Chemical Process Economics II
Chemical Process Economics II CHEE 6369
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Date Created: 09/19/15
A Short Note on Present Value Calculations John Crump for Prof Dole Sales amp Leasing Summer 2001 Topics I Present Value Concepts 0 Sample Calculations 7 With Spreadsheets and Tables II Where does the discount rate come from 7 Risk issues Problem How do we compare the value of a dollar today with that of a dollar at some future time 1quot Present Value Concepts Some Basic Observations First Basic Principle of Finance A dollar today is worth more than a dollar tomorrow gt A dollar today can be invested to earn interest starting now or it can be used for consumption now Note both exibility and growth in value over time The end goal of economic activity is consumption 7 Now or in the future whether by the actor or by someone else who is to bene t from that activity gt Consumption requires a selection or tradeoff among alternative consumable goods and services for the bene ts to be obtained from each The alternative to consumption M is future consumption whether of the same thing or of some other thing gt Some form of investment 7 including work and production or even money hidden in a mattress 7 is associated with any delayed or future consumption gt People must pay something to get you to delay consumption and to invest There are opportunity costs associated with each choice among consumption now and or in the future gt An opportunity cost is whatever the decisionmaker gives up to get the good or service to be consumed at the time it is to be consumed In general it does not have to be money There are other incentives or bene ts that depend on personal preference or utility of the outcome Often there is a mixture of incentives Here we are only interested in economic incentives The rate of return r is the reward investors demand for accepting delayed payment The rate of return is also referred to as the discount interest or hurdle and as the opportunity cost of capital CI r is an opportunity cost because it is the amount of return or the payoff foregone by investing in Investment 1 instead of Investment 2 or in consumption now The different returns that may be realized from different potential investments must be placed on a common basis to compare them for rational decision making or selection among them Net Present Value see below is the best tool to use in that comparison CI r is the ratio of the change in value per time period to the initial or present value If an actor perceives that total opportunity cost for two different things is the same the actor will have no preference between them and will be indifferent to the choice V V Note on PVdoc Page 1 12 9Aug01 0 If C0 is the initial value or cost of an investment when initially made at time to C1 is the market value of that investment at future time t1 and PV is the initial or present value at time to of that future investment value then Pvzi lr r PV the Discount Factor DF 17 and r Net Present Value N PV C0 11 r Remember that this is an investment so C0 7 the cash ow at to 7 is normally negative a cash out ow Hence the NPV is the incremental value you expect with this investment Also Note 7 Coa present value I Often C0 is known 7 The value price of a stock bond car house or other investment today I Other times it may be only an asking price to be negotiated or your guestimate What can I really get it for What will it cost to develop a new product line of squeegies Future values C1 C2 etc are your expectation as of time to I Exceptions 7 This can be a series of values stipulated to facilitate negotiations or your best guess of what your opponent really thinks or something else I However once you set your expectations those Cx values do not change for the NPV at time to C0 remains what you would pay today to get NPV I You may revise your thinking over time as circumstances change and the future slips by to become the past but that will be a review that shows whether you made a good deal relative to new circumstances at time tx or that helps inform your thinking for a further business decision 0 Where does the predicted rate of return come from How do you select r gt Part of the opportunity cost for future consumption is compensation for your forbearance or abstinence delayed gratification extra cost now perhaps some unpleasantness or pain reduced fun now This opportunity cost arises without regard to risk or other elements Another part of the opportunity cost for future consumption arises from risk and the uncertainty of the future More about this later CI Will the target good or service still be available Will the consumer still want it Will it cost more or less Will some other better cheaper alternative exist Will delivery become easier or harder Will the consumer39s ability to enjoy consumption be jeopardized somehow Will the consumer be alive to enjoy delayed consumption CI Each buyer and seller faces such risks with hisher own preferences and risk aversion Competition 7 the balance of the available supply of capital and the demand for it at any point in time 7 obviously also in uences the value of r for a given potential investment V V Note on PVdoc Page 2 12 9Aug01 NowI two guick PV Examples We can spend 5 right now on a gallon of Bluebell Ice Cream or we can invest it in a one year US government bond earning interest at 5 per year Ignoring both refrigeration and the health effects of eating all that ice cream right now and assuming that a oneyear 5 bond exists the bond will be earn 5 X 5 25 over the next year A rational decisionmaker will choose the option here that maximizes hisher satisfaction under the circumstances 7 I Spend the 5 and eat the ice cream now but have nothing thereafter or I Forego eating ice cream for now and invest the 5 but have 525 after one year N Your apartment house burned down1 leaving you with a vacant lot and a check for 200000 from the insurance company You consider rebuilding but your real estate advisor suggests putting up an office building instead Construction will cost either 200000 for a replacement apartment house or 300000 for the office building and you will have to throw in the value of the land which you could otherwise sell now for 50000 However with virtual certainty your advisor predicts a shortage of office space and predicts that the new building will be worth 400000 one year from now as compared with a forecast value of 293000 for a new apartment house Obviously the office building is expected to be worth more than an apartment house but which one offers the best use of your money and is it worth it to build anything now You would be investing 250000 now with an expectation of realizing 293000 after one year with the apartment house for a return of 43000250000 172 or 350000 now with an expectation of realizing 400000 after one year with the office building for a return of 1427 For now ignore the 100000 difference between the two real estate investments 7 assume you can fund it all out of equity 7 and assume that your only other investment choice is in US government securities You are looking for investments that 1 offer positive net present values and 2 offer rates of return r higher than the opportunity cost of capital 139 6 higher than comparable investment alternatives available to you To decide what to do you should compare three alternatives and pick the one that earns the highest return on your investment 7 a Sell the land and invest 350000 in US government securities the safest most certain investment known for 1 year b Rebuild your apartment house for 200000 and invest the remaining 100000 in US government securities for 1 year or c Spend 300000 on an office building The payoff from US government securities depends on the interest rate offered now and that interest rate establishes a benchmark for you to compare with the other alternatives This example is modified from RA BREALEY AND SO MEYERs PRINCIPLES OF CORPORATE FINANCE 16 6 ed 2000 Note on PVdoc Page 3 l2 9Aug01 a Note on PVdoc Spreadsheet tab quotRE Opportunity in Workbook quotNote on PVxlsquot attached computes solutions for three different interest rates For the given situation it shows 7 Investing in US government securities alone is never more pro table than either real estate investment unless the discount rate exceeds 1427 which would be quite extreme The apartment house alone will earn the highest rate of return so long as the discount rate is below 172 quite extreme However combined with a 100000 investment2 in Us government securities to make for a reasonable comparison with the other two alternatives it will not earn as much as the office building unless the discount rate exceeds 7 The net present value for the two real estate investments drops as the discount rate rises indicating that the future value of real estate sales is lower with higher interest rates This simple problem raises several important issues for PV computations 7 I How do we decide among competing investments for different amounts of capital3 I Discounting here was simple but what about more complex situations I The interest rate impacts on future values but how do we select the interest rate Your advisor was quotvirtually certainquot but how does that relate to the interest rate Common Notation and General Formulae There is no solid standard notation for present value and future value problems but that followed by FC Jelen4 or some similar alternative is often used 7 P present amount the value now today rn return for period n the amount that P will earn in the n3911 period R Re Uniform 139e repeated each period endofyear amount This can be positive revenue or negative costs Rb Uniform beginningofyear amount Re 1r 1 P r1P Pl r1 the future value ofP at the end ofperiod 1 Then for two periods Sz P 1 1P 1 2Pl r1 Pl r1l 1 2 Pl r2 ifrr1rz and in general5 S P1 rn PFpsym l Remember that you are funding this out of equity You have to do something with spare cash that does not go into the apartment house This issue is beyond the scope of this note but NPV is the key measure We would also use the expected rate of return from each project and other elaborations to rank potential investments See discussion of capital rationing in BREALEY AND MYERS supra note 1 55 See KK HUMPHREYS JELEN s COST AND OPTIMIZATION ENGINEERING passim 3 d ed 1991 BREALEY AND lLEYERS see supra note 1 at 3738 242246 discuss the relative values of r1 r2 etc and demonstrate why in general DFI llr1 gt llr22 DFZ or equivalently 1r22 gt lr1 Page 4 12 9Aug01 where Fpsym FPSrn l rn is the single payment future value factor or compound interest factor with discrete rather than continuous see below compounding that transforms P into S given r and n Values for Fpsym are commonly tabulated along with various other similar factors for other future and present value situations discussed below See also Spreadsheet tab quotDiscrete Factor Tablesquot in Workbook quotNote on PVXlsquot attached giving sample tables example computations and graphs for the various factors Many times analysts will use a simple bar graph to illustrate these situations 7 Future Cash Out ow P1rn Sn End of Period 0 1 2 3 l 41 5 6 7 n71 T P Initial Investment Such bar graphs are easily generated in EXCEL or other Spreadsheets See Spreadsheet tab quotDiscrete Factor Tablesquot in Workbook quotNote on PVXlsquot for this bar graph and similar illustrations for other factors that call for single or periodic payments That spreadsheet also gives sample calculations and various comments on usage 3dimensional plots of factors against rate of return r and number of time periods n and other ideas Other Factors A number of other similar factors are commonly considered in addition to the single payment future value factor the three most common are given below Workbook quotNote on PVXlsquot 7 particularly the tab quotDiscrete Factor Tablesquot 7 has been prepared to provide examples and to help users in working with future and present value problems The inverse of the single payment future value factor Fpsym is the single pament present value PV factor Fspym 7 This condition means that there is no opportunity for arbitrage no financial money machine opportunity 7 at least none longer than a fleetineg short time Failure of this general condition would wipe out any bank Also Brealey and Meyers show that while the rate of return may in fact vary from period to period over the life of an investment and may be forecast for a given situation e g variable rate home mortgages quotin practical capital budgeting a single discount rate is usually applied to all future cash flows Using the same riskadjusted discount rate for each year39s cash flow implies a larger deduction for risk from later cash flows This is because the discount rate compensates for the risk borne per period It makes sense to use a single discount rate as long as the project has the same market risk at each point in its life But look out for exceptions 7 e g a specialty chemical company developing a new product faces one set of conditions while conducting RampD and testing the market early in the project life cycle when risk is highest and another more normal set of conditions in subsequent phases such as investment in physical plant actual production distribution and sales when risk would be lower Brealey and Meyers show how to analyze such projects Note on PVdoc Page 5 12 9Aug01 F51n FSPrn 11 rquot 7 which shows how to compute the present value P equivalent to a future value Sn given r and n Note that Fspym FSPrn ll rn lFpsvm lFPSrn Finance problems commonly involve periodic payments in or out at a uniform rate These payments are called Annuities A uniform periodic series of payments quotRequot that are made or received at the end of each period is call an Ordinary Annuity or an Annuity in Arrears If the payments are made at the beginning of each period they are designated Rb and the fund is called an Annuity Due or an Annuity in Advance Examples of such payment streams include lease payments bonds certain insurance payments pensions awards paid out over time and sinking funds used to accumulate amounts that will spent in the future to replace depreciating assets Annuity Factors are commonly tabulated to relate these payment streams to the amount of an investment P that must be made today to fund the annuity and to the equivalent total future value Sn that will be paid by the end of the annuity Again Tables in Spreadsheet tab quotDiscrete Factor Tablesquot in Workbook quotNote on PVxlsquot give these factors sample calculations and other information To evaluate and ordinary annuity consider the sequence of n end of period payments quotRXquot with a rate of return quotrxquot The future value of that stream is 3n R11 r1 R21 r1l r2 R31 r1l r2l r3 Rn 1H1rn 1Rn1391rn Ifthe Rx are uniform and all equal R and only one rate of return r is used see note 6 supra then this equation is what mathematicians call a geometric series and it can be simplified to IT snRWRxFRSmr r Here FRsym is the Equal Pavment Series Future Value or Annuitv Future Value Factor with Discrete Compounding which uses the uniform end of period amount R to compute the value Sn at future end of period time n given a rate of return r This computation indicates what future sum will be available at the end of n periods 7 for example to pay anticipated educational expenses The inverse of FRSM is the Sinking Fund Factor FSRM which tells us what value of R is needed to provide anticipated future amount S given r and n The corresponding present value factor is the Equal Pavment Series Present Value or Annuitv Present Value Factor with Discrete Compounding Note on PVdoc Page 6 12 9Aug01 P R1 L R2 L R3 LNL Rn71 L Rn lr1l1r1lr2l1r1lr2lr3l 39H1r 1 39H1r 39 lrquot7l rgtltlrquot or PR RXFRPYr7n For a given n and constant values of R and r this factor is used to compute the present value of a uniform series of payments For example it might be used to compute the present value of the interest payments coupons for a bond or to compare a lump sum offered instead of a series of payments in a settlement or an installment plan or to find the present value of payments due in a lease Finally note that the single payment factors and annuity factors are all interrelated 7 FSRrn 1 FRSrn and FPRrn 1 FRPJJA and FSRJJI 1 04 X FPRJJA FSPrn X FPRrn and FRSrn FPRrn X 1 r n FRPJJI X FPSrn Several examples of single payment and uniform payment series present and future value problems are given in EXCEL Workbook quotNote on PVxlsquot Perpetuities 7 If n is set at in nity co the uniform series of payments never stops This is a perpetuity and it is a uniform series of payments R that is expected never to stop An example of a perpetuity is a 50 per year fund given by the Queen to Winston Churchill s ancestors in recognition for their service to the crown Perpetuities are also used in various financial computations where the exact duration of a payment stream cannot be defined but it is expected to continue for a long period It can also be very useful as an approximation because the geometric infinte series equation for perpetuities simplifies dramatically6 to PPerpetuity R r There is no point in going in the opposite direction to find the future value of a perpetuity 7 it is simply infinite or at least indefinite As shown in graphs at the bottom of Spreadsheet tab quotDiscrete Factor Tablesquot in Workbook quotNote on PVxlsquot a perpetuiuty can be a reasonable approximation for many annuities where the rate of return is more than say 6 and n is greater than 20 or 30 The utility of this approximation will depend of course on the circumstances and it would never be appropriate for actual payment decisions Where in ation is a concern the perpetuity equation is also very simple Where d is the annual in ation rate see text below PPerpetuitywith in ation R 139 d Note on PVdoc Page 7 12 9Aug01 For further information readers are referred to Brealey and Meyers see supra note 1 and Humphreys see supra note 4 The latter details the formulae mentioned in this note and provides equations for a variety of additional factors and economic evaluation situations On the other hand BREALEY AND MEYERS is one of the most popular texts around on corporate nance and gives many examples of applications of these concepts in that context Compounding V Simple Interest and Nominal V Effective Rates The present and future value formulae given above apply compound interest or rate of return 7 quotinterest upon interest or interest on principal plus accrued interestquot 7 as contrasted with simple interest 7 quotwhere interest earned before maturity is neither added to the principal nor paid to the lender The difference between quotsimple interestquot and quotcompound interestquot is that quotsimple interestquot does not merge with principal and thus does not become part of the base on which future interest is calculatedquot Black39s Law Dictionary 812 813 6Lh ed 1990 This can be confusing producing radically different results For example if 100 is held at an interest rate of 10 per year for 7 years it will be worth about 195 if interest compounds and only 170 if it does not The difference is nearly a 15 increase Lawyers sometimes use simple interest rather than compound interest for simplicity or convenience or when circumstances require it 7 never out of ignorance In computing compound returns the value of r applied is called the effective rate All present and future value formulae require use of effective rates By definition an effective rate of return applies to a specific duration or payment period and the rate is assumed to be on an annual basis when the duration is omitted Financial institutions frequently gloss over this point often confusing the issue Instead real estate agents banks and others often use nominal interest which is truly only a convenience arising from a the desirability of standardizing on an annual payment period and b the difficulty of converting among compounding periods With calculators and PCs that conversion is quite easy today For example if rA is the effective annual rate of return and rM is the equivalent effective monthly rate of return the two values are related since 1 rAquot 1rMXquot where X number of M compounding periods per duration A a ratio 12 here Therefore for rA 12 effective annual interest which means compounded annually rx 1 r11 7 1 1 012 12 7 1 0948879 effective monthly interest compounded monthly and rA 12 effective annual interest 1 rxX 7 1 10094887912 7 1 12000 Note on PVdoc Page 8 12 9Aug01 In contrast with effective interest nominal interest must always be quoted with its compounding frequency and it is related to the effective interest by the following definition Where i the nominal interest rate per duration Y Y the Duration associated with the nominal rate time rx the effective interest or return rate per period and X the number of compounding periods per duration Y a ratio 7 then rx i X is the effective rate of return applicable to one period compounded X times per Y duration For example 12 nominal annual interest compounded monthly equates with 1 effective monthly interest rate also compounded monthly Normally an annual interest rate refers to a nominal interest when it is given along with the frequency of compounding or a payment frequency 7 e g monthly payments at 12 annual interest This raises the further question of how the frequency of compounding affects the resulting rate of return The equation used above to equate different can be used to compare a range of compounding periods getting progressively shorter until it reaches zero in the limit for continuously compounded interest7 The result is computed in Spreadsheet tab quotCompoundingquot in Workbook quotNote on PVXlsquot attached and it shows that the difference in factors is not terribly significant until one reaches rather high rates of return In ation In the real world periodic payment values revenues costs and other amounts are constantly changing 7 increasing and decreasing Present value and future value computations can accommodate those changes but the correct amounts are difficult to predict at best Consider the changes in energy prices over the past decade Because of this uncertainty analysts frequently assume that the in ation rate d will be constant and will affect all sums in the same manner Then if an amount A is needed now and amount A1d will be needed after one period usually a year Thus A12 A11X1 d Some High Math for those with a yen for it 7 As the number of times that we compound per year goes to infinity we can use the single payment future value formula 7 S P1 rn 7 to observe that in one year ie for n 1 asX gtoltgt rA1rXX711iXX711iXXA 71e 71 because the limit of 1 i XXi 1 1 ZZ where Z iX e the Naperian constant for natural logarithms 7 m 7 7 Thus for Contlnuous compoundlng FPSJ continuous e an Fsm continuous e PS continuousn39 Note on PVdoc Page 9 12 9Aug01 where All is the amount today and A12 is the amount expected n years from now The factor 1 dn is called a de ator This equation shows how in ation erodes values over time and conversely how de ation d lt 0 has the opposite effect In ation factors can be integrated with other factors already described3 but the resulting equations can be confusing and 7 as a practical matter 7 it is much easier to handle in ation year by year in a spreadsheet model as will be described below However note that the only real return that we realize from an investment 7 the only real growth that benefits us 7 is that which exceeds in ation If an investment only keeps up with in ation its future value due to in ation alone given above will be Sin P X l d Now if the investment actually grows faster than in ation we can evaluate that real growth and future amount as S Sinfl X 1 1Areal P X1 d X 1 1Areal P X1rquot P X FPSrn where rreal lt r so long as d gt 0 because r results from a combination of in ation and real growth Rearranging we get lrldgtltlrreal r gtrrea1m 1 This shows how the market rate of return r is discounted by in ation to obtain the real return rreal Another way of looking at this is that the market rate of return r is composed of 2 factors 7 one 1 d that accounts for predicted future in ation and another 1 rreal that re ects the real rate of return that investors will demand Sample calculations are provided in Spreadsheet tab quotIn ationquot in Workbook quotNote on PVxlsquot Two more definitions that are useful along with the concept of in ation are those for Current Dollars and Constant Dollars Current Dollars 7 quotDollars of the Dayquot as they actually happen 7 are dollars in the actual amounts that were or will be paid in or out without any adjustment for in ation The purchasing power of a current dollar varies over time in response to in ation A current dollar is worth 1 at all times regardless of what that will buy Constant Dollars in contrast are adjusted to remove the effects of in ation Thus the purchasing power of a constant dollar theoretically is fixed constant A stream of constant dollars must be adjusted using some historical or predicted de ator the GDP de ator for example and must be pegged to some index year eg quotConstant 1983 See HUMPREYS supra note 4 Chapter 6 Note on PVdoc Page 10 12 9Aug01 Dollarsquot A constant year 1 dollar will likely be different from greater or less than a constant year 2 dollar Practical Considerations for Computations Fortunately the discounting systems examined above can be combined in a variety of ways to solve more demanding problems For example results from different computations can be added and multiplied as appropriate For example absent returns to scale the present value of one unit can be multiplied to nd the present value of n units and the present value of various improvements andor costs can be added or subtracted to nd the total In addition the present or future value of a widget can be multiplied to find the value for n widgets and then translated forward or backward in time to a different date to determine the expected future or present value that may result The PV and FV factors discussed above were derived years ago before the advent of handheld calculators and desktop computers Although they are useful for understanding these concepts it is often at least as easy to use the discrete single payment formulae Fpsym and Fspym than to employ the annuity equations It usually is also more transparent and easier to monitor for errors In developing spreadsheet models it is good practice to compute the component parts so as to facilitate plotting and other work that can help the analyst check for errors For the reader39s convenience a number of sample problems have been worked in Workbook quotNote on PVxlsquot to illustrate the concepts discussed above 11 Where Does the Discount Rate Come From This is easily the most interesting issue related to economic evaluations but it is a very substantial topic by itself A few quick comments are in order here Note on PVdoc First in the normal business context the discount rate applied should be specific to the company and often to the specific project as well as to economic and related circumstances at any given point in time The expected effective rate of return to apply in evaluating a given project or company the discount rate can be decomposed into several parts 7 r rf rip rpmj other where r the discount rate normally in per year an effective rate rf the riskfree interest rate rip the industry risk premium rpmj any projectspecific or companyspecific risk premium that should be applied and Page 11 12 9Aug01 Note on PVdoc other 7 might include political or country risk and any other risk category that may be relevant Frequently the interest currently charged for an appropriate new US government security the safest nancial instruments in the world is used to specify rf quotAppropriatequot would mean a bond that matches the investment being considered as well as possible eg a bond of at least roughly comparable duration 7 lyear 30year or whatever Sometimes this is easier to say than to do Theoretically the industry risk premium can be derived from the securities market Market risk rm is the risk that the securities markets apply to the peer group of rms of which the target company is a member and rm rf rip In theory this value can be obtained by examining historical market prices for a peer group collection of securities but that is often easier to say than to do9 The Capital Asset Pricing Model CAPM which most certainly has its faults is probably the analytical tool most commonly used to examine rm and rip In the end one does the best one can with the tools at hand trying to view the problem in several different ways in order to get comfortable with whatever answers will emerge Sound familiar Of course the rest of the risk factors 7 rpmj other 7 are the most difficult to quantify for inclusion in a discount factor Industry knowledge company track record personal experience judgment and other such factors normally go into this analysis From the investor39s point of view a reasonably balanced portfolio will diversify away a portion of the risk 7 theoretically at least The analogue for a lender or an insurer would be a portfolio of loans or policies that spreads the risks and assures that defaults and claims will be offset by other customers However there will virtually always be a risk premium that cannot be diversified away and that premium must be added to the riskfree rate of return rf Brealey and Meyers has a good discussion of portfolio risk and diversification BREALEY AND MEYERS supra note 1 Chapter 7 et seq The End This thing has gotten a bit out of hand 7 and I suspect you agree It s time to close it down There are manyreferences on financial risk analysis BREALEY AND IAEYERS supra note I is one of the most popular text books on corporate finance and it includes a number of chapters on risk analysis and applications See also M AMRAN AND N KULATILAKA REAL OPTIONS MANAGING STRATEGIC INVESTMENT IN AN UNCERTAIN WORLD 1999 A good managerial overview of risk issues and the real options approach to handling them Page 12 12 9Aug01
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