ࡱ> ~q mbjbjt+t+ !AAe ]( F F F F 8~ $ F z : lnnnnnn$   z    .  l d l \ ll 6 DpBbF F .lThe Discount Rate for Wrongful Death and Injury Cases Richard O. Zerbe, Jr., Ph.D. July 7, 1993 I. INTRODUCTION It is generally accepted that the rate of return on conservative investments is to be used to discount future earnings to present value in cases involving injury or wrongful death. In determining the discount rate to use the period from 1953-1990 or some sub-period is usually used. This paper suggests that the use of this period or certain of its sub-periods results in a downward bias whether one uses nominal, real or net discount rates. Real discount rates are low because during the 1950's, 60's and 70's, expected inflation was less than actual inflation so that real interest rates were lower than the long run historical pattern. Net discount rates during the period from 1950 through 1973 are low because the rate of growth of capital relative to the rate of growth of labor was unusually high by historical standards not only in the United States but in all most developed countries. These differential rates of growth produced a relatively high rate of growth of wages and a relatively low return to capital so that the net discount rate during this period contains a downward bias when judged by long term historical standards and by the experience since 1980. II. The Relationship Between Real and Nominal Rates Nominal or real interest rates and are used to discount economic loss to present value in tort cases. Nominal discount rates are market rates in current dollars, that is, unadjusted for inflation. Real rates are nominal rates adjusted for inflation. Similar definitions apply to nominal and real wage growth. The relationships are approximately as follows: Market Rate (Nominal Rate) = Real Rate + Inflation (1) Real Rate = Nominal Rate - Inflation. (2) As long as the market discount rate is used with nominal wages and the real discount rate is used with real wages, the use of the nominal and real rates will give the same answer as long as the inflation component is the same. This may be seen by writing out the expression for the net present value of a wage stream:  EMBED Unknown  (3) where G is the nominal or market growth wages in wages, Wois the wage one period before the initial period, and R is the nominal or market discount rate. The nominal growth rate, G, will equal [(1 + I)(1 + g)]-1 where I is the Inflation rate and g is the real (inflation adjusted) growth rate. Similarly, the nominal discount rate, R, will equal [(1 + I) (1+ r)]-1 where r is the real (inflation adjusted) discount rate. The expression containing the inflation components will then divide out as long as the inflation components are the same in the denominator and numerator so that equation (4) may be written as:  EMBED Unknown  (4) That is, equations (3) and (4) shows that the NPV can equivalently be expressed in real or nominal terms. III. DEFICIENCIES WITH THE USUAL PROCEDURE Suppose we decide to reduce an economic loss to present value by using realized real or nominal discount rates for say the 1953-1990 period or for some sub period. In the usual procedure there are three important deficiencies with this procedure. First, real rates are usually calculated using actual rather than expected inflation during the period. If expected inflation differs from actual inflation this will create a bias in the use of either nominal or real rates. As is well known, it is expected real rates rather than actual or realized real rates that are conceptually correct. Expected real rates are found by subtracting expected inflation, rather than actual inflation, from the nominal interest rate. Expectations of inflation play a major role in determining current interest rates, but those expectations may turn out to be substantially in error so that the realized real rate of interest may be divorced from the ex ante forces that formed it. The second and interrelated deficiency is that even a time period as long as the 1953-1990 period can contain major biases that arise from the difference between expected and actual inflation and that therefore give an incorrect figure for the discount rate. The third deficiency is that future earnings are subject to financial risk and therefore the discount rate should be adjusted for financial risk. In this paper I will not consider this third deficiency. The Expected Rate of Inflation. The expected real rate of return (ERR) is the conceptually correct measure of the discount rate to use with inflation adjusted income streams. This is equal to the market or nominal rate of return minus expected inflation. That is, the expected real rate is approximately calculated as follows: ERR = Nominal Rate - Expected Inflation (6) The actual or realized real rate of return (ARR) is different; it is the nominal rate of return minus actual inflation. That is, the actual real rate of return (ARR) is, ARR = Nominal Rate of Return-Actual Inflation (7) The difference in the expected real rate and the actual real rate is then equal to ERR - ARR = (Actual Inflation - Expected Inflation) (8) That is, the real rate of return is given by ERR = ARR + (Actual Inflation - Expected Inflation) (9) Equation (9) shows the source of the bias, namely, the difference between actual and expected inflation. If actual inflation exceeds expected inflation, the actual real rate will be lower than the expected real rate of return by the difference between actual and expected inflation. The estimate of the expected real rate of return will then be too high. If expected inflation exceeds actual inflation, the reverse is true. Systematic bias between the expected real rate and the actual or realized rate will be small in the long run; otherwise there are gains from exploiting this bias. Thus, in calculating real rates of return, the longer the time period used the better. The Real Interest Rate During the 1950-1991 Period With the above definitions in hand we can establish proposition 1 which says: Proposition 1: Real interest rates during the 1953-1990 period understate the real discount rate to be applied to future periods. From the previous discussion it is sufficient to establish that during most of the 1950-1991 period, actual inflation exceeded expected inflation by significant amounts. That this is the case is well recognized. (Barro, 1993; Theis, 1982; Walsh, 1987; Huizanga and Miskin, 1984; Nelson and Plosser, 1982.) Table 1 shows the difference between expected inflation according to the Livingston Index and actual inflation by decade. A negative number means that actual inflation exceeded expected inflation, and a positive number indicates that expected inflation was larger. Table 1: Difference Between Expected and Actual Inflation 1950-59-1.43%1960-69-0.87%1970-79-1.55%1980-890.98%1990-910.02% Calculated from Barro (1993) pp 176-177. Calculated by [(1+Ie)/(I+Ia)]-1 where Ie is expected inflation for the year and Ia is actual inflation. Table 1 suggests that a real interest rate calculated using actual inflation during the period, 1950-1979 would underestimate the real interest rate by about 1.3 percentage points. The downward bias is probably greater than this since the Livingstone index used to calculate expected inflation produces less of a difference between expected and actual interest rates than other measures (See Table 4). Real interest rates during the period 1947 to 1980 were about 1%. For the whole period 1840 through 1990, omitting the war years, they averaged 5% (See Table 6). Table 1 suggest another proposition, namely: Proposition 2: The realized real rates of the 1980's and 1990 and 1991 will be better predictors of rates in the future than earlier post-war rates. Table 1 suggests that the real interest rate calculated using actual inflation in the 1980's and 1990's has a upward bias but that this upward bias is much smaller than bias of the previous three decades. In addition, Huizinga and Mishkin, (1984) Nelson and Plosser, (1982) and Walsh , (1987) present evidence to suggest there has been a shift in the structural real rate process beginning about October 1979. This suggests that expected real rates during the post-war period, before the 1980's, understate expected real rates in the near future. Expected real rates have been higher during the 1980's than earlier in the post-war period. (Walsh, 1987). Walsh finds that a 1% change in nominal rates during the period 1979 QIV to 1984 QIII was on the average produced by a 0.8% change in expected real rates and by a 0.2% change in expected inflation. Even aside from a change in structure, to some extent, changes in rates have a random walk component. That is, if the rate goes up there is no tendency to return to any average or trend line value (See Nelson and Plosser, 1982, for example). Thus, the use of the 1953-90 period to calculate actual real rates will lead to substantial errors in the calculation of the real discount rate because it includes a substantial period with a downward bias for expected real interest rates, and because the most recent period, the period since 1979, has shown higher expected real rates which should be given greater weight given the evidence of a change in the structural rate process and in the existence of auto correlation among rates.. These considerations, taken together, suggest Proposition 3: The real discount rate during the 1980's should be given more weight than the 1953-90 period as a whole. III. The Net Discount Rate One procedure that might appear to avoid the above difficulties is to calculate the net discount rate. The net discount rate is found by subtracting the growth in earnings from the interest rate. This procedure has led to arguments for a total offset method, which is simply the use of a zero discount rate applied to the assumption of the continuation of the existing earnings level, perhaps with a life cycle earnings adjustment (e.g. Parks, 19 ). The net discount rate may be defined approximately as: Net Discount Rate = Market Interest Rate - Nominal Wage Growth (4) An exact definition can be made by referring to equation (3). If we define k as [(1+R)/(1 + G)] -1, we can more formally define k as the net discount rate. Note that the above expression for K will approximately equal R-G. If R and G contain the same inflation component, then k will also equal r -g where the lower case letters refer to real components. Equation (3) may now be written solely in terms of real components as:  EMBED Unknown  (5) Clearly the use of the net discount rate will give the same answer as the use of nominal or real rates since equation (5) is the same as equation (3). The net discount rate will be influenced by capital labor ratios and by technological progress. If the growth of capital relative to labor were especially high during some historical period the rate of growth of wages to the interest rate would be particularly large, and the use of this period as a guide to the future net discount rate would be biased downward. The use of a time period such as 1953 to 1990 to calculate net discount rates raises the question of whether or not this is reasonably representative of the future. The Problem with the 1953-90 Period for Calculation of the Net Discount Rate The period from about 1950 to about 1973 was the "golden age of growth" for a period running from about the end of the Napoleonic Wars, say from 1820, to the present for all of the developed countries. For developing nations including the United States, as shown in Table 2. Table 2: Percentage Growth in Per Capita GDP and in Non-Residential Capital Stock for Developed Countries* Time Periods1870-19131913-501950-731973-87-89*GDP Per Person**1.41.23.81.6Capital Stock*** 3.42.05.84.2 *From Maddison, Table 4.9 pg 118 **For 16 countries. A listing of these is given in Maddison, Table 3.1, pg 49. *** For 6 countries including the US. For a listing see Maddison, Table 5.4, pg. 140. GDP per person during this period increased at significantly higher rate than in other periods, an average rate of 3.8 % per person per year for all developed countries, a rate for greater than occurs elsewhere in this period. For example, the growth rate during the period from 1870 to 1950 is about 1.3% per year in GDP per person. In the U. S. this period is not quite as dramatic but nevertheless clearly stands out as is shown in Table 3. Table 3: Compound Rates of Growth of Per Capital GDP, Net Capital Stock and Labor Productivity in the US. 1Time Period1820-701870-19131913-501950-731973-892GDPa4.53.92.83.62.73GDP per Headb1.51.81.62.21.64Net Non-Residential Capital Stockc1.693.842.595Adusted Labor Inputs d0.851.671.876Capital. minus Labor growth0.842.170.727Productivity: Output per Man Houre 1.92.42.51.0a. From Maddison Table 3.2, pg. 50 b. From Maddison Table 3.1 pg. 49 c. From Maddison, Table 5.5 pg. 141 d. From Maddison Table 5.3, pg. 135 e. From Maddison Table 3.3 pg 51 Table 2 shows that the period 1950 to 1973 was a period of unusual growth in capital relative to labor and that therefore wages should be unusually high during this period relative to the return to capital, that is relative to interest rates. This, justifies proposition 4: Proposition 4: Net discount rates during the 1950-1973 period are lower than during other historical periods probably since 1820. Proposition 4 implies propostion 5 which is: Proposition 5: Net discount rates during the period 1950-73 are biased downward when applied to future net discount rates. IV. A CORRECT APPROACH The Real Discount Rate and the Net Discount Rate Expected Real Rates in the Postwar Period We have suggested that examining actual real rates in the post-war period, except for the 1980-90 period, will yield biased estimates of real discount rates and that net discount rates during this period may also be biased. Several alternatives are possible. We can examine expected instead of actual discount rates, and we can examine longer time periods. We do both of these below. Table 4 shows expected real rates calculated from several sources. TABLE 4 EXPECTED REAL RATESABCDEFG PERIOD EXPECTED REAL RATES TREAS BONDS 1 YEAR (LIVINGSTONE ADJUSTED) EXPECTED REAL RATES TREAS BONDS 1 YEAR (DRI ADJUSTED)1 percent EXPECTED REAL RATES TREAS BONDS 3 YEAR (DRI ADJUSTED)2 percent EXPECTED REAL RATES TREAS BONDS 20 YEAR (DRI ADJUSTED)3 percent EXPECTED REAL RATES PRIME RATES 15 YEARS (DECISION MAKERS POLL) (HAVRILESKSY)4 percentEXPECTED REAL RATES TREAS BONDS 1 YEAR THIES (1986) AV. OF BUYING AND SELLING PRICE EXPECTATIONS percent 1953-89 2.021953-842.851980-845.821975-893.124.171977-893.543.814.40Sept 78- Nov 874.111980-894.504.685.551. The Livingstone adjusted figures are calculated from Barro (1993) pg. 176-77. 2. Treasury yields to maturity for one year notes minus one year expected inflation as calculated from quarterly minus one year expected inflation data provided by Data Resources Inc. Treasury yields are from the Federal Reserve Bulletin. 3. Treasury yields to maturity for three year notes minus three year expected inflation as calculated from quarterly data provided by Data Resources Inc. treasury yields are from the Federal Reserve Bulletin. 4. Calculated by multiplying the ratio of the nominal yield of twenty year treasury bonds to that of 1 and treasury bonds by the yields in columns b and c. Yields are from data published by Salmon Brothers yields are arithmetic averages. 5. Data are approximately bi-weekly. There are 51 observations. 6. Treasury yields to maturity for one year notes minus one year expected inflation as calculated by Thies, for business expectations of buying and selling price inflation. The rates shown in Table 1 vary from 2.09 to 5.82%. The 2.09% real rates result from use of the Livingstone poll. Unlike all of the other polls this is based on a survey of economists. The other polls based on expectations of businessmen and may more accurately reflect what the market expects. These real rates are significantly higher than the 1% or less that is sometimes cited. Real Rates Over Long Periods TABLE 5 REALIZED REAL RATESABCDEFG PERIODPRIME COMMERCIAL PAPER (CPI ADJUSTED)1 percent AMERICAN RAILROAD BONDS (CPI ADJUSTED)2 percent 1 YEAR TREAS NOTES (CPI ADJUSTED)3 percent 3 YEAR TREAS BONDS (CPI ADJUSTED)3 percent 20 YEAR TREAS BONDS (CPI ADJUSTED)4 percentI INFLATION RATE CPI 5 percent1857-609.381865-898.861881-19154.27 [2.3]0.16 [2.1]1885-18934.62 [.13]4.170 [0]1890-19155.24 [2.3]3.76 [2.3]0.48 [2.1]1920-295.385.161953-88-891.961.92.232.461977-892.923.23.563.971980-894.27 [0.023]4.194.695.17Figures in brackets are standard deviations 1. Historical Statistics of the United States, Bicentennial Edition, Commerce, Bureau of the Census, Washington DC, 1988, pg. 996, 1001, series X-445. 2. Historical Statistics of the United States, Series X456-465, pg. 1002. 3. The Economic Report of the President, Washington, DC 1988, 1990. Federal Reserve Bulletin, selected months. 4. Analytic record of Yields and Yield Spreads from 1945, Salmon Brothers Inc. 5. Historical Statistics of the United States, Series E, pgs. 210-212. The Economic Report of the President, selected years. Table 5 shows actual real rates, that is actual rates minus actual inflation, for various yields and time periods. The actual real rates for 20-year United States Treasury Bonds varies from 2.46% to 5.17%. The rates are similar though slightly higher for 30-year Treasury Bonds. Rates for periods with little inflation and with little change in inflation have a particular appeal since rates during these periods are likely to be less influenced by expectations of inflation or by changes in inflation. Because of low levels of inflation and/or a low variance in inflation rates, three periods of particular interest are the periods 1881-1951, 1885-93, 1890-1915. During these periods, actual real rates on American Railroad bonds varied between 3.76% to 4.62%. Rates on Prime Commercial Paper in the latter period averaged 5.24%. In the period 1885-93, inflation was zero throughout. In this eight-year period, the average yield on American Railroad bonds was 4.62% with a standard deviation of 0.13%. The range of rates two standard deviations to either side of this 4.62% rate is between 4.36% to 4.88%. The smallest actual real rates mentioned in Table 5 are for the period 1953-88-89 and are about 2.5% for 20-year bonds. But, as is now well recognized, part of this period included egregious misjudgments about what the rate of inflation would be. The highest rates in Table 2 are those for the period before 1900. These should not be given as much weight as more recent rates. The rate of 4.62% for the period of greatest economic stability, 1885-1893, lays well within the range of rates derived from the expected real rates in Table 1 of 3.5% to 5.5%. A range of rates during this period of greater economic stability based on four standard deviations around the mean is included in this range. The figures here are supported by calculations from another source. Table 6 shows long term realized real interest rates calculated by Barro (1993). Table 6: Real Interest Rates (Percent) 1840-18609.11867-18809.11880-19006.31900-19163.11920-19404.91947-1960-0.21960-19801.21980-19904.8Weighted Average 1840-19904.87Weighted Average 1840-1990 without 1947-605.38Based on Rates for four to six months commercial paper and the GNP deflator From Barro (1993, Table 11.1, p. 285) Because of price controls during the Korean war the figures between for the early 50's are probably not representative on a non-price controlled period Table 6 suggests that historical real rates are much higher than the 1% often mentioned. Table 6 also supports proposition 1 and suggest proposition 6: Proposition 6: A figure of about 5% is a reasonable estimate of the real discount rate. Not only is this about the long term historical average, but it is about the rate that has prevailed in the most recent period. Long term real productivity growth has been about 1.5-2.0 percent per year which suggests a net discount rate of about 3.5 to 3.0 percent. Examining the Net Discount Rate We examine the net discount rate over a longer time period than the post war-period. Table 7 below shows the net discount rate from 1890 to 1990. Column F shows the net discount rate using wage compensation. Column E shows the net discount rate using total compensation. Total compensation figures are not reported before 1948; they differ from wages in including fringe benefits. Benefits were not increasing as a percentage of wages before W.W.II, and immediately following . This is suggested by the fact that the percentage increase in wages is the same as for total compensation for the 1953-59 period. Thus the wage should give an accurate figure for the net discount rate for the pre-war period. In the post-war period, the yearly increase in fringe benefits has been about 0.6 percentage points greater than the increase in wages. This is shown by Table 7 below. If we use the growth of wages before the war, not counting the depression, we find that real earnings grew at about 2.17 % per year. Since these are changes in real compensation, they must be subtracted from changes in real interest rates to obtain a net discount rate. Comparing this with our expected real rate of about 3.5 to 5 percent in Table 1 gives a net discount rate of about 1.3% to 2.8% as the net discount rate. A direct determination of real rates earnings and total compensation gives a similar result as shown in Tables 7 and 8. Table 7: Net Discount Rates ABCDEFPrimePercentagePrime Com.PercentagePrime Com.CommercialChange in Paper minusChange in TotalPaper minusPaperMftg. Wages% change inCompensation% Change in Total(CPI adjusted) (Average)Mfg. Wages(CPI adjusted)CompensationYEARS(CPI adjusted)Column B - CColumn B - E1890-996.90%1.21%5.69%1900-094.81%1.69%3.11%1910-143.25%1.22%2.03%1919-294.14%3.92%0.22%1930-393.86%3.41%0.45%1940-45-3.55%3.83%-7.39%1946-49-5.54%0.59%-6.13%1950-52-1.57%2.30%-3.88%1953-591.47%2.85%-1.38%2.86%-1.39%1960-692.26%1.48%0.77%2.45%-0.20%1970-790.09%0.61%-0.52%0.02%0.07%1980-894.02%-0.69%4.71%0.31%3.71%1990-912.15%-1.34%3.50%-0.14%2.29%*from Newzerbe*newzerbe*earningSeries XSeries D 802Alaska Law Rev.444-455Mfg. Average Dec. 1985 &HSUS part 2Hourly Earningseconomic rept.p. 1001of pres. '92Source: Board ofGovs. of the FRS. also, EconomicRept. of Pres. Feb. '92pg. 378 Column D in Table 7 shows the net discount rate. Column C shows the figures using the rate of return on commercial paper and manufacturing wages. Column E shows the net discount rate using total compensation. The shaded areas show the applicable net discount rate for the indicated periods. The use of Commercial paper rates give a downward bias to the rates since these are very short term rates and are used only because they are available for long time periods. Table 8 shows the net discount rate for selected periods. For the period before W.W.II , the rate of growth in earnings is used in calculating the net discount rate, and for the period after the war, the growth in total compensation is used. The war periods should be disregarded because controls on wage and interest rates do not allow the calculation of accurate figures. A depression as severe as that during the 1930's is unlikely to occur again so the depression years should also be eliminated. The argument for treating the period 1950-as special has already been made here. Finally, the period 1946-49 shows negative real interest rates and represents the effect of a continuation of war time controls so that this period also should not be included. The figure of 2.47 % appears to represent the better figure. The period producing this rate eliminates the war periods, the depression years, the period from 1950-1970 because of its special nature, and the period 1946-49 when controls led to a negative real discount rate. This rate is especially supportable because of the downward bias from using commercial paper rates. A net discount rate of zero or less than 1.0 percent appears to be without justification. Table 8: Average Net Discount Rates for Selected Periods Av. Net Discount Rate w/o War Periods 1.44 %Av. Net Discount Rate w/o War Periods & 1950-70 1.71%Av. Net Discount Rate w/o War Periods & Depression 1.91 %Av. Net Discount Rate w/o War, Depression and Period 1946-19692.47 % REFERENCES Barro, Robert J. Macroeconomics, (3rd & 4th Edition) John Wiley & Sons, New York, 1992 and 1993. Huizinga, John and Frederic S. Mishkin , Inflation and Real Interest Rates or Assets with Different Risk Characteristics," The Journal of Finance, Vol. 39, No. 3, July 1984, pp. 699-712. Reprinted as NBER Reprint No. 611. Nelson, Charles and Charles Plosser, "Trends and Random Walks in Macroeconomic Times Series: Some Evidence and Implications," Journal of Money Credit and Banking, Vol. 10, No. 2, (September, 1982) Thies, Clifford "Business Price Expectations", Journal of Money, Credit and Banking, Vol. 18, No. 3, August 1986, pp. 336-354 Walsh, Carl E."Three Questions Concerning Normal and Real Interest Rates," Economic Review, San Francisco Federal Reserve, Fall, 1987. pp. 5-19.] Zerbe, Richard O. and Dwight Dively, Benefit Cost Analysis in Theory and Practice, Harper Collins, New York, 1993 (in press) The exact definitions are, NR = [(1 +r)(1+I)]-1, and r = [(1 + R)/(1 + I] -1.where NR is the nominal rate, R is the nominal rate, r is the real rate of interest ,and I is inflation. Note that this equals I + g + Ig or approximately just I + g. See Barro (1992, pp172-180).  This is an approximation of the correct calculation which is : ERR = [(1 + nominal rate)/ (1 + expected inflation)] -1 A similar expression is the correct expression for the ARR. The expressions used in the body of the text are, however, more intuitive. A comparison of the Livingstone Index with others (see Table X) suggests that it produces a smaller difference than others between expected and acutal inflation so that the bias imay be greater than is indicasted in Table 1.  The calculations here do not take into account life cycle effects and some economists incorporate these into the net discount rate. On the other hand, the calculations here do not take into account any increase in interest rates to account for financial risk. 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dd$e+e,e]ecedeeeeeeeeeܼ-$$P0p"$$$"$$P"eUf3ggwh iiiEjjjj kklmmmmmmmmm$ ) p@ P !& ) hp@ P !mm0/ =!"#8$%Ddpp0  # A2OuϪ`,gD`!_OuϪ`,p-xW=kQ=~+Sl"c!eqe, ?B,EBH$tba#`Z7#xXv=yM zG*xTj|ֶ #1T>0 _&bZ3WK57?u,όM2OF6?vkriYO !!S'*LHU#Kq_:bGqᾋ*lc'.rb'N|Wȉླ?PD;>oJ{吊FI [Js=u%9ErEyʌ)aLVþx!UjCiۂ.~ᅙVW Jx'δ/*[L=Js(מî8>(iᓒ:3w'&yPI{S.'zګay:K^I{5 ƩCg85Ik:&g]LkO>q Dd  0  # A2OuϪ`,~`!OuϪ`, VxV=KA}OXr50XXX؈D "~ "`aBlli8ˑ;Nc.lf9;̾7s i WEWHoU4f1F4fHew^!Y)}჊ƽjw<d:W qm",1Ӹ5f ZT4SRӌl)fR 3լ鈙iu׿3>nr e^a7P;ܰ݇)?-'5ռ  ֓b`Jj5 b1IM}+u5-NA:܈׵ K*3]|Qƚ[oHNz< ԫ_OSIrmNw_䣕:IN>[_} J9|)Ct4'd  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuwxyz{|}Root Entry" F#%Yb>Data vWordDocument!!ObjectPool$Ћb%Yb_919846056FЋbpbOle PIC LPICT G   !"#$%&'(*-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQSTUXYZ[^`abcdefghijklmnopqrstuvwxyz{|}~L ip G dxpr  " currentpoint ",Times .+NPV, Symbol)=( (W ) o )()1)+) G) )(/()1)+) R))"'.(Y+( cW ) o )()1)+) G) ) (2 (j()1)+) R)) (2"c3 (+)......( W ) o )()1)+) G) ) (n (()1)+) R)) (n"49/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7552 div 1024 3 -1 roll exch div scale currentpoint translate 64 40 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1206 469 moveto 1513 0 rlineto stroke 3108 469 moveto 1675 0 rlineto stroke 5758 469 moveto 1675 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (NPV) 13 568 sh (W) 1225 322 sh (G) 2291 322 sh (R) 2109 869 sh (W) 3127 322 sh (G) 4193 322 sh (R) 4011 869 sh (W) 5777 322 sh (G) 6843 322 sh (R) 6661 869 sh 224 ns (o) 1544 322 sh (o) 3446 322 sh (o) 6096 322 sh (n) 7265 152 sh (n) 7031 699 sh 384 /Symbol f1 (=) 876 568 sh (+) 2003 322 sh (+) 1798 869 sh (+) 2809 568 sh (+) 3905 322 sh (+) 3700 869 sh (+) 4873 568 sh (+) 6555 322 sh (+) 6350 869 sh 384 /Times-Roman f1 (\() 1659 322 sh (\)) 2578 322 sh (\() 1454 869 sh (\)) 2344 869 sh (\() 3561 322 sh (\)) 4480 322 sh (\() 3356 869 sh (\)) 4246 869 sh (......) 5146 568 sh (\() 6211 322 sh (\)) 7130 322 sh (\() 6006 869 sh (\)) 6896 869 sh 384 /Times-Roman f1 (1) 1762 322 sh (1) 1557 869 sh (1) 3664 322 sh (1) 3459 869 sh (1) 6314 322 sh (1) 6109 869 sh 224 ns (2) 4614 152 sh (2) 4380 699 sh end MTsave restore dMATH) NPV=W o (1+G)(1+R)+W o (1+G) 2 (1+R) 2 +......W o (1+G) n (1+R) n sFMicrosoft Equation Editor 2.0DNQE Equation.2TCIP iG G dxpr  " currentpoint ",Times .+NPV, Symbol)=( (W ) o )()1)+) G) )(/()1)CompObj )YObjInfo+OlePres000 ,o Ole10NativeR+) R))"'.(Y+( cW ) o )()1)+) G) ) (2 (j()1)+) R)) (2"c3 (+)......( W ) o )()1)+) G) ) (n (()1)+) R)) (n"49/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7552 div 1024 3 -1 roll exch div scale currentpoint translate 64 40 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1206 469 moveto 1513 0 rlineto stroke 3108 469 moveto 1675 0 rlineto stroke 5758 469 moveto 1675 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (NPV) 13 568 sh (W) 1225 322 sh (G) 2291 322 sh (R) 2109 869 sh (W) 3127 322 sh (G) 4193 322 sh (R) 4011 869 sh (W) 5777 322 sh (G) 6843 322 sh (R) 6661 869 sh 224 ns (o) 1544 322 sh (o) 3446 322 sh (o) 6096 322 sh (n) 7265 152 sh (n) 7031 699 sh 384 /Symbol f1 (=) 876 568 sh (+) 2003 322 sh (+) 1798 869 sh (+) 2809 568 sh (+) 3905 322 sh (+) 3700 869 sh (+) 4873 568 sh (+) 6555 322 sh (+) 6350 869 sh 384 /Times-Roman f1 (\() 1659 322 sh (\)) 2578 322 sh (\() 1454 869 sh (\)) 2344 869 sh (\() 3561 322 sh (\)) 4480 322 sh (\() 3356 869 sh (\)) 4246 869 sh (......) 5146 568 sh (\() 6211 322 sh (\)) 7130 322 sh (\() 6006 869 sh (\)) 6896 869 sh 384 /Times-Roman f1 (1) 1762 322 sh (1) 1557 869 sh (1) 3664 322 sh (1) 3459 869 sh (1) 6314 322 sh (1) 6109 869 sh 224 ns (2) 4614 152 sh (2) 4380 699 sh end MTsave restore dMATH) NPV=W o (1+G)(1+R)+W o (1+G) 2 (1+R) 2 +......W o (1+G) n (1+R) n s NPV=W o (1+G)(1+R)+W o (1+G) 2 (1+R) 2 +......W o (1+G) n (1+R) n Equation NPV=W o (Ole10FmtProgID  V Equation Native W _919846077F#b#bOle \1+G)(1+R)+W o (1+G) 2 (1+R) 2 +......W o (1+G) n (1+R) nL`  x'dxpr  '"'PIC ]LPICT _x CompObjYObjInfo currentpoint ",Times .+NPV, Symbol)=((W ) o )()1)+) g))(/()1)+) r))"',(W+( bW ) o )()1)+) g)) (2 (i()1)+) r)) (2"a2 (+).....( W + 0(n (( )1)+)g )) (n (#()1)+)r))"2D/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7328 div 1248 3 -1 roll exch div scale currentpoint translate 64 40 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1214 597 moveto 1438 0 rlineto stroke 3047 597 moveto 1634 0 rlineto stroke 5568 597 moveto 1636 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (NPV) 13 696 sh (W) 1233 469 sh (g) 2313 469 sh (r) 2106 906 sh (W) 3098 347 sh (g) 4178 347 sh (r) 3955 967 sh (W) 5587 402 sh (g) 6684 448 sh (r) 6461 1093 sh 224 ns (o) 1552 469 sh (o) 3417 347 sh (n) 7067 152 sh (n) 6765 797 sh 384 /Symbol f1 (=) 880 696 sh (+) 2014 469 sh (+) 1813 906 sh (+) 2745 696 sh (+) 3879 347 sh (+) 3662 967 sh (+) 4774 696 sh (+) 6417 448 sh (+) 6200 1093 sh 384 /Times-Roman f1 (\() 1667 469 sh (\)) 2511 469 sh (\() 1466 906 sh (\)) 2274 906 sh (\() 3532 347 sh (\)) 4376 347 sh (\() 3315 967 sh (\)) 4123 967 sh (.....) 5050 696 sh (\() 5885 1093 sh (\)) 6629 1093 sh 576 ns (\() 6043 448 sh (\)) 6877 448 sh 384 /Times-Roman f1 (1) 1770 469 sh (1) 1569 906 sh (1) 3635 347 sh (1) 3418 967 sh (1) 6205 448 sh (1) 5988 1093 sh 224 ns (2) 4511 181 sh (2) 4258 797 sh (0) 5869 498 sh end MTsave restore dMATH , NPV=W o (1+g)(1+r)+W o (1+g) 2 (1+r) 2 +.....W 0 * n ( e1+g ) * n  e(1+r) rFMicrosoft Equation Editor 2.0DNQE Equation.2OlePres000 Ole10NativeOle10FmtProgID  Equation Native 'TCIP`x x'dxpr  '"' currentpoint ",Times .+NPV, Symbol)=((W ) o )()1)+) g))(/()1)+) r))"',(W+( bW ) o )()1)+) g)) (2 (i()1)+) r)) (2"a2 (+).....( W + 0(n (( )1)+)g )) (n (#()1)+)r))"2D/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7328 div 1248 3 -1 roll exch div scale currentpoint translate 64 40 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1214 597 moveto 1438 0 rlineto stroke 3047 597 moveto 1634 0 rlineto stroke 5568 597 moveto 1636 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (NPV) 13 696 sh (W) 1233 469 sh (g) 2313 469 sh (r) 2106 906 sh (W) 3098 347 sh (g) 4178 347 sh (r) 3955 967 sh (W) 5587 402 sh (g) 6684 448 sh (r) 6461 1093 sh 224 ns (o) 1552 469 sh (o) 3417 347 sh (n) 7067 152 sh (n) 6765 797 sh 384 /Symbol f1 (=) 880 696 sh (+) 2014 469 sh (+) 1813 906 sh (+) 2745 696 sh (+) 3879 347 sh (+) 3662 967 sh (+) 4774 696 sh (+) 6417 448 sh (+) 6200 1093 sh 384 /Times-Roman f1 (\() 1667 469 sh (\)) 2511 469 sh (\() 1466 906 sh (\)) 2274 906 sh (\() 3532 347 sh (\)) 4376 347 sh (\() 3315 967 sh (\)) 4123 967 sh (.....) 5050 696 sh (\() 5885 1093 sh (\)) 6629 1093 sh 576 ns (\() 6043 448 sh (\)) 6877 448 sh 384 /Times-Roman f1 (1) 1770 469 sh (1) 1569 906 sh (1) 3635 347 sh (1) 3418 967 sh (1) 6205 448 sh (1) 5988 1093 sh 224 ns (2) 4511 181 sh (2) 4258 797 sh (0) 5869 498 sh end MTsave restore dMATH , NPV=W o (1+g)(1+r)+W o (1+g) 2 (1+r) 2 +.....W 0 * n ( e1+g ) * n  e(1+r) r NPV=W o (1+g)(1+r)+W o (1+g) 2 (1+r) 2 +.....W o (1+g) 2 (1+r) 2 Equation  NPV=W o (1+g)(1+r)+W o (1+g) 2 (1+r) 2 +.....W 0 * n ( e1+g ) * n  e(1+r)L@h R dxpr  " currentpoint ",Times .+NPV, Symbol)=( .Wo('()1)+) k))"&(I+( ZWo (j2 (R()1_919846098F_4b_4bOle PIC LPICT R)+) k)) (q2"R# (x+).).).( W ) o (()1)+) k)) (n"#30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5632 div 1024 3 -1 roll exch div scale currentpoint translate 64 -1779 translate 13 2387 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (NPV) show 864 2387 moveto 384 /Symbol f1 (=) show 1426 2148 moveto 384 /Times-Italic f1 (Wo) show 1196 2680 moveto 384 /Times-Roman f1 (\() show 1302 2680 moveto 384 /Times-Roman f1 (1) show 1546 2680 moveto 384 /Symbol f1 (+) show 1842 2680 moveto 384 /Times-Italic f1 (k) show 2039 2680 moveto 384 /Times-Roman f1 (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1182 2288 moveto 998 0 rlineto stroke 2273 2387 moveto 384 /Symbol f1 (+) show 2819 2148 moveto 384 /Times-Italic f1 (Wo) show 3343 1977 moveto 224 /Times-Roman f1 (2) show 2589 2680 moveto 384 /Times-Roman f1 (\() show 2695 2680 moveto 384 /Times-Roman f1 (1) show 2939 2680 moveto 384 /Symbol f1 (+) show 3235 2680 moveto 384 /Times-Italic f1 (k) show 3432 2680 moveto 384 /Times-Roman f1 (\)) show 3567 2509 moveto 224 /Times-Roman f1 (2) show 2575 2288 moveto 1162 0 rlineto stroke 3798 2387 moveto 384 /Symbol f1 (+) show 4010 2387 moveto 384 /Times-Roman f1 (.) show 4118 2387 moveto (.) show 4226 2387 moveto (.) show 4723 2148 moveto 384 /Times-Italic f1 (W) show 5042 2148 moveto 224 ns (o) show 4374 2680 moveto 384 /Times-Roman f1 (\() show 4480 2680 moveto 384 /Times-Roman f1 (1) show 4724 2680 moveto 384 /Symbol f1 (+) show 5020 2680 moveto 384 /Times-Italic f1 (k) show 5217 2680 moveto 384 /Times-Roman f1 (\)) show 5353 2509 moveto 224 /Times-Italic f1 (n) show 4360 2288 moveto 1162 0 rlineto stroke end FMicrosoft Equation 2.01ELO Equation.2CompObjRObjInfoOlePres000(Ole10Native  NPV=Wo(1+k)+Wo 2 (1+k) 2 +...W o (1+k) n EquationOh+'0 ,8Ole10FmtProgID  1TableaSummaryInformation(#DocumentSummaryInformation8 T ` lx6The Discount Rate for Wrongful Death and 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