 The process of selecting asset investments for the firm follows a logical sequence of activities:

• Development and submission of possible capital budgeting projects
• Determine relevant cash flows and assess the associated risk of those cash flows
• Selecting a required rate of return consistent with the risks being assumed
• Evaluating the estimated cash flows using accepted "metrics" of evaluation
• Selecting a project or other investment considering non-quantitative factors
• Performing a post-audit to appraise the performance of the selected projects

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Time Value Concepts

All time value calculations build from the concept of compound interest which was initially proposed by an anonymous Italian writer in 1476 under the rubric "commercial arithmetic". The first formal presentation was by Pacioli in his Summa in 1478 where he discussed both double entry accounting and commercial arithmetic although no time value tables were provided in that publication. Those tables initially appeared in the mid-16th century and were in relatively wide spread usage throughout Europe by the 17th century so this concept had been around for a considerable time before it was presented as a means of evaluating proposed new investments in the mid to latter part of the twentieth century. While its applicability to evaluating investment opportunities was recognized and promoted, the advent of the computer and finance calculator caused it to be more widely adopted since the computations were not so onerous.

(The first hand-held finance calculator (HP-80) was introduced in February, 1973 by Hewlett Packard at a price of \$395.00. After doing discounted cash flow calculations using published tables and a slide rule as a student, it was worth skipping some meals to pay for that machine! My current students pay a tenth of that price for a reliable machine with more power and still have difficulty with time value.)

The critical assumption in all time value formulas is that the reinvestment rate is inherently calculated as the stated rate. That is, you promise to always reinvest the cash flows being received immediately at the same interest rate for the remaining period of time. That is what makes it "compound" as opposed to "simple" interest. This reinvestment assumption is much more demanding than most users recognize and failure to accomplish the reinvestment at the stated rate is the reason the majority of investments do not perform as well as expected.The formula shown in the above link are identical to those programmed into a finance calculator.

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Valuation Methods

There are numerous "metrics" or valuation methodologies used to evaluate the worth of investment opportunities. Most finance professionals use Net Present Value (NPV) as their first choice methodology, but the other evaluation tools are in widespread use and each has its devotees. Therefore, it is likely that most of these valuation methods will be calculated for each individual investment being considered in order to satisfy the preferences of different decision makers. They are discussed below using the following two investment opportunities:

 Project A Project B Time Period Cash Flows Cash Flows 0 \$100,000 \$120,000 1 20,000 58,000 2 30,000 45,000 3 45,000 30,000 4 30,000 15,000 5 15,000 5,000 Payback

The payback period is computed exactly as the name implies; how long until you have "paid back" the cost of the project. There is typically no time value associated with payback although there are a few adherents to a measure of payback that utilizes the present value of future cash flows. However, that has always seemed a bit silly since finding the payback using the present value of future cash flows at an appropriate discount rate is identical to using the net present value approach except the measure of worth is in terms of time rather than dollars. Since the objective of the firm is to maximize shareholder wealth, the appropriate measure when evaluating new investment opportunities is the dollars of value added rather than the time period required to return the initial investment. The following illustrates the determination of the payback period for the example investments:

 Project A Project B Time Cash Flows Cumulative Flows Cash Flows Cumulative Flows 0 -100,000 -120,000 1 20,000 20,000 58,000 58,000 2 30,000 50,000 45,000 103,000 3 45,000 95,000 30,000 120,000 4 30,000 100,000 15,000 5 15,000 5,000 Payback 3.1667 years 2.5667 years

Project A has a payback of 3.1667 years (The .1667 is the last \$5,000 required to finish paying back the \$100,000 investment divided by the \$30,000 to be received for year 4.) or 3 years and 2 months.

Project B has a payback of 2.5667 years (Again, the .5667 is the last \$17,000 needed to cumulate sufficient funds to pay back the initial \$120,000 divided by the \$30,000 to be received in year 3.) Another way to express the result is that the project is "paid back" in 2 years and 6.8 months.

There are two noteworthy observations about the results. First, it is obvious that we are not adjusting for the time value of money so the last dollars needed to complete the return of the initial investment are considered to have the same value to the firm as the initial dollars invested today or at time zero. Second, the method stops counting dollars to be received as soon as the initial investment is "paid back" which means it ignores the remaining \$40,000 to be received in years 4 and 5 from Project A (the remaining \$25,000 in year 4 and the \$15,000 in year 5) and the remaining \$33,000 to be received in years 3, 4 and 5 from Project B!

The best one can say about this method of evaluating investments is that you frequently see it used even though it has some major flaws in terms of time value and accounting for the entire sums to be received from the investment. Some decision makers use payback as a filtering device wherein they adopt the position that experience has taught them not to bother doing the more challenging calculations of net present value or internal rate of return when the payback is greater than some minimum number of periods. There is no magic theoretical number, but experienced managers often develop an insight that the project must meet a minimum payback standard based on looking at similar projects.

Average Rate of Return (AROR)

The average rate of return (AROR) solves the problem in payback of not counting all of the cash inflows although it maintains the illusion that money does not have any time value. Sum the cash inflows and divide by the number of periods to get an "average cash flow". This value is then divided by the cash outflow to get an AROR.

 Project A Project B Time Period Cash Flows Cash Flows 0 \$100,000 \$120,000 1 20,000 58,000 2 30,000 45,000 3 45,000 30,000 4 30,000 15,000 5 15,000 5,000 Total Inflows \$140,000 \$153,000 Average Inflow \$28,000 \$30,600 AROR 28% 25.5%

In the case of the two example projects; divide the average annual cash flow for Project A, \$28,000, by the initial investment of \$100,000 to arrive at an AROR of 28%. The same process for Project B (divide the average cash flow of \$30,600 by the investment of \$120,000) yields an AROR of 25.5%. Notice that we are already beginning to get a conflict in terms of project desireability since the payback period would rank Project B over Project A whereas the AROR indicates that Project A is the better investment. A more critical point is that the computed average periodic cash flows do not match the forecasted cash flows for any period! Granted, an arithmetic average often does not match the numbers used in its calculation, but we went to a great deal of effort to estimate cash flows as accurately as possible. Now, all that effort is being neutralized through the use of a "pretend" equivalent average cash flow. If the decision maker is prone to "pretending" concerning the cash flows relevant to a project, lets just head straight to the ouija board and avoid all of this effort! Seriously, it is a critical flaw in this method of evaluating projects since both the amount and pattern of cash flows are crucial to any investment and this approach does not take the cash flow pattern into account. It goes without elaboration to point out that the choice to not consider the time value of money is another major concern with this method of project evaluation.

Net Present Value (NPV)

Net Present Value (NPV) is the most commonly accepted metric to evaluate investments since it considers the time value of money and measures the addition to shareholder wealth created by a postive net present value. The "net" in NPV is the difference found by subtracting the present value of the cash outflows from the present value of the cash inflows. The simple rule is that the net present value at an appropriate discount rate must be zero or positve before making an investment or "NPV greater than or equal to zero". The challenge lies in estimating the appropriate discount rate since every project will have slightly different risk characteristics. Since we live in a world where risk and return are related, it is critical that the chosen rate of discount be consistent with the risks being taken when making the investment. The calculations below assume that the firm has selected 10% as the relevant required rate of return for projects bearing the risks assessed for our two projects. The selection of this discount rate often begins with looking at the firm's cost of funds or cost of capital and adjusting for any risks being taken that differ from the risks of existing assets.

Note that the concept of EVA or Economic Value Added (sometimes called SVA or Shareholder Value Added) is essentially a similar process wherein the cost of funds employed for assets are subtracted from the returns on those assets to see if anything is left over to add value to the shareholder's equity.

 Project A Project B Time Cash Flows Present Value Cash Flows Present Value 0 -100,000 \$-100,000.00 -120,000 \$-120,000.00 1 20,000 18,181.82 58,000 52,727.27 2 30,000 24,793.39 45,000 37,190.08 3 45,000 33,809.17 30,000 22,539.44 4 30,000 20,490.40 15,000 10,245.20 5 15,000 9313.82 5,000 3,104.61 Present Value Inflows \$106,588.60 \$125,806.60 Present Value Outflows -100,000.00 -120,000.00 Net Present Value (NPV) \$6,588.60 \$5,806.61 Present Value Index (PVI) 1.0659 1.0484

We observe from this table that Project A is considerably more attractive since it will add the amount of net present value to shareholder's wealth over and beyond what the shareholders expect as a return on their equity investment. In effect, it is the potential additional shareholder value created by the discovery and implementation of an investment with greater cash inflows than a normal 10% return project.

The present value numbers for any given period can be found using the time value formula for present value or entering the following example into your calculator:

• n = 3
• i = 10
• PMT = 0
• FV = 45000
• PV = ? -33,809.1660 [Remember the sign convention, FV entered as a positive number results in a negative PV]

The alternative is to use the cash flow registers on the calculator and entering all of the problem information to arrive at the NPV. More information on this procedure is available in the calculator section of this material. Calculator Tutorial

Present Value Index (PVI)

The Present Value Index (PVI) is found by dividing the present value of the cash inflows by the present value of the cash outflows rather than subtracting as was done for NPV. Project A has an index of 1.0659 as opposed to the PVI for Project B of 1.0484 which again makes "A" the more attractive investment. Present Value Index is used to rank mutually exclusive investments of unequal size where the firm needs only one of the alternatives to accomplish a specific task. Management can look at the higher ranked of the investments and choose the option providing the most value for the investment dollar. The evaluation rule now becomes "PVI greater than or equal to 1".

Another use of PVI occurs when a firm is capital rationed which means it has limited the size of its annual capital budget either by choice or by the reality that available funds are limited to existing equity and debt credit lines. In that situation, it is useful to rank projects in their order of relative attractiveness which PVI accomplishes much better than NPV. Since all projects are shown in terms of their relative size, it is easy to look for combinations of investment that most effectively uses the rationed or constrained capital budget.

Internal Rate of Return (IRR)

When working with Net Present Value, the rule was: "assume the rate and find the NPV". When working with Internal Rate of Return (IRR), the rule is; "assume the NPV=0 and find the rate". In other words, at what rate of discount does the present value of the inflows equal the present value of the outflows or an NPV of 0? This is where you finally get the full value from your purchase of a finance calculator since you could accomplish most of the other evaluation metrics either by hand or with a much more basic calculator.

In pre-finance calculator times, we found the IRR using the "hunt, search and curse" method wherein you would pick a rate, compute the present value of the cash inflows and compare it to the present value of the cash outflows. More often than not, they would not be the same so you would quietly curse, change the discount rate in the appropriate direction and try again. This would be followed by further cursing and adjustments to the rate until finally the two present values were sufficiently close that you could not bear making any further adjustments. (It is worth noting that this scenario is the finance prof's equivalent to the "when I was a boy I walked two miles through chest high snow to get to school" fable!)

The process is similar to the example shown in the calculator section of this chapter for a yield to maturity on a bond except the cash flows are different each period.

 Project A Project B Time Cash Flows Cash Flows 0 CF0 -100,000 -120,000 1 CF1 20,000 58,000 2 CF2 30,000 45,000 3 CF3 45,000 30,000 4 CF4 30,000 15,000 5 CF5 15,000 5,000 IRR 12.5567% 12.6522%

Be sure to enter CF0 as a negative number since the calculator needs that value as the reference point for the present value of the cash outflows. Solving for IRR yields the values shown above which would indicate that Project "B" is now our first choice.

(You need to enter the above cash flow values into the "cash flow" registers of your calculator. Each machine does it slightly differently so you might want to access the links suggested below or ask someone familiar with the machine to show you how to use the cash flow feature. Due to the fact that the problem is not stated in terms of a series of level payments or annuities, it means you cannot use the five basic finance keys unless you determined to experience the "hunt, search and curse" approach. Reading the manual that came with your machine will both frustrate and exhaust you so is not highly recommended.)

Modified Internal Rate of Return (MIRR)

One of the problems with IRR is the implied assumption that you will reinvest the cash flows at the calculated rate. Therefore, this evaluation metric tends to prefer projects with stronger early cash flows since it can reinvest them for a longer time period. If the calculated IRR seems too good to be true, then it probably is too good to be true! There is nothing wrong with your calculator or how you entered the information, but efficient markets are not inclined to let us have above normal returns on a consistent basis which means you typically reinvest interim cash flows at the expected rate of return based on the history of your firm and its assets. The Modified Internal Rate of Return (MIRR) addresses this issue by allowing you to control the rate of return on reinvested cash flows during the life of the project. In the example projects, we assumed that you used a 10% discount rate when we did the NPV calculation and that rate is taken as a reasonable estimate of the expected returns on new projects in the future.

MIRR returns us to the basic principles of time value by first calculating the expected future value of the cash inflows at the end of the project using our 10% rate. In other words, if you reinvest each year's cash flows in projects earning the 10% rate for the time remaining to the end of this project, how much would be in your "account".

 Remaining Years Project A Future Value Project B Future Value Time Period Cash Flows Cash Flows 0 \$100,000 \$120,000 1 4 20,000 29,282.00 58,000 84,917.80 2 3 30,000 39,930.00 45,000 59,895.00 3 2 45,000 54,450.00 30,000 36,300.00 4 1 30,000 33,000.00 15,000 16,500.00 5 0 15,000 15,000.00 5,000 5,000.00 Future Value \$171,662.00 \$202,612.80 MIRR 11.4127% 11.0445%

At this point, you know the future value of the cash inflows under the assumption that you will reinvest each of them at the "normal" rate of 10% and you also know the cost of the investment that will produce that value. You can use the financial calculator to find the implied rate of return or MIRR for Project A by entering n=5, PV = -100000, PMT =0 and FV = 171662 followed by solving for "i" which should be 11.413%. In the case of Project B, it will be 11.045% which is slightly less than Project A. Notice that by controlling the reinvestment rate at the more realistic 10% rate, we have reversed the order of attractiveness for the projects so that Project A is again rightfully in the lead. This is consistent with the findings of the NPV metric so the conflict in ranking created by the IRR and its implied reinvestment assumption has been eliminated.

Granted, this all seems a bit fussy if you are simply looking at a single project and trying to determine if it will create shareholder value by earning more than it costs. However, most organizations either are or tend to see themselves as capital rationed which creates a need to rank projects from best to worse. Rates of return are a popular ranking device since many of us are extremely "rate driven" and saying a project has an 11.4% return when we only require 10% has more emotional impact than saying it has a positive NPV of \$6,588.60 at the same discount rate. The information contained in the NPV results is actually more valuable, but does not have the same presentation impact as working in percentages or rates of return. If you are going to use rates of return as a ranking device, it is highly recommended to use the MIRR since it avoids the implied reinvestment assumption problem.

In review, the two projects would be ranked as below:

 Project A Project B Winner Payback 3.1667 years 2.6667 years B AROR 28% 25.5% A NPV \$6,588.60 \$5,806.61 A PVI 1.0659 1.0484 A IRR 12.5567% 12.6522% B MIRR 11.4127% 11.0455% A

It is apparent that not all of the evaluation methods provide the same signal which is not a problem when a project is viewed in isolation without consideration of funds availability. Either project would presumably be accepted since the NPV of both is positive and the rate of return is in excess of the required return of 10%. However, if you only needed one of these two alternatives to accomplish a company goal or if the firm had chosen to ration its available funds the ranking issue comes to the fore along with the conflicts between the various methods of evaluating the investments. It is advisable to rely primarily upon net present value since you are controlling the rate of reinvestment, you have properly accounted for all of the relevant cash flows and the timing of their occurrence and it measures the creation of shareholder value beyond the expected return embedded in the discount rate. However, you are well advised to be willing and able to determine all of the metrics discussed since different decision makers have different preferences. So long as the NPV of the selected project is equal to or greater than zero, the project is worth pursuing at which point you make your presentation using whichever alternative method seems most appropriate for your audience.

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Incremental Cash Flows

Estimating the cash flows associated with a possible investment is the most challenging part of the capital budgeting process since it necessarily involves forecasting events into an uncertain future. We are interested only in the "incremental" cash flows or the amount of net change in cash flow created by the proposed project. It is relatively simple to consider incremental flows when the project involves an entirely new activity with a new product or new market. The proposal indicates the amount and timing of the new investment in assets and considers the net flows to the firm over the life of the project. Most proposals, however, are not quite so simple in that the new assets will either supplant existing assets or the new product will either cannibalize or complement the cash flows of existing products.

Sunk costs are those expenditures which have already been made and cannot be recovered regardless of whether or not the new proposed investment is made. Therefore, they are not incremental since they are not altered by a positive or negative decision regarding the new asset being considered and would not be part of the cash flow projection. It is difficult to walk away from money already spent, but the point is that it cannot be recovered so is not considered. Cash flows from the sale of a new product can reduce sales of existing products and this 'cannibalization' needs to be considered as an incremental cash flow. If, for example, a new product is introduced that is expected to reduce the sales of existing products, those lost sales and their associated net cash inflows are incremental and need to be part of the analysis. The new product must pay for itself and compensate the firm for the lost revenues and associated cash flows from the existing product line.

If the new product or market requires additional working capital in the form of inventory or accounts receivable, that investment is incremental to the project and needs to be included as a cash outflow. It will be "recovered" at the end of the project, but the firm has lost the opportunity cost of investing those funds elsewhere and the project needs to compensate for that cost. This typically involves spending the investment in working capital during the life of the new project and then recovering it at the end of the analysis period. The use of time value based evaluation metrics will ensure that the opportunity cost of those funds has been accomodated. For example, assume a firm estimates it will require an additional \$100,000 investment in working capital to service the expected sales of a new product during the next five years. At a discount rate of 10%, the net cost of \$37,907.87 (present value of \$100,000 five years from now at 10% = \$62,092.13; subtracting the investment of \$100,000 today results in a "cost" of \$37,907.87) which must be compensated by the incremental cash inflows from the new product.

It can become even more challenging to estimate incremental flows when evaluating a replacement for an existing asset. Only the differences in cash inflows and cash outflows is considered in the analysis since the cash flows of the existing asset are already part of the firm. In many instances, replacement decisions are made on the basis of net cost savings which is much less than looking at total revenue less operating costs. The new asset is only going to lower some of the costs and making the investment is often more difficult to justify since the entire cash flows are not available.

There are two primary approaches to estimating cash flows. First, the net profit is calculated in the traditional manner and the depreciation charge is added back to create "cash flow". Since depreciation is a book entry taken solely for tax savings, it does not involve actual cash so the depreciation charge is typically added back to net profit to obtain cash flow. Second, you could examine the two components of cash flow independently: the first component is the Earnings Before Interest, Tax and Depreciation (EBITD) and the second component is the depreciation and its associated tax shelter. Example

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Horizon Values

A common challenge is evaluating new investments is to estimate the horizon or terminal value of the project. If the proposal involves a simple purchase of a new machine, it is relatively simple to estimate its salvage value at the end of its physical life and adjust that value for any changes to cash flow in the last period of the analysis. In most instances, however, investment opportunities do not suddenly end at some finite point. Instead, the project could continue into the future still producing cash inflows albeit at a somewhat reduced level given competitive changes or new products being introduced. It is likely that the firm would choose to either sell the assets to another party that might find them of value or choose to switch the assets to the production of another product. The point is that the analysis period rarely matches the physical life of the investment and we need to end the analysis by considering that terminal or horizon value as accurately as possible. It often is far into the future and estimating its economic worth at that future date is not easy, but some estimate needs to be made since we are looking at all of the incremental flows associated with the proposed new investment.

A simple example is the point raised earlier concerning working capital invested to support a proposed new product. It is unlikely that a firm would suddenly stop the sale of a product; use up the remaining inventory producing the last few units to be sold and collect all of the accounts receivable outstanding. Instead, a new product will likely be introduced and it will take over the investment in inventory and receivables needed to support its sales. However, in our analysis of the first product investment, we need to consider the "recovery" of that working capital investment at that future time point by requiring that the new product "pay for" for its working capital. Otherwise, the new product gets a free ride in the sense that it acquires working capital at no cost and it is possible the firm would make an inappropriate investment in a new product proposed since it is underestimating the amount of required investment.

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Notes on Equivalent Annual Cost

What it is: The face value of an annuity with the same length of life as the investment and the same present value.

Example: A machine is purchased today for \$15,000, and lasts for three years, costing \$5000 to operate each year. (For simplicity treat the operating costs as paid at the end of the year.) The discount rate is 6%.

 Cash Flows at year 0 1 2 3 -15 -5 -5 -5

Net Present Value: -28.37

A 3 year annuity paying -10.61 has the same NPV:

 Cash Flows (costs) at year 0 1 2 3 -10.61 -10.61 -10.61

How to calculate it: The simplest way is to start by calculating NPV of a \$1 annuity for the required number of years; this is sometimes called the "annuity factor." Dividing the NPV of the investment by the annuity factor yields the equivalent annual cost.

Example: At 6%, a 3-year annuity with face value \$1 has a NPV of 2.673; this is the three-year annuity factor. 28.37 / 2.673 = 10.61

What it’s good for: This is the tricky part. The equivalent annual cost is useful in an analysis only if the alternatives being examined involve (or at least can be approximated as) re-investing in an identical project every n years forever.

Example #1: Long lived vs Short lived equipment

We are choosing between two pieces of equipment. Machine A has an NPV (cost) of -28.37 and requires replacement every three years. Machine B has an NPV (cost) of -21.00 and requires replacement every two years. Which is better? We could compare the NPV of the following infinite cash flows:

 Year 0 1 2 3 4 5 6 7 … A -28.37 0 0 -28.37 0 0 -28.37 0 … B -21 0 -21 0 -21 0 -21 0 …

Or we could do the same thing by using the equivalent annual cost to smooth out these cash flows: as we saw before 28.37 is equivalent to a 3 year annuity of 10.61; while 21.00 is equivalent to a 2 year annuity of 11.45. Thus the following smooth cash flows have the same NPV’s as the originals:

 Year 0 1 2 3 4 5 6 7 … A 0 -10.61 -10.61 -10.61 -10.61 -10.61 -10.61 -10.61 … B 0 -11.45 -11.45 -11.45 -11.45 -11.45 -11.45 -11.45 …

Clearly machine A is cheaper.

Note that the number of years to use in the annuity calculation is the number of years between repetitions of the investment. Thus if there were, say, additional costs of disposal for some years after the machine was replaced with the new machine, so that each new machine involved expenditures over 5 years, that would not affect the fact that the annuity should be calculated over the number of years before a new cycle of expenditures had to begin (2 or 3 years in this example).

Example #2: When to replace a machine

The current machine provides 4000 this year (i.e. immediately). If not replaced, it will also provide 4000 next year. Beyond that point the current machine will provide 0 and will have to be replaced. Thus we either replace it this year or next year. The replacement machine will last 3 years, and then will have to be replaced (presumably with an identical model). The machine costs –15 up front and yields 8 each year for three years. Should we replace the existing machine immediately, or wait?

Here are the cash flows in each case:

Immediate replacement:

 Year 0 1 2 3 4 5 6 7 8 … Old 4 New -15 8 8 8 Next -15 8 8 8 Next -15 8 8 … Total -11 8 8 -7 8 8 -7 8 8 …

Delayed replacement:

 Year 0 1 2 3 4 5 6 7 8 … Old 4 4 New -15 8 8 8 Next -15 8 8 8 Next -15 8 … Total 4 -11 8 8 -7 8 8 -7 8 …

Note the difference between the two arrangements is the obtaining an extra 4 from the old machine in year 1 in return for delaying the entire future payoffs from the series of new machines by a year.

Again, either we can calculate the NPV of each of these total cash flows and compare directly, or we can note that the cycle of payoffs from a replacement machine is equivalent to a payment of 2.387 annually for three years. Thus the two cash flows are equivalent to the following:

 Year 0 1 2 3 4 5 6 7 8 … Immediate Replacement 4 2.387 2.387 2.387 2.387 2.387 2.387 2.387 2.387 … Delayed Replacement 4 4 2.387 2.387 2.387 2.387 2.387 2.387 2.387 …

Delay is better.

Conclusion: Although the equivalent annual cost method is often used and can legitimately be used in cases like these, you may find it easier just to be explicit about the interrelated projects you are comparing, spell them out completely and simply calculate net present values. The results are identical.

Calculator Issues

The majority of calculators have five basic finance function keys:

• n = number of periods
• i = interest rate per period
• PV = Present Value of flows
• PMT = Level period payments
• FV = Future Value of flows

It is less confusing if you will set your calculator for one payment per period (see your owner's manual) and end of period payments. You can then control both the number of periods and the periodic interest rate. All finance calculators have a sign convention that permits them to perform computations of implied interest rates based on the entered cash flows. For example, if you wanted to know the current yield to maturity of a bond that pays six percent coupon interest per year for the next five years, will return the face value (usually done on the basis of \$1000 par value) at maturity and presently sells for \$962.50; you would enter it in the calculator as follows: (In most finance calculators, you enter the value first and then press the key with appropriate letter to enter the values.)

• n = 10 (most bonds are evaluated on a semi-annual basis so 5 years times 2 equals 10 semi-annual periods)
• PV = -962.50 (notice it was entered as a negative number since it is the outflow you are investing)
• PMT = 30 (remember we are working in semi-annual terms so 6% times \$1000 face value divided by 2 equals \$30)
• FV = 1000 (you will receive the maturity value at the end of the time period)

Press "i" or "CPT and I/Y" (when using a Texas Instruments calculator) and you should get the answer of 3.45. This is the semi-annual yield to maturity which is multiplied by 2 to get the annual yield on this bond of 6.90%. It earns a bit more than the 6%coupon rate since you bought it for less than par value.

If you neglected to change the sign on the PV to a negative number, you will get a message "Error 5" which is the calculator's way of telling you this deal is too good to be true since all of the cash flows are postive and it costs you nothing! Simply change the PV or purchase price to a negative number and it will compute the correct yield.

Further information on using a finance calculator can be found at the following web sites:

Useful Web Sites: