interest rate of 8%. This demonstrates the benefits of compounding. As interest is compounded more frequently, you are earning more interest on interest, which increases the value of your investment.
Excel also has a formula to convert nominal interest rates to effective annual interest rates:
=EFFECT(nominal_rate,npery)
where npery represents the number of compounding periods per year.
Using our same example with quarterly compounding, we would input:
=EFFECT(0.08,4) = 0.08243216 = 8.24%
You can also find effective rates other than the effective annual rate. For these cases, where the payments are periodic and interest is compounded at a non-annual period, use the periodic effective rate formula:
[pic]
In the periodic effective rate equation, m represents the frequency in which interest is compounded per year, and n represents the number of interest compounding periods in one payment period. For instance, if you receive semi-annual payments with quarterly compounding, m would be 4 since interest is compounded quarterly. Further, n is 2 since there are two quarterly compounding periods in one semi-annual payment period. Thus, if your nominal rate is 12%, your effective semi-annual rate with quarterly compounding is 6.09%.
[pic]
This means your payments are compounded 6.09% every 6 months.
SECTION 10.8: GROWING ANNUITIES
There are special cases in which the dollar amounts of an annuity can be expected to increase over time at some growth rate, g. This is called a growing annuity. Growing annuity problems become very easy to solve using Excel.
For example, suppose your sister plans to start college four years from now. She wants to begin saving for her college education today, t=0, so that at the start of year 4, she has enough money to pay for the four years of college. She expects tuition to cost $15,000 at year 4 and to increase at a rate of 5% per year. How much should she deposit each year (t=0 to t=3), if she makes her first payment today? Assume she can invest her money at an 8% nominal rate.
First, we need to determine the PV of the tuition payments. To do this, we need to grow each tuition payment at 5% as seen in column B of Figure 10.13. To find the PV at year 4, when you sister starts college, discount each payment to year 4, as seen in column D. This amount will let you know how much your sister’s deposits must be worth at year 4. To find the cost of college as of today, we discount each tuition payment to year 0, as seen in column E.
Figure 10.13
[pic]
From Figure 10.13, we find that the PV at year 4 of the growing tuition payments equals $57,545.97 and the PV at year 0 equals $42,298.01.
Then, we solve for the amount your sister needs to deposit each year - we can treat this as an annuity due problem. Using Excel’s PMT function, we find that your sister needs to deposit $11,824.68 per year for four years, if her deposits start today.
=pmt(rate,nper,pv,fv,type)
=pmt(0.08,4,-42298.01,0,1) = $11,824.68
Note that this is an annuity due (type=1) because your sister will make her first deposit today.
We can confirm this amount by compounding the four annual deposits of $11,824.68 to year 4 and summing these FV amounts, as seen in Figure 10.14.
Figure 10.14
[pic]
This shows that depositing $11,824.68 for four years, with the first payment occurring today, will result in a FV of $57,545.97 at year 4. This matches with the previously stated assumption that deposits will have to be worth $57,545.97 at year 4 in order for your sister to be able to pay the tuition payments during all four years of college.
SECTION 10.9: PERPETUITIES
Unlike annuities, where payments are made for some finite period, there exists another type of investment, called a perpetuity. A perpetuity pays a stream of equal payments for an infinite period. For example, the British government issues bonds, called consols, that entitle the bondholder to forever receive interest payments annually[iii]. Although this concept may seem illogical at first, perpetuities do have a present value.
[pic]
Perpetuities do not have a future value because they are never-ending. Additionally, the PV that you calculate is the value one period before the first payment. Before we look at real-world perpetuity examples, it is important for you to understand why the formula of the PV of a perpetuity is its payment divided by its interest rate. Since a perpetuity is a series of infinite payments, it has the same properties as an infinite sum series. As you can recall from calculus, infinite sum series are either convergent or divergent. In simplistic terms, convergent infinite sum series is an infinite summation that approaches a number. Divergent sum series is an infinite sum that keeps growing forever. Let us look at some examples.
Suppose you want to find the value of [pic], where t denotes a period. If we expand this equation we see that:
[pic] = (1/2) + (1/4) + (1/8) + (1/16) + (1/32) + … = 1
This is an example of a convergent series because it approaches 1. As time increases, the denominator of the period’s fraction increases, therefore the period’s fraction decreases. To keep our focus within the scope of this textbook, if you are interested on how the value approached by the series is determined, you can consult any calculus textbook and find this concept explained under infinite series.
In a divergent series, such that[pic], the series will be equal to 1 + 2 + 3 + 4 + etc., which equals infinity.
Now that we have refreshed your knowledge on infinite sum series, let us see how this concept relates to perpetuities. Suppose that you will be paid $100 every year forever at an interest rate of 10%. To determine the PV of the perpetuity:
1. Discount each $100 payment back to year 0
2. Add these values together to find the PV of the perpetuity at year 0
Based on our infinite sum series we can write the formula of this perpetuity as:
[pic] = ($100/1.10)1 + ($100/1.10)2 + ($100/1.10)3 +… = $100/.10 = pmt/i= $1,000
As you can see from Figure 10.15 below, the PV of each payment gets smaller as time progresses.
Figure 10.15
After period 48, the PV of these payments is so small that it will not make a significant contribution to the PV of the perpetuity (which is the summation of the PV of each yearly payment). When we add the PV of the payments for years 1-400, we get that the PV of the perpetuity is $999.999999999999. This explains the use of the pmt/i equation that gives us the same $1,000 approximation.
Now that you understand the concept behind the present value of a perpetuity, let us take a look at a real-world example. Suppose that today you purchase a perpetuity that pays you $1,000 per year, beginning next year, with an interest rate of 8%. What is the present value of this perpetuity?
[pic]
Because of the simplicity of the formula, Excel does not have a function for finding the PV of a perpetuity. Therefore, we calculate the PV by dividing the payment by the interest rate, as shown in Figure 10.16.
Figure 10.16
[pic]
What if this perpetuity paid you $1,000 per year starting in year 5? Using the formula above, dividing $1,000 by the 8% interest rate would give you the value of the perpetuity at year 4. Therefore, you would then discount this value back to year 0 using the same 8% interest rate. The PV at year 0 of the perpetuity would be $9,187.87.
The value of a company’s stock can also be found using the perpetuity PV equation. This is because dividends of a stock are expected to be constant and infinite, unless the firm is liquidated or sold to another concern[iv]. Because the only source of cash flows that a stock will provide to its stockholders is dividends, a stock’s value is the PV of its dividend stream. To better understand this concept, assume that you buy a stock today, receive dividends, and expect to sell it at the end of year 1 at P1. How do we decide what we should pay for this stock today, P0? P0 is determined by the PV of the expected dividend streams plus the price of the stock at the end of year 1, P1.This process will continue infinitely for P2 to ∞ and therefore explains why:
[pic]
where Dt is the next period’s dividend and is is the interest rate on the stock
For example, if a stock will pay dividends of $1.50 per year for an infinite period at a 7% interest rate, how much is the stock worth today? The answer is $21.43.
[pic]
SECTION 10.10 GROWING PERPETUITIES
The infinite cash streams of investments will increase, decrease, or remain constant over time.
The infinite sum series explained above also holds true for perpetuities that have constant growth. For example, if starting today you will receive a cash flow, CF, of $100 every year forever, but this cash flow will grow at a rate of 5% per year, g, at an 8% interest rate, i, the infinite sum series will be:
[pic]= $105/1.081 + $110.25/1.082 + … = $105/(.08-.05) = CF1/(i-g)
If we were to recreate Figure 10.15 for constant growth perpetuities, we would see that every period the PV of each cash flow decreases, just as we saw in the PV of the cash flows of non-growth perpetuities. After some period, the PV of the cash flows is so minute, that the infinite sum series of the PV of the cash flows is convergent to a number, CF1/(i-g).
Therefore, when the dividends Dn are set to grow at a constant rate, the PV perpetuity function will be:
[pic]
where D1 = D0(1+g)1
Assume that today you want to purchase a stock that has D1 = $1.10, dividends are growing at 10% per year, and the required rate of return on the stock is 13%. How much should you be willing to pay for this investment today? The answer is $36.67.
[pic] = $36.67
For growing perpetuity investments, the rate of return on the stock should be greater than the growth rate of the dividends, i > g, and the growth rate should be expected to grow forever; if not, the answer will return a negative PV of the stock. In the example above, the dividends are growing at a rate of 10% but are being discounted to time 0 at a rate of 13%. Therefore, as shown in Figure 10.17, the PV of each sequential dividend is decreasing. If we were to sum up all of the PV of the dividends, the answer would be $36.67, which is the price of then stock, P0.
Figure 10.17
[pic]
SECTION 10.11: INSTALLMENT LOANS
Installment Loans are loans in which you make equal payments every period to your lender until it is paid off. However, because interest is being charged on the outstanding balance of the loan, interest will be much higher when you first start paying off then loan. Let us look at an example that uses the Loan Amortization Schedule.
Suppose you have just bought a $45,000 car and will take out an installment loan on this amount. Your lender charges you an annual interest rate of 7.5% per year and the loan needs to be paid off in 15 years (that is, in 180 monthly installments). We can construct a loan amortization table in Excel so that we can see how much interest and principal you will pay every month until the loan matures.
First, we need to calculate the amount you will need to pay each month. Since the payments are to be made monthly, we divide the annual interest rate by 12 months and get a periodic monthly interest rate of 0.625%. We can now input the PMT formula into excel and find that the monthly payments will be $417.16.
=PMT(.00625,180,-4500,0)
Note that the future value of the loan is 0. This is because you expect to pay out the loan by the end of the 15 years.
Now we need to calculate the monthly interest you will pay on the loan. Every month, you will pay interest on the remaining balance of the loan. During the first month, interest will be charged on the entire $45,000 because interest is always charged on the outstanding principal. We expect monthly interest to be .625% (7.5%/12 months). Therefore, interest for the first month is $281.25, which means that $135.91, ($417.16 – $281.25) is the payment amount toward the principal. To find the ending balance at each period, we subtract the amount of principal you paid during the particular period from the beginning balance of that same period.
As shown in Figure 10.17, you can set up a loan amortization table in Excel to see your monthly payments and to see how much interest and principal comprise each payment. From the table you can see that the amount you are paying for interest is decreasing every month whereas the amount of principal that you are paying every month is increasing.
Figure 10.17
[pic]
Using the Loan Amortization Table, it is now very easy to determine how much interest-paid-to-date, how much principal-paid-to-date, and how much remaining balance you have on the loan at any period. In this example, because we take 15 years to pay the loan and interest is being charged on the outstanding principal, we end up paying a total of $75,088.00 to the lender for a $45,000 loan.
Bibliography
Financial Modeling. 2nd ed. Cambridge, 2000. Print.
Fundamentals of Financial Management. 10th ed. Mason: Thomson South-Western, 2007. Print.
Fundamentals of Financial Management. 11th ed. Mason: Thomson South-Western, 2007. Print.
" - Statitudes - Statitudes: A-Rod makes history with huge deal - Tuesday December 19, 2000 12:49 AM." Breaking news, real-time scores and daily analysis from Sports Illustrated. Web. 15 Nov. 2009. .
"Inside Alex Rodriguez's record deal." News, Travel, Weather, Entertainment, Sports, Technology, U.S. & World - . Web. 15 Nov. 2009.
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" - Statitudes - Statitudes: A-Rod makes history with huge deal - Tuesday December 19, 2000 12:49 AM." Breaking news, real-time scores and daily analysis from Sports Illustrated. Web. 15 Nov. 2009. .
-----------------------
[i]
[ii]
[iii]
[iv] Fundamentals of Financial Management, Third Edition. Brigham & Houston.
-----------------------
Annuity Due:
t = 0 1 2 3 4
CF = $100 $100 $100 $100
Ordinary Annuity:
t = 0 1 2 3 4
CF = $100 $100 $100 $100
Year 1
Period 1
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