Discounted Cash Flow Theory

How much money must I invest? How much money will I make? When do these “cash flows” happen? The immediate concern of an individual considering an investment is to judge how attractive that investment is. In other words, he would like to “measure” the attractiveness of this particular investment compared to other opportunities he may have now or in the future. There are many such measures which have been developed to respond to this need. Some are peculiar to the field of Real Estate. For instance, if a salesman says that a property is selling for five times its gross income and the investor knows that most of the comparable property is going for seven times, then he might make an internal judgment that the property is a good investment. However, the investment value of the property would also depend on the quality of the financing available, deferred maintenance, and the level of expenses, to mention but a few factors. In other words, the Gross Income Multiplier is an incomplete and inaccurate measure of an investment because it does not consider all of the relevant factors, and it does not weigh those factors properly.

Another weakness of measures which are limited to Real Estate is that it is extremely difficult to compare them with investments in other fields. For instance, how can you compare an apartment house selling at five times gross income with a ten year bond with a nine percent coupon selling at a ten percent discount from face? For this reason, financial analysts have developed several methods of measuring investments which do not depend on the particular type of investment involved.

While there are many measures which have been devel- oped over time, virtually all of them are based on the answers to the three simple questions shown here which, in a financial sense, are the most important things an investor is concerned with. The first two of these ques- tions are the most commonly asked, but the last question is typically the most important. Consider an investment of $1,000 which returns $1,100 in one year. Most people would say that this investment has a 10% “rate of return”. However, if the time period is expanded from one to ten years, that 10% rate of return vanishes, and we are left with a rate of return under one percent. In other words, while the answers to the first two questions are the same in this case, the answer to the third question makes all the difference. For this reason, financial analysts and sophisticated investors have come to rely on investment measures known in the literature of the field as the “Net Present Value” the “Internal Rate of Return”, and the “Modified Internal Rate of Return”. These measures are based, in turn, on a technique known as “Discounted Cash Flow Analysis”.

The basic premise of Discounted Cash Flow Analysis is that the value of money is related to time. That is, a dollar in hand today is worth more than a dollar which is received one year from now. For instance, the investor could take the dollar he has today and put it in a savings account at six percent interest. One year from now he would have $1.06 in the bank. In other words, a dollar today is worth $1.06 one year from now. Expressing this another way, the “Present Value” of a dollar one year from now is $.9434 discounted at 6%, since an investor placing $.9434 in the bank at 6% would have a dollar in the bank at the end of a year. (The 0.9434, or 1 divided by 1.06, is known as the “Present Value Discount Factor”.)

This concept of Present Value is most useful, since it enables us to express the value of money received in the future in terms of today’s dollars. For example, in the investment returning $1,100 one year from now, the Present Value of $1,100 discounted at 6% is $1,100 divided by 1.06, or $1,038. Since this is greater than the $1,000 investment, the investor would be better off by making the investment than by taking the alternative of putting his money in the bank at 6%.

This leads to the concept of Net Present Value. If we subtract the $1,000 investment from the $1,038 Present Value of the future cash flow, the difference, or Net Present Value, is $38. Since this difference is positive, we know that receiving $1,100 a year from now is better than investing the $1,000 in the bank at 6%. Another way to interpret the Net Present Value is as follows: the investor could afford to pay $38 more than the $1,000 for this investment and still make 6% interest on his money. All the Present Values shown in planEASe analyses are actually Net Present Values. For this reason, a positive Net Present Value in the analysis means that the particular stream of cash flows is attractive, as compared with other investments which earn interest at the discount rate.

A major difficulty with using Net Present Values in order to make investment decisions is determining what discount rate to use in the calculations. Theoretically, the proper discount rate is the rate at which alternative investments may be made. Thus, if a savings account is the investor’s alternative, the six percent discount rate may be appropriate. Other investors may feel that they have different alternatives, however. For this reason, the individual investor’s discount rate is an assumption in the analysis so that it may be varied for each investor. This difficulty in determining the proper discount rate is eliminated, however, when the investor uses the Internal Rate of Return to evaluate the investment.

It is a small step from Net Present Values to the Internal Rate of Return for an investment. In the $1,000 investment example, we used a 6% discount rate to obtain the $1,038 Present Value of the future cash receipts. If we had used an 8% discount rate, the corresponding Present Value would have been $1,019, and the Net Present Value would have been a positive $19. At a 10% discount rate, the Present Value of $1,100 is exactly $1,000, so the Net Present Value of the investment is exactly zero.

The particular discount rate which gives us a Net Present Value of zero is called the “Internal Rate of Return.” The meaning of this number may be expressed in many ways, but the most useful definition for our purposes is as an effective interest rate. For example, if you had placed $1,000 in a savings account and removed $1,100 from the account a year later and the account had no more money in it, then the bank would have paid you 10% interest compounded annually on your money. A calculation quite similar to this is used to compute the Annual Percentage Rate (APR) for loans, the yield to maturity for bonds, and the annual yield for savings certificates. This is what makes the Rate of Return so useful as an investment measure: it may easily be compared to alternative investments because the rates of return for those investments are typically expressed in the same terms.

The foregoing introduction to the concepts of Rates of Return and Net Present Values is not meant to be complete, but rather is intended to provide sufficient background in the subject to enable you to use these measures to judge the attractiveness of the investment being analyzed. If you are interested in delving further into these concepts, there is a substantial body of literature dealing with this subject in the areas of business economics and finance.

This box shows the formula for computing Net Present Values in planEASe. planEASe obtains the Internal Rate of Return for cash flow streams by finding the discount rate (r) in this formula which causes the formula to be equal to zero. One major convention in all the system models affecting the calculation of rates of return and net present values is the treatment of time. planEASe models assume that inflation occurs continuously over time, even though the analysis is conducted in terms of discrete time periods, such as years. The method for handling this situation is best shown by example. In the case of the Sample Apartments analysis shipped with the system, the assumption is that the property is acquired in April of 2001, so there are nine months of operations which will occur in 2001. Revenues and expenses occur continuously during that nine months, but one could say that the average time of their occurrence is half that time, or 4.5 months from the beginning of operations. Accordingly, planEASe

Example 1 2002 15.5% –––––––––– = 0.8351655 Discount Factor (1+.155) will increase all inflating items for 4.5 months of inflation in 2001, and the time of that cash flow (“i” in the formula) will be 4.5 months, or.375 years. Similarly, the average revenue and expense in 2002 will occur on July 1, 2002, or 15 months from the April 1, 2001 start of operations. Therefore the inflating revenues and expenses for that year are increased by 15 months of inflation from the starting values as of April 1, 2001, and “i” in the formula is 15 months (or 1.25 years) in this case.

This is a rather dry mathematical formula, and perhaps it is difficult to see how it relates to practicality. Let’s take, as an example, the 15.4% Rate of Return Before Tax for the Sample Apartments in the Model Documentation. In order to verify this rate of return, and also to show the means of calculation, it is only necessary to show that the Net Present Value of the Cash Flow Before Tax is approximately zero at a 15.4% discount rate.

The calculation is shown in this “IRR Verification Table”. The first two columns show the Year and Cash Flow for the Cash Flow Before Tax. The third column is the time, measured from the Acquisition Date, when the Cash Flow occurred. The next two columns show the discount factor and the Present Value of the Cash Flows at a discount rate of 15.5%, and the last two columns show the same calcula- tions at a 15.4% rate... the Rate of Return Before Tax for the Sample Apartments. The Net Present Value at 15.4% is a positive 64.8, so we next calculate the Net Present Value at 15.5% and find that it is a negative 689.7. Thus we know that the Internal Rate of Return is between 15.4% and 15.5%, and we could interpolate to find a close estimate between those two rates. planEASe reports Internal Rates of Return accurate to the nearest.1%, or 15.4% in this case.

The calculation of the 2002 discount factor is shown here to demonstrate how the 15.5% discount rate and the 1.25 year timing of the cash flow affect the calculations. Of course, the difference between the way we calculate the IRR here, and the way you may have seen it done elsewhere is in the way in which we measure TIME. If you look at the mathematics, it’s clear that the IRR depends on only two factors --- Cash Flow AND the Time the cash is received. Virtually all analysts take pains to forecast the cash results of an investment with great accuracy. Indeed, much of that accuracy is totally spurious, such as the 5 in the $1,275 Cash Flow above. But the same analyst will throw away all of that accuracy by measuring time to one digit accuracy. That is, many analysts will use a time value of one year for the first year’s cash flows, two years for the second, et cetera. Before computers there was some excuse for this, because discount tables using fractional time factors were rare. But with computers, the increase in accuracy gained by measuring time accurately costs nothing. Unfortunately, many programmers have imitated the practices of the past in developing their systems, so their financial planning systems perpetuate this inaccuracy.

As the real estate industry (and the measurement of its returns) has progressed, events have combined to make this model of reality inaccurate in many cases. When looking at Development Projects, an assumption that cash

flows occur in the middle of the year is grossly inaccurate. Consequently, no banker or developer will even look at annual projections, insisting (properly) on monthly numbers, at least during the development phase. Likewise, consideration of the results of cost reimbursement and re-leasing space in Office and Retail properties again causes cash flows to occur in a non-symmetric fashion during any year for such properties. This inaccuracy in the measurement of time in the calculation of Net Present Values, and Internal Rates of Return, is typically expressed in an unusual fashion in the trade... sophisticated analysts ask for a “ monthly ” IRR, dividing both cash flows and time into monthly increments. These people know that dividing time into gross increments of a year is simply not adequate to measure an investment properly. Indeed, if you insist on measuring a key variable (time in this example) to one digit accuracy, the answer you obtain (the NPV and/or IRR) will also be accurate to one digit as any mathematician will tell you!!

The proper measurement of time will, in general, improve the IRR on positive cash flow properties, and correspondingly, lower the IRR on negative cash flow properties. This is because the cash flows are being considered to be received or disbursed earlier than with the simpler (year-end) method. The difference is generally quite noticeable. Depending on the amount of cash flow between the buy and sell, the difference between IRR’s measured in the two different ways can range from 0 to 30%. That is, a 15% IRR measured one way, can be transformed into 20% measured the other way. This considers only the effect of mis- measuring time. Consider the effect of misplacing development and vacancy costs in time and you have gross mis-measurement of the returns!

For these reasons, planEASe measures cash flows and their occurrence on a monthly basis when computing IRR’s, NPV’s and the other measures discussed in this section, whether you have the optional planEASe Monthly Extension or not. All planEASe reports, capabilities and measures documented here operate in the same way with the Monthly Extension except that the cash flows and times involved are monthly rather than yearly. Monthly or quarterly analyses are useful for other reasons beyond accuracy in IRR’s. Such analyses can expose otherwise hidden cash flow problems within a year, and can greatly aid in the interpretation of a yearly planEASe analysis by showing discrete events such as loan draws and repayments, lease steps, and capital expenditures in substantially greater detail than possible when all cash flows within a year are combined into one number.

Relatively recently, theoreticians have developed the Modified Internal Rate of Return (MIRR) in response a perceived need to measure the growth in net worth due to an investment. The IRR calculation does not consider what happens to positive cash flows thrown off during the life of an investment. In order to do that, you need to assume a “Reinvestment Rate” at which these cash flows are reinvested and then measure the total amount of cash generated by both the investment and the reinvestment of the cash generated during the life of the investment. Similarly the MIRR calculation assumes that the cash necessary to fund any negative cash flows after the initial investment is invested at a “Safe Rate” at the beginning of the investment so that the necessary funds will be available at the time of the negative cash flow. The “Safe Rate” would represent the rate at which the investor could actually place the money in an account where it would be liquid at the appropriate time.

The methodology used to correct the IRR for these deficiencies requires that the investor specify his “ Reinvestment Rate ” for positive cash flows during the life of the investment, and his “ Safe Rate ” for funding the negative cash flows during that life. The Safe Rate is used to discount all negative cash flows to the present (the Acquisition Date in this case), yielding a Present Value of the invested amounts which represents the sum of the initial investment plus the amount necessary to fund the future negative cash flows, assuming that the funding amount is invested at the Safe Rate on the Acquisition Date. The Reinvestment Rate is used to compound all future positive cash flows to the end of the investment

period. This yields a Future Value which represents the sum of the cash received from the sale of the investment plus the amount that would be available from the reinvested positive cash flows.

The calculations lying in back of the determination of the Modified Internal Rate of Return for the Cash Flow Before Tax for the Sample Apartments analysis are shown in this MIRR Calculation Table for a Safe Rate of 8% and a Rein- vestment Rate of 10%. Since there are no negative cash flows after the Acquisition Date in this case, the Safe Rate is irrele- vant to the calculation, as shown. Below the table, the MIRR is computed as the interest rate which will yield the Future Value from the Present Value over the Holding Period, according to the MIRR formula shown above.

This example Future Value Factor calcu- lation shows how the FV Factors in the table are determined. The 2.75 exponent comes from the fact that the cash flow occurs at 1.25 years, which is 2.75 years from the end of the four year holding period.

There is another measure sometimes used, called variously Net Future Value (NFV) by some, and Capital Accumulation by others. It is simply the difference between the Future Value and the Present Value in the MIRR calculation, or $396,907.3 - $225,000 or $171,907.3, which represents the addition to the investor’s net worth at the end of the four year holding period, expressed in dollars discounted to the end of the holding period. The Cash Flow Analysis function in the optional Financial Utilities allows you to import investment cash flow streams from any planEASe analysis and compute the IRR, MIRR, NPV, CpA (or NFV) for those cash flows.

In rare instances a correction to the cash flows must be made in calculating the MIRR. Consider the following simplified (but real life) example: A developer plans to float $10M of bonds to develop a property over three years, establish it as an operating investment over the next two years, and then sell it for $14M at the end of five years, paying off the bonds, and netting a $4M sale profit. For simplicity, we’ll assume that the bond interest rate is zero.

His cash flows, then, would look like those in this “Zero Investment Development” table. The IRR for these cash flows is infinite. That makes sense. After all, the investor is never “out of pocket” in such an investment, and when you’re smart enough to make a profit without investing anything, your rate of return is logically infinite. As we have discussed the MIRR calculations so far, the MIRR for these cash flows at an 8% Safe Rate and a 10% Reinvestment Rate is 23.4%!! If that relates to any type of reality, it escapes us. The reason for this anomaly is that the calculation ignores the fact that the earlier $10M positive cash flow would be used to fund the later negative cash flows

To correct for this problem, the practice is to adjust the cash flows before computing the MIRR by removing as many negative cash flows as possible when they are preceded by positive cash flows. The removal is

accomplished by computing the Present Value of the negative cash flow at the time of the earlier positive cash flow and subtracting that amount from the positive cash flow while eliminating the negative cash flow. This simulates setting aside enough of the earlier positive cash flow to fund the later negative flow at the safe rate. When this is done to the cash flows above the MIRR is infinite, as it should be.

The Modified Internal Rate of Return has two basic advantages over the Internal Rate of Return. First, it is much easier to compute, and takes less time, whether done by machine or by hand. Secondly, it is more realistic in that it considers the Reinvestment and Safe Rates, whereas the IRR does not.

The major disadvantage to the MIRR is that the computation is different from that for other investments, so that you are typically comparing apples to oranges when you compare a Bond Yield to Maturity (which is computed like the IRR) to an MIRR on a real estate investment. A second disadvantage is that the calculation depends on the individual investor’s situation (his Reinvestment and Safe Rates).

But these advantages and disadvantages are relatively minor. The real determining point must be the preference of the individual investor or user of the system. planEASe makes both measures available, and only you can choose which measures you want to compute for the investment.