Welcome to Financial Management. In this lesson we will discuss the time value of money.
The following topics will be covered in this lesson:
Finding the interest rate, I;
Finding the number of years, N;
Future value of an ordinary annuity;
Future value of an annuity due;
Present value of ordinary annuities and annuities due;
Finding annuity payments, periods, and interest rates;
Uneven, or irregular, cash flows;
Future value of an uneven cash flow stream;
Solving for I with irregular cash flows;
Semiannual and other compounding periods;
Fractional time periods;
Amortized loans; and,
Recall, the primary objective of financial management is to maximize the value of the firm’s stock. Moreover, the value of the firm’s stock depends in part on the timing of the cash flows investors expect to receive from investing in the firm. Hence, it is very important that the financial manager have an understanding of the time value of money and how it impacts the firm’s stock price. Time value of money is also referred to as discounted cash flow, or DCF, analysis. As we study this concept it is important to remember that there is no other concept in finance that is more important than time value of money or DCF.
When we analyze time value of money it is important to draw a timeline because this helps us visualize what is happening in a particular problem and helps us solve the problem. Consider the timeline shown on the slide.
Time zero is today;
Time one is one from today, or the end of period one;
Time two is two time periods from today, or the end of period two and so on.
Many times the periods are measured in years, but that is not a requirement. Time can be measured in semiannual periods, quarters, months, or days. Look that time period one. The tick mark at time one represents the end of period one and it also represents the beginning of time two since time one has just passed. Cash flows are placed directly underneath the tick marks. Suppose a lump sum or single amount of cash outflow in the amount of one hundred dollars is invested at time zero. The five percent is the interest rate for each of the three time periods. Look at time period three. At time three the cash flow is unknown. Note that in time periods one and two there are no cash flows and the interest rate is constant for all three time periods.
A dollar today is worth more than a dollar in the future primarily because of inflation. We refer to the value of a dollar today as present value or PV. If we invest money today at some interest rate we refer to the value received in the future as the future value or FV. The process of going from present value to future value is referred to as compounding. I is the interest rate the bank pays on the account each year. INT is the dollar amount of interest earned during the year. We calculate this amount by multiplying the beginning amount by I. Therefore, INT equals PV times I. FV sub N is the future value, or ending amount, in the account at the end of N years. PV is the value today but FV sub N is the value N years in the future after the interest earned is added to the account.
There are four ways we can solve this problem. First we can use a step by step approach. This method requires that we calculate the future value for each year and then sum the results.
The second method we can use is called the formula approach. The formula approach uses a mathematical equation to solve time value of money problems. In general, FV sub N equals PV times one plus I raised to the Nth power.
Next we can use a financial calculator to solve time valued money problems. Financial calculators have five keys corresponding to the five variables in the time value equations:
Specifically, N is the number of time periods;
I divided by YR is the interest rate per period;
PV is the present value and since we began by making a deposit this number is an outflow and must have a minus sign in front of it;
PMT is the payment –this key is used only if there is a series of equal payments; in our example this value should be entered as a zero; and
FV is future value which is automatically determined by the calculator.
The last method we can use to solve a time value of money problem is an Excel spreadsheet. To calculate future value we locate the FV function which is given by FV. This formula calculates the FV. We can set up this formula by using either numbers or cell references from an Excel spreadsheet and the results are the same. Using spreadsheets to solve a time value money problems has two advantages over the other methods. First, it is easy to verify the inputs. Second the analysis is more transparent.
Recall when interest is earned on interest earned in prior periods it is referred to as compound interest. If instead interest is earned solely on the principal it is referred to as simple interest. Mathematically with simple interest is total interest is given by PV times I times N or principal times interest times the number of time periods. Then the future value equals PV plus PV times I times N.
Present value is the opposite of future value. To see that this is true consider the following example. Assume we have money to invest and a broker offers to sell us a bond that pays one hundred fifteen dollars and seventy-six cents in three years. Assume that banks offer a three year certificate of deposit, or CD, at five percent and if you don’t purchase the bond you’ll purchase the CD. The five percent paid on the CD is called the opportunity cost or the rate of return we would earn on a different investment of similar risk. We want to know how much we should pay for the bond today. To determine this amount we must calculate the present value which means we are discounting a future sum.
Recall, to find FV we use the formula FV sub N equals PV times the quantity one plus I raised to the Nth power. To find the PV we rearrange the formula and find that PV equals FV sub N divided by the quantity one plus I raised to the Nth power. We know FV sub N equals one hundred fifteen dollars and seventy-six cents and I equals five percent.
Then PV equals one hundred fifteen dollars and seventy-six cents divided by one point zero five cubed which equals one hundred dollars.
This amount is referred to as the fair price of the bond. If we could purchase the bond for less than one hundred dollars we should buy the bond instead of the CD. If we must pay more than one hundred dollars for the bond we should purchase the CD. If the price of the bond is exactly one hundred dollars we are in different between the bond and the CD.
The one hundred dollars is the present value of one hundred fifteen dollars and seventy-six cents due in three years when the interest rate is five percent. It is important to remember that time value of money problems can be solved using more than one method. Additionally always keep in mind that the goal of financial management is to maximize the company’s intrinsic or fundamental value. This value is the present value of the firm’s expected future cash flows.
Finding the Interest Rate, I
So far we’ve calculated FV and PV. But notice that the equation has four variables. If we know the values of three of the variables we can easily calculate the fourth. Note that the variables are PV, FV, I, and N. Suppose we know the values for TV, FV and N and we want to find I. How do we do this?
Now we know PV, FV, N, and must determine I. To calculate I we solve the following equation: FV equals PV times the quantity one plus I raised to the Nth power. We should use either a financial calculator or the RATE function in Excel to solve the problem since any other method would prove to be very difficult and very time-consuming.
Finding the Number of Years, N
Now suppose we have five hundred thousand dollars to invest when the interest is four point five percent. We want to calculate how long it will take five hundred thousand dollars to accumulate to one million dollars. To determine N we solve the following the equation:
One million dollars equals five hundred thousand dollars times the quantity one plus zero point zero four five raised to the Nth power. We can solve for N by using a financial calculator, the NPER function in Excel or by working with natural logarithms using natural logarithms. Regardless of the method used, the result is the same.
An annuity is a series of equal payments made at fixed intervals. If the payments are made at the end of each period the annuity is called an ordinary annuity or deferred annuity. If the payments are made at the beginning of each period the annuity is called an annuity due. In finance ordinary annuities are more common than annuities due. It is important to observe that in the case of an annuity due, each payment is shifted back one time period.
Check Your Understanding
Future Value of an Ordinary Annuity
Suppose we have an ordinary annuity where we deposit one hundred dollars at the end of each year for three years and earn five percent per year. We want to calculate the future value of the annuity or FVA sub N. To solve for FVA sub N we can use a step by step formula approach, a financial calculator or the FV function in Excel. If we use the step by step approach we set up the problem in the following way:
FVA sub N equals PMT times the quantity one plus I raised to the N minus one power plus PMT times the quantity one plus I raise to the N minus two power plus PMT times a quantity one plus I raised to the N minus three power.
This equation tells us that the first payment earns interest for two periods, the second for one period, and the third earns no interest because the payment is made at the end of the annuity’s life. It follows that FVA sub N equals one hundred dollars times one point zero five squared plus one hundred dollars times one point zero five plus one hundred dollars which equals three hundred fifteen dollars and twenty-five cents. In general, the future value of an annuity is given by FVA sub N equals PMT times the quantity one plus I raised to Nth power divided by I minus one divided by I.
Future Value of an Annuity Due
The future value of an annuity due is larger than that of an ordinary annuity because in the case of an annuity due payments are made at the beginning of each time period and for this reason each payment occurs one period earlier and therefore the payment earns interest for one additional period. These types of problems are solved by using either a financial calculator where we set the calculator to begin mode or the FV function in Excel where we set Type equal to one.
Additionally, FVA sub due equals FVA sub ordinary times the quantity one plus I.
Present Value of Ordinary Annuities and Annuities Due
To calculate the present value of an annuity, with PVA sub N we can use the step by step approach, the formula approach, a financial calculator, or the spreadsheet method. Let’s look at the present value of an ordinary annuity. The PV of an ordinary annuity can be written as PVA sub N equals PMT times the quantity one divided by I minus I divided by I times the quantity one plus I raised to the Nth power. Additionally, we can use a financial calculator or the PV function in Excel to solve this problem.
If instead we want to calculate PVA sub due we can use the following formula:
PVA sub due equals PVA sub ordinary times one plus I. We use this formula because each payment occurs one period earlier.
Finding Annuity Payments, Periods, and Interest Rates
Assume we need ten thousand dollars in five years. If we earn six percent interest per year on our money. How much must we deposit to earn this amount? In other words, we need to calculate PMT. We know that FV equals ten thousand dollars, PV equals zero, N equals five and I equals six percent. We can use either a financial calculator or the PMT function in Excel to solve this problem. In the case of an ordinary annuity we would need to deposit seventeen hundred seventy three dollars and ninety-six cents per year. In the case of an annuity due we would need to deposit sixteen hundred seventy-three dollars and fifty-five cents at the beginning of each year.
Continuing with our example, assume we need ten thousand dollars and decide to make end of year deposits but can only deposit twelve hundred dollars per year. Assuming we earn six percent per year how long would it take to accumulate ten thousand dollars?
In this case it is not advisable to use the step by step approach since it would require a trial and error procedure to determine N or I for that matter. Hence, we should use either a financial calculator or the NPER function in Excel. It turns out that N equals six point nine six years. If instead we make deposits at the beginning of each time period and equals six point six three years.
Now assume we save twelve hundred dollars annually but need ten thousand dollars in five years. We need to calculate the rate of return we have to earn in order to achieve our goal. In this case we should use either a financial calculator or the RATE function in Excel. It turns out we would have to earn twenty-five point seventy-eight percent on our deposits to accumulate ten thousand dollars by the end of five years!
A perpetuity is a bond that promises to pay interest forever. Sometimes perpetuities are called consols. To find the PV of a perpetuity we use the following formula: PV for a perpetuity equals PMT divided by I. Assume a consol or perpetuity pays twenty-five dollars per year and the going interest rate is two point five percent. In this case the original value or present value of the consol is given by twenty-five dollars divided by zero point zero two five which equals one thousand dollars. What happens to the original value of the console if the interest rate increases to five point two percent?
Now the PV of the perpetuity equals twenty-five dollars divided by zero point zero five two which equals four hundred eighty dollars and seventy-seven cents. If instead, the interest rate drops to two percent the present value of the consol is twenty-five dollars divided by zero point zero two which equals one thousand two hundred fifty dollars. These examples illustrate a very important point about the relationship between bonds and interest rates. Specifically there is an inverse relationship between the price of outstanding bonds and interest rates. Hence if interest rates rise, the price of outstanding bonds decline and if interest rates decline the price of outstanding bonds increases. This rule holds true for both consols and bonds with finite maturities.
Uneven, or Irregular, Cash Flows
Recall the definition of an annuity requires that the payments are identical over a given number periods. Many times financial decisions involve uneven or irregular cash flows. When we work with uneven or irregular cash flows we label them CF sub t where t denotes the period in which the cash flow occurs. There are two types of uneven cash flows that are important in finance. The first is one in which the cash flows stream is composed of a series of annuity payments plus a lump sum paid in year N. A bond is an example of this type of uneven cash flow. The second is one in which all the cash flows are uneven. Stocks and capital investments are examples of this type of uneven cash flow.
To solve problems in which we have an annuity payment plus a lump sum we use the following formula:
PV equals summation t equal one to T CF sub T divided by the quantity one plus I raised to the tth power.
Solving problem like these is a two-step process. First we calculate the present value of the annuity. Then, we calculate the present value of the final payment. Last we add these numbers together to find a present value of the income stream. To calculate these values we can use a financial calculator or the PV function in Excel.
In cases where the cash flows are all uneven, we can use a step by step approach, a financial calculator, or the NPV function in Excel. If we use a financial calculator we must remember that the cash flows must be entered into the cash flow register in order to solve the problem.
Future Value of an Uneven Cash Flow Stream
Now let’s look at how to calculate the future value of stream of uneven cash flows. Sometimes this value is referred to as the terminal or horizon value. We calculate it by compounding each payment to the end of the term and then adding them together. The mathematical equation we use to calculate the future value has the form FV equals summation from t equals zero to N CF sub t times the quantity one plus I raised to the N minus t power.
Alternatively, we can use a financial calculator or Excel.
If we use Excel, calculating the FV is a two-step process. First we use the NPV function to calculate NPV. Second, we use the FV function to compound the NPV to obtain the future value.
Solving for I with Irregular Cash Flows
Now let’s look at how to determine I if we know the values of the other inputs. If we have an annuity plus a lump sum it’s easy to determine I. However it is considerably more difficult to determine I if we have irregular or uneven cash flows. When all cash flows are irregular or uneven we use a financial calculator the internal rate of return, or IRR function in Excel to solve this problem. Using a financial calculator requires that we enter the cash flows into the cash flow register and press the IRR key to obtain the value for I. This is also called the rate of return on the investment. Additionally it is important to remember that the initial investment at t equals zero must be entered as a negative number since it is a cash outflow.
Semiannual and Other Compounding Periods
Up to this point we assumed that interest has compounded annually. This is referred to as annual compounding. Assume we deposit one hundred thousand dollars into a bank account.
The interest paid on the deposit is six percent but it is paid every six months. This is referred to as semiannual compounding. If we leave the funds in the account how much will we have at the end of year one?
Since the bank pays six percent interest we receive sixty dollars at the end of one year. We receive thirty dollars at the end of six months and another thirty dollars at the end of the year. With semiannual compounding we earn interest on the first thirty dollars during the second six month period. For this reason, the total amount of interest earned is more than sixty dollars. Interest can also be paid quarterly, monthly, weekly or daily. It is very important to understand nonannual compounding because many financial instruments pay or charge interest on a nonannual basis.
If interest is not compounded on an annual basis we must deal with four types of interest rates, namely, nominal annual rates, I sub NOM, annual percentage rates, APR, periodic rates, I sub per, and effective annual rates, EAR or EFF percent. The nominal or quoted rate, I sub NOM is the rate quoted by bankers, brokers, and other financial institutions. Additionally, when the nominal rate is quoted it must include the number of confounding periods per year. The nominal rate is never shown on a timeline, nor is it entered into a financial calculator unless compounding occurs only once per year.
The periodic rate, I sub PER is the rate charged by a lender or paid by a borrower each period. We calculate the periodic rate using the formula I sub PER equals I sub NOM divided by M where I sub NOM is the nominal annual rate and M is the number of compounding periods per year. Hence, a six percent nominal rate with semiannual payments yields a periodic rate of I sub PER equals zero point zero six divided by two which equals zero point zero three. The periodic rate is the rate shown on timelines and used in calculations.
The effective annual rate EAR or EFF percent is the annual rate that yields the same result as compounding at the periodic rate for M times per year. This rate is determined using the following equation: EAR equals EFF percent equals the quantity one plus I sub NOM divided by M raised to the M power minus one where I sub NOM divided by M is the periodic rate and M is the number of periods per year. The EFF percent is used to compare the effect of costs on loans or rates of return on investments when the payment periods are different. They’re rarely used in calculations.
Fractional Time Periods
So far we’ve assumed that payments occur either at the beginning or at the end of the time periods but not within the time periods.
Solving these types of problems is three-step process.
First, we calculate the periodic rate which yields the interest rate paid per day.
Second, we calculate the number of days the money will be invested.
Last, we calculate the final value.
A very important application of compound interest is in the case of installment loans which are paid overtime. These loans are repaid in equal amounts on a monthly quarterly or annual basis and are referred to as amortized loans. Problems like these require that we determine PMT and we solve them by using either a financial calculator or the PMT function in Excel. Each payment is broken into two parts, that part which is interest and the second part which is a repayment of principal. This breakdown is typically shown in an amortization schedule.
A growing annuity is a series of payments that grows a constant rate. One example of a growing annuity is a situation, in which an individual wants to determine the maximum constant real or inflation-adjusted withdrawals he or she can make over a given number of years. There are two ways in which to solve this problem. First we can set up a spreadsheet in Excel’s Goal Seek function which is found under the What If tab in the program. Second, we can use a financial calculator. If we use a financial calculator we must first calculate the expected real rate of interest. Using the real rate of interest we can solve an annuity due problem. There’s a third method which we could use to solve this problem however it is very complicated and time consuming to use. The preferred method is either Excel or a financial calculator.
Suppose instead we want to accumulate a certain sum over given time period. We plan to make a deposit at time zero and then made nine more payments at the beginning of each of the next nine years. If we know the interest rate earned on the deposit and the expected inflation rate, we can calculate the real rate of interest and the amount of the initial deposit. In this case it is easier to use a financial calculator to solve the problem. The key is to remember that all variables must be expressed in real not nominal terms.
Check Your Understanding
We have now reached the end of this lesson. Let’s review what we’ve covered.
First, we identified that the time value of money is an extremely important concept in the field of financed. This principle was demonstrated in the concept of timelines.
Next, we continued with time value of money through an discussion on future values. Because of inflation, a dollar today is worth more than a dollar in the future. This lead to presenting four methods we can use to solve time value of money problems.
Then, we defined present value as the opposite of future value. To demonstrate we determined the possible amount on a bon by calculating the present value and allowing for discounting the future sum. This followed with identifying the interest rate and number of years and how they affect the future value of money.
Also, we defined an annuity as a series of equal payments made at fixed intervals. If the payments are made at the end of each period the annuity is called an ordinary annuity or deferred annuity.
Next, we discussed the future value and present value of ordinary and an annuity due. As an example, the future value of an annuity due is larger than that of an ordinary annuity because in the case of an annuity due payments are made at the beginning of each time period and for this reason each payments occurs one period earlier and therefore the payment earns interest for one additional period.
Then, we defined perpetuities as a bond that promises to pay interest forever. At times, perpetuities are called consols. We examined some examples that illustrate a very important point about the relationship between bonds and interest rates.
Also, we covered uneven or irregular cash flows. Many times financial decisions involve uneven or irregular cash flows. Solving problems related to cash flows we utilize several possible formulas. This included calculations for future value and solving for I with irregular cash flows.
Next, we learned about semiannual and other compounding periods. This followed with solving problems in fractional time periods.
Finally, we discussed amortized loans and growing annuities. Amortized loans are a very important application of compound interest is in the case of installment loans which are paid overtime. These loans are repaid in equal amounts on a monthly quarterly or annual basis. Growing annuities are a series of payments that grows a constant rate. Excel provides a Goal Seek function as way to work through problems related to growing annuities.
This concludes this lesson.
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