CHAPTER 3: FINANCIAL ANALYSIS WITH INFLATION


 Amy Fletcher
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1 Up to ow, we have mostly igored iflatio. However, iflatio ad iterest are closely related. It was oted i the last chapter that iterest rates should geerally cover more tha iflatio. I fact, the amout of iterest eared over iflatio is the oly real reward for a ivestmet. Cosider a ivestmet of $1,000 that ears 4% over a oeyear period, ad assume that durig that year the iflatio rate is also 4%. At the ed of the year, the ivestor receives $1,040. Has the ivestor gaied aythig from this ivestmet? O the surface, he has eared $40, so it seems like he has. However, because of iflatio, the purchasig power of the $1,040 that he ow has is the same as the purchasig power that the iitial ivestmet of $1,000 would have had a year earlier. Essetially, the ivestor has eared othig. I this case, we would say that the real rate of retur, the rate of retur after iflatio, was zero. It is easy to cofuse discoutig ad compoudig, which accout for the time value of moey, with deflatig ad iflatig, which accout for the chagig purchasig power of moey. Iflatio reduces the purchasig power of a uit of currecy, such as the dollar, so a dollar at oe poit i time will ot have the same purchasig power as a dollar at aother poit i time. Iflatig ad deflatig are used to adjust a amout expressed i dollars associated with oe poit i time to a amout with equivalet amout of purchasig power expressed i dollars associated with aother poit i time. Chages over time i the purchasig power of a uit of currecy such as the dollar are ot the same as differeces i the values of costs ad reveues that result from their timig. To assure you that iflatig ad deflatig are truly differet from compoudig ad discoutig, try to imagie a world without iflatio; i.e., assume a world where the prices of goods ever chage. If the iterest rate merely accouts for iflatio, the the iterest rate would also be zero i such a world. But what would happe if the iterest rate was also zero? Would people still save ad ivest their moey? O first thought, it seems like some people would. They would wat to save so they would have moey for future purchases, right? But would that be ecessary? If they eed moey for future purchases, they would be able to borrow the moey whe they eed it with o iterest after all, the iterest rate is zero. Thikig about it this way, it seems likely that i a world with o iflatio ad with a iterest rate of zero, few would bother to save ad early everyoe would wat to borrow. There would be little to o supply of savigs ad a very high demad for moey to borrow. This would be a classic case of a market out of equilibrium. I order to have a equilibrium i the market for moey, the price of usig moey ad the reward for ledig moey would have to rise. The rate of iterest that is eared i excess of the iflatio rate is this price of moey. This rate, called the real iterest rate, is determied by the balace betwee the demad for moey to borrow ad the supply of moey from leders. Eve without iflatio it would still be ecessary to charge iterest to achieve a balace i the market for moey. I a world with iflatio, the iterest rate must cover both iflatio ad the cost of capital. FOREST RESOURCE MANAGEMENT 40
2 Because the true profitability of a ivestmet is determied by the amout eared over iflatio, it is very importat for you to be able to accout for iflatio whe doig a fiacial aalysis. Sice iflatio is early always preset i realworld situatios, you will ot be able to reliably apply the fiacial aalysis techiques you have leared so far without accoutig for iflatio. Geerally, the best way to accout for iflatio is to remove it from all of the values i your aalysis by covertig all values to curret dollar values ad by usig a real iterest rate that is, a iterest rate with iflatio removed from it. Furthermore, removig iflatio usually simplifies the aalysis of forestry ivestmets. 1. What is Iflatio? Iflatio is a icrease i the average price level, reducig the purchasig power of the dollar. The iflatio rate is the average aual rate of icrease i the price of goods. Iflatio is measured by a variety of idexes. The broadest ad most commolyused are the cosumer price idex (CPI) ad the producer price idex (PPI). The CPI tracks the average icrease i the cost of a stadard collectio of cosumer goods, ad the PPI tracks the average icrease i the cost of a stadard collectio of productio iputs. The stadard collectios of goods that defie these idices called market baskets by ecoomists are modified ad updated over time to reflect chages i techology ad cosumptio ad productio patters. The CPI ad the PPI are price idexes whose values are arbitrarily set to 100 at some poit i time. The Uited States Bureau of Labor Statistics (BLS) is resposible for calculatig the values of these idices for the U.S. dollar. You ca fid the values of these idices for differet poits i time by visitig the BLS web page at To calculate the iflatio rate over a period of time, you eed to choose a appropriate idex ad obtai the idex value at the begiig of the time period ad at the ed of the time period. The ratio of the two idex values idicates how much the average price level icreased over the time period betwee the two idex values. The average aual iflatio rate over the period betwee the two idex values is calculated usig a formula that is aalogous to solvig for the iterest rate i the sigle value discoutig formula: k = ( t t ) CPI CPI t t I the above equatio, k represets the average iflatio rate betwee times t 1 ad t 2. The PPI would be used i a aalogous way to calculate the average iflatio rate for producer goods. Example: Calculatig the average aual iflatio rate over a give time period. What was the average iflatio rate, as measured by the CPI, betwee 1977 ad 1997? Aswer: If you visit the BLS web page at you will fid that FOREST RESOURCE MANAGEMENT 41
3 there are may versios of the CPI. We will use the ew (revised) Cosumer Price Idex All Urba Cosumers (Series ID: CUUR0000SA0). The CPI was 60.6 i 1977 ad i Note that this meas that the average price level wet up more tha 150% durig this 20year period. Let k represet the iflatio rate ad CPI 97 ad CPI 77 represet the Cosumer Price Idex i 1997 ad 1977, respectively. The iterest rate versio of the siglevalue discoutig formula ca be used to calculate the aual iflatio rate, as follows: k CPI = 97 = CPI ( 97 77) = = %. 77 Thus, the average aual iflatio rate averaged almost exactly 5% over this period. 2. Compoets of the Iterest Rate It may help you develop a better uderstadig of the relatioship betwee iterest rates ad iflatio to thik about breakig the iterest rate dow ito the differet compoets that ifluece iterest rate levels. This discussio will also develop some of the basic termiology related to iterest rates. The iterest rate that most people usually talk about is called the omial iterest rate. The omial iterest rate (sometimes simply called the omial rate) is the iterest rate that is quoted by baks, credit cards, stock brokers, etc. The omial rate icludes both the cost of capital ad iflatio. It is the rate that is used to discout actual, iflated future values. Part of the omial iterest rate goes to cover iflatio, ad the rest is what is really eared o a ivestmet. What is left over after iflatio is called the real iterest rate. The real iterest rate (also called the real rate) is the rate eared o a capital ivestmet after accoutig for iflatio. Iflatio has bee removed from the real iterest rate. The real iterest rate should be used to discout future values that are expressed i curret dollar values. A omial iterest rate ca be broke out ito two compoets: the iflatio rate ad the real iterest rate. That is, the omial rate. the iflatio rate + the real rate. (The "." symbol is used here to idicate that this relatioship is oly approximately correct.) This relatioship is a critical oe, as it shows simply ad ituitively how omial iterest rates iclude compoets to cover both iflatio ad the cost of capital. FOREST RESOURCE MANAGEMENT 42
4 The real iterest rate is the real retur for ivestig oe s moey. As metioed i the last chapter, all ivestmets are ot alike, ad differet ivestmets should be expected to ear differet real rates of retur. Five factors that affect the expected real rate of retur of a ivestmet are: risk the degree of ucertaity regardig the retur o the ivestmet, illiquidity how difficult it is to withdraw cash from the ivestmet outside the ivestmet s usual paymet schedule, trasactios costs appraisal fees, loa origiatio fees, commissios, etc., taxes differet ivestmets have differet tax implicatios: such as capital gais treatmet, property taxes, etc., ad time period loger ivestmets ted to be less liquid ad more ucertai. Each of these factors geerally adds to the real rate of retur that a ivestmet must ear i order to be attractive to ivestors. That is, all other thigs beig equal, a riskier ivestmet must promise a higher expected real rate of retur tha a safer ivestmet. Similarly, a ivestmet that locks up ivestors moey would be expected to produce a higher real rate of retur, relative to a ivestmet where ivestors ca get their moey out at ay time. The real rate of retur o a ivestmet ca be broke dow ito compoets that accout for each of these factors, plus a compoet reflectig the pure cost of usig moey the pure iterest rate. The pure iterest rate is the real rate of retur eared by a imagiary, riskfree, perfectly liquid, taxfree ivestmet with o trasactios costs ad a very short time period. Obviously, o such ivestmet exists, but it is useful to thik of such a ivestmet as a stadard of compariso for other ivestmets. The closest actual ivestmet to this idealized stadard is the 3moth U.S. Treasury bill. The U.S. Treasury is ulikely to default o it s debt; there is a active market for tradig these bods; the oly taxes o the returs are icome taxes; i large deomiatios, the trasactios costs o these bods are small; ad the timeperiod of the bods is relatively short. Average historical real rates of retur o 3 moth treasury bills have raged from mius oe percet to three percet. All ivestmets ivolve some risk, varyig degrees of illiquidity, taxes, trasactios costs, ad varyig time periods. The expected real rate of retur of a ivestmet cosists of a pure rate of retur plus differig compoets, or premiums, for each of these factors. That is, the real rate. the pure rate + a risk premium + a illiquidity premium + a tax premium + a trasactios cost premium + a time period premium. Each of these premiums is additioal compesatio ivestors require i order to assume the additioal risk, illiquidity, taxes, trasactios costs, ad time period of a particular ivestmet. For example, riskier ivestmets require a risk premium to compesate ivestors for assumig a greater level of risk associated with those ivestmets. The more risky a ivestmet, the higher this risk premium will have to be. Without this premium, people would ot be willig to place their moey i more risky ivestmets. FOREST RESOURCE MANAGEMENT 43
5 I summary, omial iterest rates are made up of may compoets, each of which is ecessary to compesate ivestors for differet aspects of the ivestmet. A key compoet is iflatio. After iflatio, the remaiig iterest is called the real iterest rate, as this is the real retur o the ivestmet. The real rate, i tur, icludes several compoets, icludig a pure rate which accouts for the opportuity costs of usig moey plus premiums to accout for the riskiess, illiquidity, taxes, trasactios costs, ad time period of the particular ivestmet. 3. Combiig Iterest Rates So far i this chapter the compoets of iterest rates have bee treated as if they were additive. For example, it was stated that the omial iterest rate is basically the sum of the iflatio rate ad the real iterest rate. This view, however, is oly approximately correct. Just as the compouded iterest over two years is ot exactly 2 times the aual iterest rate, whe two iterest rate factors are combied to give aother iterest rate, the combied iterest rate is ot exactly the sum of the two other iterest rates. Thus, while the omial rate is a combiatio of the real rate ad the rate of iflatio, it is ot precisely correct to say that the omial rate is equal to the sum of the real rate ad the iflatio rate. While this is a pretty good approximatio, it is ot exactly correct. To see how iterest rates should be combied properly, cosider first the case of the twoperiod iterest rate, i 2. For two periods, the rate eared is: 2 2 i = i) i) 1 = 1+ 2i + i 1 = 2i + i 2i 2 Note that i 2 is approximately equal to 2i, but ot quite. There is the extra term i 2. Note that i 2 will geerally be small, however. Combiig two rates (such as the real rate ad the iflatio rate) to get aother rate (i this case, the omial rate) requires a aalogous formula. Before itroducig the formula, some ew otatio is eeded. Let: i = the omial iterest rate, r = the real iterest rate, ad k = the iflatio rate. Now, the formula for combiig the real iterest rate ad the iflatio rate to get the omial iterest rate is: i = r) k) 1 Note the similarity of this relatioship to the relatioship above for i 2. This expressio ca be expaded as follows: i = r) k) 1 = r + k + rk r + k FOREST RESOURCE MANAGEMENT 44
6 So, just as with the twoperiod rate, the combied rate here i this case, the omial iterest rate equals the sum of the two compoet rates (r+ k), plus the product of the two compoet rates (rk). The simplest form of the equatio expressig the relatioship betwee the omial rate, the real rate, ad iflatio, is: i) = r) k) This ca be rearraged to solve for i: i = r) k) 1 or, to solve for r: r = or, to solve for k: k = i) 1 k) i) 1 r) Example: Solvig for the Nomial Rate. If you wish to ear a real rate of at least 3% o a ivestmet ad you expect the iflatio rate will be 4% durig the ivestmet period, what is the miimum omial rate you must ear? Aswer: i = r) k) 1 = ( 103. )( 104. ) 1 = = 712%. Note that this also equals (.03) + (.04) + (.03)(.04) = Example: Solvig for the Real Rate. If the omial iterest rate eared o a ivestmet was 8% ad the iflatio rate was 3% durig the ivestmet period, what real rate was eared o the ivestmet? Aswer: r = i) ( 108. ) 1 = 1 = = 4. 85% k) ( 103. ) FOREST RESOURCE MANAGEMENT 45
7 4. Nomial Values ad Real Values You ve probably heard the phrase learig the value of a dollar. However, as you also have probably leared, the value of a dollar is ot costat. Whe my father started his career i the late 1940s, he eared a salary of about $4,000. At the time, this was a fairly respectable salary for someoe just graduatig from college. I terms of buyig power, it correspoded roughly to a startig salary of $26,000 today. That is, the buyig power of $26,000 i 1997 is roughly the same as the buyig power that $4,000 had i If i 1948 someoe was cosiderig a ivestmet that would pay them $26,000 i 1997, they would wat to accout for the fact that the purchasig power of $26,000 i 1997 would be about the same as the purchasig power of $4,000 i I other words, they would wat to kow that the promised omial value i 1997 dollars of $26,000 would have a correspodig real value i 1948 dollars of oly $4,000. A omial value is a value expressed i the currecy of the year i which the value occurs, i.e., a value that is expressed i dollars that have the purchasig power of dollars i the year whe the value occurs. For example, a omial value that will be paid or eared i the year 2010 is expressed i terms of dollars with the purchasig power that dollars will have i the year Because of iflatio, it is likely that a dollar will have less purchasig power i 2010 tha it does today. A real value is a value that is expressed i terms of dollars with the same purchasig power as dollars today, or at ay other meaigful referece poit i time. Note that from the perspective of people livig i 1948 a real value would be oe that is expressed i the dollars of their time 1948 dollars. Geerally, however, real values are expressed i curret dollars i.e., dollars with a equivalet amout of purchasig power as dollars today. Curret dollars are the relevat referece poit for most aalyses sice they are the dollars we are familiar with. Because of iflatio, whe speakig of future values, it is geerally ecessary to clarify whether values are real future values i.e., values expressed i dollars with the same purchasig power as dollars today or omial future values i.e., values expressed i terms of future dollars. Although both situatios are commo, if someoe does ot expressly state whether they are referrig to real or omial future values, you should assume that they are referrig to omial future values. For example, i legal cotracts such as the mortgage o a house future dollar trasactios are early always spelled out i omial terms. Thus, the mothly paymet o my house is a fixed, omial amout. If I am still makig paymets o it i 29 years whe it is scheduled to be paid off, the omial dollar value of my paymets will be the same as they are today. O the other had, we ofte thik of future values i real terms. For example, i estimatig the future reveue from a timber sale possibly twety, thirty, or eve more years from ow FOREST RESOURCE MANAGEMENT 46
8 it is commo to assume that the reveue ca be calculated by multiplyig the expected future harvest volume times a stumpage price similar to today s price. It would be very difficult to predict stumpage prices may years from ow, so it is commo to simply assume that the future price is likely to be similar to the price today. This may be a reasoable assumptio, although ot ecessarily the best. It is oly reasoable, however, if we mea that the real future price will likely be similar to today s price. Because of iflatio, the omial future price is almost certai to be higher tha today s price. I fact, over the last decade or so stumpage prices i may areas have icreased much faster tha iflatio. It may, therefore, be more reasoable to assume that stumpage prices will icrease faster tha iflatio. The ext sectio discusses ways to hadle this. I ay case, it is highly ulikely that stumpage prices will remai costat i omial dollars. That would be the same as assumig that real stumpage prices will declie. It is useful to covert omial values to real values so that we ca judge the value of a dollar amout relative to the currecy that we are familiar with today s dollars. For example, suppose you are promised $1,000 (a omial value) that you will receive i the year Sice this is a omial value, it is expressed i terms of actual dollars that you will receive i Because of iflatio, it is likely that $1,000 will have less purchasig power i 2010 tha it does today. It may be hard for you to judge, the, the true value of the promised amout. It would be useful for you to at least be able to estimate what the $1,000 will be worth i today s dollars a currecy you are familiar with. 5. Deflatig ad Iflatig Deflatig ad iflatig are used to covert omial values to real values ad vice versa. Deflatig ad iflatig are differet from discoutig ad compoudig because they do ot chage the time at which a value is assumed to occur; they oly chage the kids of dollars that those values are expressed i. Deflatig is the process of covertig a value expressed i the currecy of a give poit i time ito a value with a equivalet amout of purchasig power expressed i the currecy of a earlier time; for example, covertig a value expressed i 2010 dollars to a equivalet value expressed i 1998 dollars. Iflatig is the process of covertig a value expressed i the currecy of a give poit i time ito a value with a equivalet amout of purchasig power expressed i the currecy of a later time; for example, covertig a value expressed i 1998 dollars to a equivalet value expressed i 2010 dollars. There is a fudametal differece betwee iflatig past values to the preset ad deflatig future values to the preset. For past values, we have a history of the iflatio rate sice the past value occurred, ad a idex such as the CPI or the PPI ca be used to covert past dollar values to curret dollar values or vice versa. These idices accout for variatios i FOREST RESOURCE MANAGEMENT 47
9 the iflatio rate over time ad idicate the appropriate amout of iflatio for ay historical period. I order to iflate a past value to a curret dollars, the past value should be multiplied by the ratio of the curret price idex over the price idex for the time at which the past value occurred. For example, to covert the $4,000 that my father eared i 1948, use the CPI for ad the CPI for as follows: CPI'. Salary ($1997) = Salary ($1948) $4, $26, 97 = CPI = For future values, we ca oly guess what the iflatio rate will be betwee ow ad the time whe the future value will occur future values of the CPI or PPI obviously are ot available. Sice we ca hardly guess what the average future iflatio rate will be, it does t make sese to project a differet iflatio rate for each year. Thus, a sigle, average iflatio rate is usually projected. A mathematical formula which looks a lot like the formula for calculatig the preset value of a sigle sum is used to deflate future values. First, some otatio; let: V = k) V = V = k) V V k) ' 48 = a omial future value occurrig i year, ad = a real future value occurrig i year. The formula for deflatig a omial future value (i.e., calculatig the equivalet real future value) is: This formula ca be rearraged to give the formula used to iflate a real future value ad covert it ito a omial future value: These equatios are very similar to the differet versios of the basic formula for discoutig ad compoudig a sigle sum, but they are ot the same. First, these equatios do ot chage the time at which the values are assumed to occur. That is, the values o both the left ad the right sides of the equatios V ad V are future values; oe is a real future value ad the other is a omial future value. Thus, while deflatig ad iflatig appear to be mathematically the same as discoutig ad compoudig, they are coceptually quite differet. The secod key differece betwee the formulas preseted here ad the siglesum discoutig formulas is the rate variable. The iflatio rate is used to deflate ad iflate values, ad a discout rate (i.e., iterest rate) is used to discout or compoud values. Example: The Price of Milk i 1972 A gallo of milk cost about $1.00 i What did a gallo of milk cost the, i curret (1997) dollars? FOREST RESOURCE MANAGEMENT 48
10 Aswer: First, you will eed the CPI for 1997 ad The idex values ca be obtaied from the BLS web site, ad they are: CPI 72 = 41.8 ad CPI 97 = Now, the real price ($1997) of milk ca be calculated as follows: P CPI = Pm $1. 00 = CPI = 418. m, 72, $3. 84($1997) Example: The Real Value of $1,000 i 2010 If iflatio averages 3% per year betwee 1997 ad 2010, what will be the real value, i $1997, of $1,000 i 2010? Aswer: Deflate the $1,000 value, usig a 3% iflatio rate, for 13 years: V ' 10 = V' 10 ) k $1, 000 = 13 = ( 103. ) $ Accoutig for Iflatio Whe Discoutig Sice there are two kids of future values omial future values ad real future values it is ecessary to discuss the appropriate way to discout each. The rule is straightforward: L Discout real future values with a real iterest rate, ad discout omial future values with a omial iterest rate. The corollary to the last rule is that whe a preset value is compouded with a omial iterest rate the result is a omial future value, ad whe a preset value is compouded with a real iterest rate the result is a real future value. These rules should seem fairly obvious, but they are amog the most commoly violated rules i fiacial aalysis. Ufortuately, whe these rules are broke, the resultig aalysis is likely to be quite wrog. Usually such mistakes occur because the aalyst is ot absolutely clear whether the future value beig discouted is a real value or a omial value. Serious errors will occur if this distictio is ot clear ad explicit. Table 3.1 summarizes the use of the differet rates for differet types of operatios. (Figure 3.1, below, also helps show the relatioships betwee these operators ad the differet rates, preset ad future values.) Here are a few basic poits to keep i mid about real future values, omial future values, ad accoutig for iflatio i discoutig ad compoudig: L Real future values are uiflated; omial future values are iflated. L Covertig a omial future value ito a real future value is a example of deflatig. L Covertig a real future value to a omial future values is a example of iflatig. FOREST RESOURCE MANAGEMENT 49
11 L To covert real future values to omial future values or vice versa, you must use the iflatio rate. L To discout real future values, use a real iterest rate. L To discout omial future values, use a omial iterest rate. L Compoudig a preset value with a real iterest rate results i a real future value. L Compoudig a preset value with a omial iterest rate results i a omial future value. Table 3.1. Formulas for deflatig, iflatig, discoutig, ad compoudig real ad omial future values. Formula Variables (replace x, y, ad z i the formula) k, V, V r, V 0, V i, V 0, V y = z / (1 + x) Deflatig Discoutig real future values z = (1 + x) y Iflatig Compoudig; results i real future values Discoutig omial future values Compoudig; results i omial future values As metioed earlier, sometimes the omial price of a specific product will icrease at a rate differet tha the geeral iflatio rate. So, thigs ca, ad will, get eve more complicated. For ow, make sure you uderstad these relatioships. Chagig real prices are discussed i a later sectio. Example: Iflatio Example 1 A bod that you buy today for $ matures i 10 years, whe it will be worth $1,000. (Ca you verify that the bod ears 7%?) This $1,000 is a actual amout that you will get paid; therefore it is a omial amout, ad the 7% you will ear o the bod is a omial rate. a. If you expect iflatio to average 3% durig the 10year period util the bod matures, what real rate of retur will you will ear o the bod? Aswer: r = i). 1 = = = 38835%. k) 103. b. What will be the real future value of the bod? Aswer: Note that there are two ways to calculate the real future value of the bod: 1) by deflatig the omial future value ($1,000): FOREST RESOURCE MANAGEMENT 50
12 V V = k) $1, 000 = 10 = $ ( 103. ) or 2) by compoudig the preset value usig the real iterest rate we calculated above: 10 V = r) V = ( ) $ = $ I summary, the ivestmet of $ today ears $1,000 i omial dollars te years from ow. Due to expected iflatio durig that period, the buyig power of the $1,000 eared i te years will have the same buyig power as $ would have today. I other words, i te years a omial value of $1,000 will be eeded to equal a real value of $ Figure 3.1 summarizes these relatioships. Example: Iflatio Example 2 Assume a shortrotatio hybrid poplar platatio ca be established for about $600 per acre. Good lad will be required, which ca curretly be reted for $100 per acre per year (paid at the begiig of the year). You expect the lad ret to go up at about the same rate as the iflatio rate, which you expect to be about 4% a year. After 7 years, you project the platatio will produce 20 tos of chips per acre. The curret price for chips is $100 per to, ad you expect this price to go up at about the rate of iflatio. You would like to ear a real rate of 8% o your ivestmet. At this iterest rate, what is the preset value per acre of the hybrid poplar platatio veture? Aswer: At time zero, $700 dollars will be required to establish the platatio ad pay the first year s ret. Sice the lad ret is expected to go up at the same rate as iflatio, the real ret paid i years 1 through 6 will be costat, at $100/ac. (Thik about this for a miute, as it is a importat cocept.) I year 7, $2,000/ac (20 tos of chips at $100/to) will be eared. The table below shows the et reveue stream over time, i real ad iflated values. It also shows the preset value of each period's et reveue. Below the table are some example calculatios. To calculate the values for year 5: The real aual ret is $100 per acre. Because of the 4% iflatio rate, it has become (omially): 5 V = k) V = 104. $100 = $ FOREST RESOURCE MANAGEMENT 51
13 Figure 3.1. The relatioship betwee the real future value, the omial future value, ad the preset value usig the values from Example 1. This value is a omial future value, so it should be discouted with a omial discout rate. The omial discout rate is calculated as follows: i = k) r) 1 = ( 104. )( 108. ) 1 = Now, discout the omial value of $ dollars for five years usig this omial discout rate: $ = = $ ( ) V 0 5 Thus, the preset value of the ret for year 5 is $ Note that you should get the same preset value if you discout the real future value ($100) usig the real discout rate (8%): = $100 = $ ( 108. ) V 0 5 The preset value of the project is just the sum of the preset values from each year. You should verify some of the other umbers i the table for yourself, goig through these same steps. FOREST RESOURCE MANAGEMENT 52
14 Table 3.2. Real ad iflated cash flows for hybrid poplar example, ad preset values. Year Real Net Reveue Iflated Net Reveue Preset Value $2,000 $2, $1, Net Preset Value $4.70 Before leavig this example, cosider for a momet what the preset value of $4.70 per acre meas. This seems like a pretty small umber. Does it mea that this ivestmet is ot very profitable? No. The fact that the preset value of the project is positive idicates that the ivestmet does produce at least the desired alterate rate of retur; i.e., it ears at least a real rate of retur of 8%. This is a good real alterate rate of retur, so it meas the ivestmet is actually quite profitable. The fact that the et preset value is small idicates that the ivestmet does ot ear much more tha the 8% real alterate rate of retur. I other words, the ivestmet ears almost exactly 8% (real). If you worked through the above example, you may have oticed that it is easier to work with real values rather tha omial values. For example, because the real value of the ret is the same i each year the formula for a fiite aual series could be used to calculate the preset value of all of the ret paymets at oce. Also, it is easier to work with real values rather tha omial values i this example because they are easier to predict. It is really pretty hard to say what iflatio will be over the seve years of the project, ad it is ot likely to be costat every year. However, for most products, it is relatively safe over the log ru to assume that the prices of those products will icrease at about the same rate as iflatio whatever it turs out to be. This should geerally be your default assumptio. For some products, you may have reasos to expect that their prices may icrease somewhat faster, or slower, tha iflatio this is discussed i the ext sectio. I ay case, it is usually easier to FOREST RESOURCE MANAGEMENT 53
15 predict the chages i the future prices of products relative to the iflatio rate (i.e., to predict real future prices) tha to predict how the actual future prices of the products will chage (i.e., to predict omial future prices). It is also easier to work with real values because it is easier to judge whether real future prices ad costs seem reasoable tha to judge the reasoableess of omial future values. Real prices are expressed i today s dollars, ad we are familiar with the value of a dollar today. We are just as ufamiliar with the omial value of a dollar i 2020 as we are with the value of a ye or a frac. (For example, is 2,400 a reasoable price for a meal at a restaurat?) Fially, as discussed above, it is useful to work with real future values ad real iterest rates because, after all, it is the real rate of retur from our ivestmet that really matters. It does't do much good to make a omial rate of retur of 10% o a ivestmet if the iflatio rate is 12%. O the other had, a omial rate of retur of 4% ca be pretty good whe there is o iflatio. 7. Real Chages i Prices ad Costs Iflatio refers to the chage i the average level of prices. As is usually the case with averages, the prices of some goods go up faster tha this average, while the prices of other goods go up slower tha the average. Uiversity tuitio is a example of a price that has goe up faster tha iflatio. Oe study estimated that the average rate of icrease i tuitio rates was about 8 percet durig the decade of the 1980's. Durig the same time period, the geeral rate of iflatio was about 4 to 5 percet. Thus, uiversity tuitio wet up at a rate that was about 3 to 4 percetage poits faster tha the average rate of icrease i the price of everythig else. I other words, uiversity tuitio icreased at a omial rate of about 8 percet ad at a real rate of about 3 to 4 percet. This "real rate" is the rate tuitio "really wet up" relative to the price of everythig else. I order to assess the likely profitability of projects, it will usually be ecessary to estimate future costs ad output prices. As discussed earlier, i may cases the best assumptio is that real costs ad prices will stay moreorless the same as they are ow. (Note that because of iflatio it is ecessary to say real costs ad prices here. Otherwise, the phrase prices will stay the same would be ambiguous, ad, furthermore, it is geerally ot a good assumptio that omial prices will stay the same.) Sometimes, however, historical price series show cosistet treds that suggest that a alterative assumptio should be used. I these cases, it geerally is best to project real price treds rather tha omial price treds because iflatio has historically bee quite variable ad is difficult to predict. Projectig real price treds allows you to avoid havig to predict iflatio rates. Thus, price projectios will ofte use phrases such as prices (or costs) will stay costat i real terms, or prices will icrease at X% faster tha iflatio. FOREST RESOURCE MANAGEMENT 54
16 The followig termiology will help clarify the discussio of real price chages. The omial future price of a product is the expected price that will be paid for the product at a future date, expressed i future dollars i.e., dollars with the purchasig power that dollars will have at that future date. The real future price of a product is the expected price that will be paid for the product at a future date, expressed i curret dollars. The omial rate of price chage for a product is the expected or historical rate of chage i the omial price of the product over a particular period. The real rate of price chage is the expected or historical rate of chage i the real price of a product. I other words, it is the rate at which the price for that product has chaged, or is expected to chage, relative to the geeral rate of iflatio. Because iflatio is almost always positive, omial price chages are more likely to be positive tha egative. That is, the omial prices for most products usually icrease. Real prices, o the other had, are just as likely to go dow as up. Sice the geeral rate of iflatio is the average rate of chage for all products, the prices of about half of all products will go up faster tha the iflatio rate ad the prices of about half will go up slower tha the iflatio rate. Real price chages have bee adjusted for iflatio, ad, o average, real prices stay the same. Thus, the average real price chage must be zero. I aalyzig forestry ivestmets, a key price is the price that is received for stumpage, that is, the price of trees o the stump. I may areas, stumpage prices have bee goig up faster tha the rate of iflatio. I Louisiaa, for example, average oak sawtimber prices wet up at a real, aual rate of over 20% betwee 1988 ad Betwee the first quarter of 1992 ad the last quarter of 1997, average black cherry sawtimber prices i orthwester Pesylvaia wet up at a real, aual rate of about 7%. 1 Such shortterm icreases are ulikely to cotiue for log periods of time, but most logterm sawtimber stumpage price series show real price icreases of 1 to 2%. Pulpwood prices have ot geerally goe up faster tha iflatio for log periods of time. Recetly, however, hardwood pulpwood prices have made some relatively dramatic gais i the South. Just because the price of a product has bee goig up faster tha iflatio for the last few years does ot mea this tred must cotiue. Sometimes the real price of a product will go up for a while ad the go dow. Depedig o the period of time oe looks at, the price of gasolie has icreased at some times faster tha iflatio ad at other times slower tha iflatio. I 1976, I had a job pumpig gas. The price the was 59.9 per gallo. Today, i 1998, the price of gas is about $1.05 per gallo. This represets a omial price icrease of 1 For Pesylvaia stumpage prices, see the Timber Market Report, published by the Pe State Cooperative Extesio Service ( FOREST RESOURCE MANAGEMENT 55
17 about 2.6% per year. 2 If the 1976 price of gas is coverted to 1998 dollars, however, it is clear that the real price of gas has falle sice The 1976 price of gas ca be iflated to 1998 dollars usig the curret PPI ad the PPI for : P g, 1976($1998 /gal) = ($1976/gal) $1. 22/gal = 611. The 1976 price of gas, P g, 1976, i 1998 dollars, was $1.22/gallo, which is about 15% higher tha the price today. I other words, the real price of gasolie has falle at a aual rate of about mius 0.6% from $1.22/gal ($1998) i 1976 to $1.05/gal ($1998) today. A similar calculatio for the period betwee 1976 ad 1984, however, would show that the real price of gasolie icreased durig that period. Questio: What other products ca you thik of whose real prices have falle? Ca you thik of ay products whose omial prices have falle? We have discussed real ad omial discout rates, the geeral rate of iflatio, ad real ad omial rates of price chage for particular products. Some otatio would be useful for keepig track of all these rates; let: r = real discout rate, i = the omial discout rate, k = the expected (geeral) iflatio rate, k p = the expected omial rate of price chage for the price of a particular product (product p), ad r p = the expected real rate of price chage for the price of a particular product (product p). These differet rates are related to each other. First, ote that the omial rate of price chage for a particular product is approximately equal to the iflatio rate plus the real rate of price chage for the product. That is, k p. k + r p. The exact relatioship betwee these differet rates is give by the followig equatios: or, or, r p = k p) 1 k) 2 The formula for this is: k g = = ( ) FOREST RESOURCE MANAGEMENT 56
18 Now, let = the omial future price of product p i period, = the real future price of product p i period, ad = the curret price of product p. To calculate the expected omial future price of product p i period, it is ecessary to iflate the curret price usig the expected omial iflatio rate for product p for periods. The formula for this operatio is: P = ( + k ) P p, 1 p p, 0 To calculate the expected real future price of product p i period, iflate the curret price usig the expected real iflatio rate for product p for periods. The formula for this operatio is: P = ( + r ) P p, 1 p p, 0 These last two equatios ca be rearraged to calculate the omial ad real rates of price chage, respectively, whe you kow the prices at ay two poits i time (t 1 ad t 2 ). That is, ad k r p p = = P ( t t ) p, t2 P p, t ( t t ) P P p, t p, t Example: Calculatig Real Hard Maple Stumpage Price Chages i NW Pesylvaia I the first quarter of 1992, hard maple stumpage i orthwester Pesylvaia sold for a average price of $104/mbf. I the first quarter of 1997, the average hard maple stumpage price i the regio was $293. The PPI for the first quarter of 1992 was 115.9, ad i the first quarter of 1997 it was a. What were the real ad omial rates of chage i hard maple prices i the regio over this 5year period? Aswer: First, outlie what is kow: P HM,' 92 P HM,' 97 = $104/mbf = $293/mbf PPI 92 = PPI 97 = FOREST RESOURCE MANAGEMENT 57
19 Note that the first subscript, HM, stads for hard maple. The secod subscript idicates the year i which the price was observed. The asterisk () idicates that a price is a omial price. Calculatig the omial price chage is a straightforward applicatio of the equatio give just above: k HM = $ = % $104 Thus, omial hard maple prices icreased at a average rate of 23% per year! Several approaches could be used to calculate the real rate of price chage. Oe approach would be to calculate the average iflatio rate over the 5year period usig the two PPI values. The, the followig relatioship ca be used to solve for the real rate of price icrease: r HM k = khm ) = 1 k) Calculate the average iflatio rate as follows: = % Now, calculate the rate of real price chage: r HM = = %. Thus, the real stumpage price for hard maple sawtimber icreased, o average, by 20.5% over the fiveyear period. I other words, hard maple sawtimber stumpage prices icreased at a rate of 20.5% over iflatio a pheomeal rate! Note that the real rate of price icrease could also have bee calculated by covertig the 1992 price to 1997 dollars, as follows: P PPI = PHM,' = $104/mbf = $ /mbf ($1997) PPI HM,' Now, the real price chage ca be calculated usig the followig equatio: r HM = P HM,' 97 (' 97 ' 92) 5 P HM,' 92 = $ = % $ FOREST RESOURCE MANAGEMENT 58
20 Note that it was assumed here that 1997 is the base year, ad that real prices are expressed i 1997 dollars. However, we could just as well have assumed that 1992 is the base year. I that case, we could have coverted the price i 1997 to 1992 dollars. The we could have used the same formula, with prices for both years expressed i 1992 dollars. The result would have bee the same (except for roudoff error). b. What will the omial ad real hard maple stumpage prices be i 2005 if these treds cotiue? Aswer: The iformatio you will eed is: The hard maple stumpage price was $293/mbf i Iflatio is assumed to be %. The omial hard maple stumpage price is assumed to icrease by % each year. The real hard maple stumpage price is assumed to icrease by % each year. To fid the omial expected price i 2005, iflate the curret price by the projected omial rate of price icrease, as follows: 8 P = $293 /mbf ( ) = $1, /mbf ($2005) HM,' 05 To fid the real expected price i 2005, iflate the curret price by the projected real rate of price icrease, as follows: 8 P HM,' 05 = $293 /mbf ( ) = $1, /mbf ($1997) Example: The Hybrid Poplar Project with Chagig Real Prices Here we will recosider the price assumptio from the hybrid poplar example from the previous sectio. Recall that the platatio ca be established for about $600 per acre ad the curret lad ret is $100 per acre per year. After 7 years, the stad is projected to produce 20 tos of chips per acre. The curret price for chips is $100 per to, ad iflatio is expected to be about 4% each year. This time, we wo t assume that the price per to of chips will rise at the same rate as iflatio, however. Istead, cosider what will happe if the price of chips goes up 1 percet faster tha iflatio. (I.e., what if there is a 1 percet real aual icrease i the price of chips?) The real alterate rate of retur o the ivestmet is still 8%. What is the preset value of the hybrid poplar platatio veture per acre uder this ew assumptio? Aswer: Table 3.3 shows the ew et reveue stream over time, i real ad iflated values. It also shows the preset value of each period's et reveue. FOREST RESOURCE MANAGEMENT 59
21 Compare the values i the table with those i the earlier example. Note that oly the values i row 7 ad the NPV have chaged. This is because oe of the assumptios cocerig the et reveues for years 0 through 6 have chaged. The oly thig that chaged was the expectatio regardig the price of chips i year 7. To calculate the values for year 7, cosider first what will happe to the real price of chips. The price of chips is ow assumed to go up at a rate 1 percet over the rate of iflatio. This is the same thig as sayig that the real price of chips will go up by 1% each year. Thus, to calculate the real future price, iflate the curret price of chips by 1% over 7 years (ote that the subscript c is for "chips" i.e., P 7 c is the real price of chips i year 7): 7 P = r ) P = 101. $100 = $ c c 0c Table 3.3. Real ad iflated cash flows ad preset values for the hybrid poplar example with a 1% real price icrease. Year Real Net Reveue Iflated Net Reveue Preset Value $2, $2, $1, Net Preset Value $88.87 Thus the projected real price of a to of chips i year 7 (P 7c ) is $ The real future value of the reveue from the sale of chips is this price times the projected yield (Y 7 ): V 7 = Y 7 P 7c = 20 $ = $2, This is the real future value of the reveue from the sale of chips i year 7. FOREST RESOURCE MANAGEMENT 60
22 Now, cosider the omial future value of the reveue i year 7. First, calculate the omial price of chips i year 7. Sice the price of chips is assumed to go up at a rate 1 percet over the rate of iflatio, the expected omial rate of price icrease for chips is approximately 4% + 1%. The followig formula gives the precise rate of icrease: k = k) r ) 1 = ( 104. )( 101. ) 1 = = 5. 04% c c, 7 c The price of chips is assumed to go up at a omial rate of 5.04%. Now calculate the omial price of chips i year 7 (P c, 7 ) uder these assumptios: P = k ) $100 /to = $100 /to = $ /to c 7 7 Thus, the projected omial price of a to of chips i year 7 is $ The omial reveue i year 7 ca be obtaied by multiplyig this price times the projected yield: V 7 = Y 7 P 7c = 20 $ = $2, Now, to calculate the preset value of the reveue i year 7, either discout the omial future value ( $2,821.71) usig the omial discout rate (12.32%): $2, = = $1, ( ) V 0 7 Or, discout the real future value ($2,144.27) usig the real discout rate (8%): $2, = = ( 108. ) V 0 7 $1, As always, you should calculate these umbers for yourself. As before, the et preset value of the project is foud by summig the preset values from each year which results i a et preset value of $88.87/ac. Notice what the chage i the assumptios did to the et preset value i the example. With a assumed real price icrease of oe percet, the et preset value per acre is $88.87, compared with $4.70 with o assumed price icrease. Which is the "right" et preset value? It all depeds which sceario eds up beig closer to the truth! I fact, either of them is likely to be exactly correct because it is ulikely that either oe will be a perfect predictio of the future. 8. Study Questios 1. I a world with o iflatio, would iterest rates still be positive? Why or why ot? FOREST RESOURCE MANAGEMENT 61
23 2. I a world with o iflatio, would it still be ecessary to discout future values? Why or why ot? 3. What is iflatio? 4. How is iflatio measured? 5. What is the differece betwee a real iterest rate ad a omial iterest rate? 6. What is the differece betwee deflatig ad discoutig? 7. What characteristics of a ivestmet ifluece the real iterest rate that the ivestmet must ear i order to attract ivestmet fuds? 8. Explai why a riskier ivestmet would be expected to ear a higher rate of retur tha a safer ivestmet. 9. What is the differece betwee a omial future value ad a real future value? 10. Why is it ot ecessary to distiguish betwee real preset values ad omial preset values? 11. List some situatios where future values are likely to be expressed as omial values. List some situatios where future values are likely to be expressed as real values. 12. Explai why assumig that future omial stumpage prices will remai costat is equivalet to assumig decliig real stumpage prices. 13. Why is it ot realistic i most cases to assume that the omial price of a good will stay the same? 14. The formula for deflatig a omial future value is very similar to the formula for discoutig a sigle future value. How are these formulas differet? 15. Why is it usually simpler i fiacial aalyses of forestry problems to express all future values i real terms? 16. Explai why the average real price chage for all products is zero. 17. Explai why the real prices of particular products are just as likely to go dow as up. FOREST RESOURCE MANAGEMENT 62
24 9. Exercises 1. If you wat to ear a real rate of 7% o a 5year ivestmet, ad you expect iflatio to average 3% over the ext five years, what omial rate of retur must you ear o the ivestmet? 2. You have ivested $1,000 at a omial rate of 9%. a. What will be the omial value of your ivestmet after 20 years? b. If iflatio averages 3.5% over the ext 20 years, what will the real value of your ivestmet be after 20 years? c. What will be the real rate of retur o your ivestmet? 3. The CPI for 1960 was 29.6 ad the CPI for 1999 (so far) is a. What was the average rate of iflatio for cosumer goods betwee 1960 ad 1999? b. A loaf of bread sold for about 20 cets i What did a loaf of bread cost the, i curret (1999) dollars? 4. Assume that you ivested $1,000 te years ago ad that ow you have doubled your moey i the ivestmet; i.e., it is ow worth $2,000 (omial). a. What (omial) rate of retur have you made o your ivestmet? b. If iflatio has averaged 3.5% over the last 10 years, what real rate of retur have you eared? 5. You wish to edow your alma matter with a fud that will geerate a real value of $1,000 each year, forever, for scholarships. The fud is expected to ear a omial rate of 8%, ad iflatio is expected to average 3.5%. a. What real rate of retur is the fud expected to ear? b. How much moey will you eed to place i the fud to esure that a real value of $1,000 ca be withdraw each year? 6. Your fatherilaw is due to retire this year. He has saved up $250,000 for his retiremet. He has the moey ivested i a accout that ears, o average, a omial rate of 9% per year. He wats to assume that either he or your motherilaw will live for aother 25 years, so he wats to spread the moey out over a 25year period. He has asked you to help him determie how much he ca withdraw from this accout each year so that the purchasig power of his withdrawals will remai approximately costat each year ad so that the fud is used up after 25 years. a. How much should he withdraw i the first year if he expects iflatio to be about 4% o average over the ext 25 years? FOREST RESOURCE MANAGEMENT 63
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