Present Values, Investment Returns and Discount Rates

Transcription

1 Preset Values, Ivestmet Returs ad Discout Rates Dimitry Midli, ASA, MAAA, PhD Presidet CDI Advisors LLC May 2, 203 Copyright 20, CDI Advisors LLC

2 The cocept of preset value lies at the heart of fiace i geeral ad actuarial sciece i particular. The importace of the cocept is uiversally recogized. Preset values of various cash flows are extesively utilized i the pricig of fiacial istrumets, fudig of fiacial commitmets, fiacial reportig, ad other areas. A typical fudig problem ivolves a fiacial commitmet (defied as a series of future paymets) to be fuded. A fiacial commitmet is fuded if all paymets are made whe they are due. A preset value of a fiacial commitmet is defied as the asset value required at the preset to fud the commitmet. Traditioally, the calculatio of a preset value utilizes a discout rate a determiistic retur assumptio that represets ivestmet returs. If the ivestmet retur ad the commitmet are certai, the the discout rate is equal to the ivestmet retur ad the preset value is equal to the sum of all paymets discouted by the compouded discout rates. The asset value that is equal to this preset value ad ivested i the portfolio that geerates the ivestmet retur will fud the commitmet with certaity. I practice, however, perfectly certai future fiacial commitmets ad ivestmet returs rarely exist. While the calculatio of the preset value is straightforward whe returs ad commitmets are certai, ucertaities i the commitmets ad returs make the calculatio of the preset value aythig but straightforward. Whe ivestmet returs are ucertai, a sigle discout rate caot ecompass the etire spectrum of ivestmet returs, hece the selectio of a discout rate is a challege. I geeral, the asset value required to fud a ucertai fiacial commitmet via ivestig i risky assets the preset value of the commitmet is ucertai (stochastic). While the aalysis of preset values is vital to the process of fudig fiacial commitmets, ucertai (stochastic) preset values are outside of the scope of this paper. This paper assumes that a preset value is certai (determiistic) a preset value is assumed to be a umber, ot a radom variable i this paper. The desire to have a determiistic preset value requires a set of assumptios that "assume away" all the ucertaities i the fudig problem. I particular, it is geerally ecessary to assume that all future paymets are perfectly kow at the preset. The ext step is to select a proper measuremet of ivestmet returs that serves as the discout rate for preset value calculatios. This step the selectio of the discout rate is the mai subject of this paper. Oe of the mai messages of this paper is the selectio of the discout rate depeds o the objective of the calculatio. Differet objectives may ecessitate differet discout rates. The paper defies ivestmet returs ad specifies their relatioships with preset ad future values. The key measuremets of ivestmet returs are defied i the cotext of retur series ad, after a cocise discussio of capital market assumptios, i the cotext of retur distributios. The paper cocludes with several examples of ivestmet objectives ad the discout rates associated with these objectives. Preset Values, Ivestmet Returs ad Discout Rates 2 05/02/203

3 . Ivestmet Returs This sectio discusses oe of the most importat cocepts i fiace ivestmet returs. Let us defie the ivestmet retur for a portfolio of assets with kow asset values at the begiig ad the ed of a time period. If PV is the asset value ivested i portfolio P at the begiig of a time period, ad FV is the value of the portfolio at the ed of the period, the the portfolio retur R P for the period is defied as R P FV PV (.) PV Thus, give the begiig ad edig values, portfolio retur is defied (retrospectively) as the ratio of the ivestmet gai over the begiig value. Defiitio (.) establishes a relatioship betwee portfolio retur R P ad asset values PV ad FV. Simple trasformatios of defiitio (.) produce the followig formula: P FV PV R (.2) Formula (.2) allows a forward-lookig (prospective) calculatio of the ed-of-period asset value FV. The formula is usually used whe the asset value at the preset PV ad portfolio retur R P are kow (this explais the otatio: PV stads for Preset Value ; FV stads for Future Value ). While defiitio (.) ad formula (.2) are mathematically equivalet, they utilize portfolio retur R P i fudametally differet ways. The retur i defiitio (.) is certai, as it is used retrospectively as a measuremet of portfolio performace. I cotrast, the retur i formula (.2) is used prospectively to calculate the future value of the portfolio, ad it may or may ot be certai. Whe a portfolio cotais risky assets, the portfolio retur is ucertai by defiitio. Most istitutioal ad idividual ivestors edeavor to fud their fiacial commitmets by virtue of ivestig i risky assets. The distributio of ucertai portfolio retur is usually aalyzed usig a set of forward-lookig capital market assumptios that iclude expected returs, risks, ad correlatios betwee various asset classes. Later sectios discuss capital market assumptios i more detail. Give the preset value ad portfolio retur, formula (.2) calculates the future value. However, may ivestors with future fiacial commitmets to fud (e.g. retiremet plas) face a differet challege. Future values the commitmets are usually give, ad the challege is to calculate preset values. A simple trasformatio of formula (.2) produces the followig formula: Preset Values, Ivestmet Returs ad Discout Rates 3 05/02/203

4 PV FV (.3) RP Formula (.3) represets the cocept of discoutig procedure. Give a portfolio, formula (.3) produces the asset value PV required to be ivested i this portfolio at the preset i order to accumulate future value FV. It must be emphasized that retur R P i (.3) is geerated by the actual portfolio P, as there is o discoutig without ivestig. Ay discoutig procedure assumes that the assets are actually ivested i a portfolio that geerates the returs used i the procedure. Formulas (.2) ad (.3) are mathematically equivalet, ad they utilize portfolio retur i similar ways. Depedig o the purpose of a calculatio i (.2) or (.3), oe may utilize either a particular measuremet of retur (e.g. the expected retur or media retur) or the full rage of returs. 2 The desirable properties of the future value i (.2) or preset value i (.3) would determie the right choice of the retur assumptio. Future ad preset values are, i a certai sese, iverse of each other. It is iformative to look at the aalogy betwee future ad preset values i the cotext of a fudig problem, which would explicitly ivolve a future fiacial commitmet to fud. Thik of a ivestor that has $P at the preset ad has made a commitmet to accumulate $F at the ed of the period by meas of ivestig i a portfolio that geerates ivestmet retur R. Similar to (.2), the future value of $P is equal to FV P R (.4) Similar to (.3), the preset value of $F is equal to PV F (.5) R The shortfall evet is defied as failig to accumulate $F at the ed of the period: FV F (.6) The shortfall evet ca also be defied equivaletly i terms of the preset value as $P beig isufficiet to accumulate $F at the ed of the period: P PV (.7) I particular, the shortfall probability ca be expressed i terms of future ad preset values: Shortfall Probability = Pr FV F Pr PV P (.8) Preset Values, Ivestmet Returs ad Discout Rates 4 05/02/203

5 If the shortfall evet happes, the the shortfall size ca also be measured i terms of future ad preset values. The future shortfall F FV is the additioal amout the ivestor will be required to cotribute at the ed of the period to fulfill the commitmet. The preset shortfall PV P is the additioal amout the ivestor is required to cotribute at the preset to fulfill the commitmet. Clearly, there is a fudametal coectio betwee future ad preset values. However, this coectio goes oly so far, as there are issues of great theoretical ad practical importace that distiguish future ad preset values. As demostrated i a later sectio, similar coditios imposed o future ad preset values lead to differet discout rates. Ucertai future values geerated by the ucertaities of ivestmet returs (ad commitmets) play o part i fiacial reportig. I cotrast, various actuarial ad accoutig reports require calculatios of preset values, ad these preset values must be determiistic (uder curret accoutig stadards, at least). Therefore, there is a eed for a determiistic discoutig procedure. Covetioal calculatios of determiistic preset values usually utilize a sigle measuremet of ivestmet returs that serves as the discout rate. Sice there are umerous measuremets of ivestmet returs, the challege is to select the most appropriate measuremet for a particular calculatio. To clarify these issues, subsequet sectios discuss various measuremets of ivestmet returs. 2. Measuremets of Ivestmet Returs: Retur Series This sectio discusses the key measuremets of series of returs ad relatioships betwee these measuremets. Give a series of returs r,, r, it is desirable to have a measuremet of the series a sigle rate of retur that, i a certai sese, would reflect the properties of the series. The right measuremet always depeds o the objective of the measuremet. The most popular measuremet of a series of returs r,, r is its arithmetic average A defied as the average value of the series: rk k A (2.) As ay other measuremet, the arithmetic average has its pros ad cos. While the arithmetic average is a ubiased estimate of the retur, the probability of achievig this value may be usatisfactory. As a predictor of future returs, the arithmetic average may be too optimistic. Aother sigificat shortcomig of the arithmetic retur is it does ot coect the startig ad edig asset values. The startig asset value multiplied by the compouded arithmetic retur factor ( + A) is ormally greater tha the edig asset value. 3 Therefore, the arithmetic average is iappropriate if the objective is to coect the startig ad edig asset values. The Preset Values, Ivestmet Returs ad Discout Rates 5 05/02/203

6 objective that leads to the arithmetic average as the right choice of discout rate is preseted i Sectio 5. Clearly, it would be desirable to coect the startig ad edig asset values to fid a sigle rate of retur that, give a series of returs ad a startig asset value, geerates the same future value as the series. This observatio suggests the followig importat objective. Objective : To "coect" the startig ad edig asset values. The cocept of geometric average is specifically desiged to achieve this objective. If A 0 ad A are the startig ad edig asset values correspodigly, the, by defiitio, A0 r r A (2.2) The geometric average G is defied as the sigle rate of retur that geerates the same future value as the series of returs. Namely, the startig asset value multiplied by the compouded retur factor G is equal to the edig asset value: A G A (2.3) 0 Combiig (2.2) ad (2.3), we get the stadard defiitio of the geometric average G: G rk (2.4) k Let us re-write formulas (.2) ad (.3) i terms of preset ad future values. If A is a future paymet ad r,, r are the ivestmet returs, the the preset value of A is equal to the paymet discouted by the geometric average: A 0 A A r r G (2.5) Thus, the geometric average coects the startig ad edig asset values (ad the arithmetic average does ot). Therefore, if the primary objective of discout rate selectio is to coects the startig ad edig asset values, the the geometric average should be used for the preset value calculatios. To preset certai relatioships betwee arithmetic ad geometric averages, let us defie variace V as follows: 4 k k 2 (2.6) V r A Preset Values, Ivestmet Returs ad Discout Rates 6 05/02/203

7 If V = 0, the all returs i the series are the same, ad the arithmetic average is equal to the geometric average. Otherwise (if V > 0), the arithmetic average is greater tha the geometric average (A > G). 5 There are several approximate relatioships betwee arithmetic average A, geometric average G, ad variace V. These relatioships iclude the followig relatioships that are deoted (R) (R4) i this paper. G A V 2 (R) 2 2 G A V (R2) G Aexp V A 2 2 (R3) 2 2 G A V A (R4) These relatioships produce differet results, ad some of them work better tha the others i differet situatios. Relatioship (R) is the simplest, popularized i may publicatios, but usually sub-optimal ad teds to uderestimate the geometric retur. 6 Relatioships (R2) (R4) are slightly more complicated, but, i most cases, should be expected to produce better results tha (R). The geometric average estimate geerated by (R4) is always greater tha the oe geerated by (R3), which i tur is always greater tha the oe geerated by (R2). 7 Loosely speakig, (R2) < (R3) < (R4) I geeral, iequality (R) < (R2) is ot ecessarily true, although it is true for most practical examples. If A V 4, the the geometric average estimate geerated by (R) is less tha the oe geerated by (R2). 8 There is some evidece to suggest that, for historical data, relatioship (R4) should be expected to produce better results tha (R) (R3). See Midli [200] for more details regardig the derivatios of (R) (R4) ad their properties. Example 2.. 2, r %, r 2 5%. The arithmetic mea A, geometric mea G, ad variace V are calculated as follows. A % 5% 7.00% 2 Preset Values, Ivestmet Returs ad Discout Rates 7 05/02/203

8 2 2 Note that G % 5% 6.70% 2 2 V rk A 0.64% 2 k G A V, so formula (R2) is exact i this example. Give $ at the preset, future value FV is FV % 5%.385 If we apply arithmetic retur A to $ at the preset for two years, we get 7% which is greater tha future value FV =.385. If we apply geometric retur G to $ at the preset for two years, we get 6.70% which is equal to future value FV, as expected. Give $ i two years, preset value PV is PV % 5% If we discout $ i two years usig geometric retur G, we get 6.70% which is equal to preset value PV, as expected. If we discout $ i two years usig arithmetic retur A, we get Preset Values, Ivestmet Returs ad Discout Rates 8 05/02/203

9 7.00% which is less tha preset value PV = Capital Market Assumptios ad Portfolio Returs This sectio itroduces capital market assumptios for major asset classes ad outlies basic steps for the estimatio of portfolio returs. It is assumed that the capital markets cosist of asset classes. The followig otatio is used throughout this sectio: m i mea (arithmetic) retur; s i stadard deviatio of retur; c ij correlatio coefficiet betwee asset classes i ad j. A portfolio is defied as a series of weights w i, such that the fractio of the portfolio ivested i the asset class i. wi. Each weight wi represets i Portfolio mea retur A ad variace V are calculated as follows: A w m (3.) i i i V w w s s c (3.2) i, j i j i j ij Let us also defie retur factor as + R. It is commo to assume that the retur factor has logormal distributio (which meas l( + R) has ormal distributio). Uder this assumptio, parameters µ ad σ of the logormal distributio are calculated as follows: 2 2 l V A (3.3) Usig σ calculated i (3.3), parameter μ of the logormal distributio is calculated as follows: 2 l A (3.4) 2 Give parameters μ ad σ, the P th percetile of the retur distributio is equal to the followig: Preset Values, Ivestmet Returs ad Discout Rates 9 05/02/203

10 RP Preset Values, Ivestmet Returs ad Discout Rates 0 05/02/203 exp P (3.5) where is the stadard ormal distributio. I particular, if P = 50%, the P 0. Therefore, the media of the retur distributio uder the logormal retur factor assumptio is calculated as follows. R0.5 exp (3.6) Example 3.. Let us cosider two ucorrelated asset classes with mea returs 8.00% ad 6.00% ad stadard deviatios 20.00% ad 0.00% correspodigly. If a portfolio has 35% of the first class ad 65% of the secod class, its mea ad variace are calculated as follows. A 8.00% 35% 6.00% 65% 6.70% 2 2 V 20.00% 35% 0.00% 65% 0.925% It is iterestig to ote that the stadard deviatio of the portfolio is 9.55% ( 0.925% ), which is lower tha the stadard deviatios of the uderlyig asset classes (20.00% ad 0.00%). Assumig that the retur factor of this portfolio has logormal distributio, the parameters of this distributio are 0.925% l % 2 From (3.5), the media retur for this portfolio is l 6.70% From (3.5), the 45th percetile for this portfolio is R 0.5 exp % R 0.45 exp % 4. Measuremets of Ivestmet Returs: Retur Distributios The previous sectio preseted the relatioships betwee the arithmetic ad geometric averages defied for a series of returs. This sectio develops similar results whe retur distributio R is give.

11 I this case, the arithmetic average (mea) retur A is defied as the expected value of R: A E R (4.) The geometric average (mea) retur G is defied as follows: G exp E l R (4.2) These arithmetic ad geometric average returs are the limits of the arithmetic ad geometric averages of appropriately selected series of idepedet idetically distributed returs. r be a series of idepedet returs that has the same distributio as R. Let us Specifically, let k defie arithmetic averages A ad geometric averagesg for r, r :, A rk k (4.3) G rk (4.4) k Accordig to the Law of Large Numbers (LLN), k A coverge to E. Also, from (4.4) we have l G l rk (4.5) r Agai, accordig to the LLN, l k coverge to the expected value E l R k (4.5), l G coverges to E l R exp E l R To recap,, which, accordig to (4.2), is equal to G. as well. Cosequetly, G coverge. From A coverges to A ad G coverges to G whe teds to ifiity. As discussed above, the approximatios (R) (R4) are true for A ad G, where V is defied as i (2.6): k k 2 (4.6) V r A Sice V coverge to the variace of returs V whe teds to ifiity, the approximatios (R) (R4) are true for A ad G as well. As was discussed before, if the primary objective of discout to Preset Values, Ivestmet Returs ad Discout Rates 05/02/203

12 rate selectio is to coects the startig ad edig asset values, the the geometric mea is a reasoable choice for the discout rate. This coclusio, however, is valid over relatively log time horizos oly. Over shorter time r has o-trivial volatility ad caot be cosidered horizos, the geometric average of series k approximately costat. More importatly, the ivestor may have objectives other tha coectig the startig ad edig asset values. All i all, additioal coditios of stochastic ature may be required to select a reasoable discout rate. Such coditios are discussed i the ext sectio. For large, the Cetral Limit Theorem (CLT) ca be used to aalyze the stochastic properties of the geometric average. Accordig to the CLT applied to l rk, the geometric average k retur factor defied as G G r k exp l rk k k is approximately logormally distributed. If the mea ad stadard deviatio of l r ad correspodigly, the the parameters of the geometric average retur factor are ad Preset Values, Ivestmet Returs ad Discout Rates 2 05/02/203 k are Assumig that the retur factor has logormal distributio, it ca be show that relatioship (R4) is exact: G A V A (4.7). A importat property of logormal retur factors is the geometric mea retur is equal to the media retur. Ideed, if ad are the parameters of the logormal distributio, the l R is ormal ad G exp E l R exp (4.8) which is the media of the retur distributio accordig to (3.6). Thus, if a discout rate were chose at radom (ot that this is a great idea), the there would be a 50% chace for the discout rate to be greater tha the geometric mea ad a 50% chace to be less tha the geometric mea. Similarly, if a preset value were calculated usig radomly selected discout rate, the there would be a 50% chace that a preset value is greater tha the preset value calculated usig the geometric mea. 0

13 Give arithmetic mea A ad variace V, formula (4.7) produces geometric retur G. If there is a eed to calculate the arithmetic mea whe the geometric mea ad the variace are give, the the arithmetic mea is calculated as follows: 4V A G (4.9) 2 2 G 2 Example 4.. This example is a cotiuatio of Example 3.. I this example, A 6.70% ad V 0.925%. Accordig to (4.7), G % which is equal to the media retur calculated i Example 3.. Note that the geometric returs for the idividual asset classes are 6.9% ad 5.53%. It is oteworthy that the geometric retur for the portfolio that has 35% of the first class ad 65% of the secod class is 6.27%, which higher tha the geometric returs of the idividual classes. Let us take a look at the stochastic properties of the geometric average for this portfolio. Uder the logormal retur factor assumptio, the parameters of the retur distributio are ad (see Example 3.). If = 0, the the geometric average retur factor G is approximately logormally distributed with parameters ad The mea, media ad stadard deviatio are 6.32%, 6.27% ad 3.00% correspodigly. Note sigificat decreases of the mea ad stadard deviatio of the geometric average compared to the origial retur distributio (6.32% vs. 6.70% ad 3.00% vs. 9.55%), while the media remais the same. 5. Examples of Discout Rate Selectio As was discussed i the previous sectio, the ivestor may have objectives other tha coectig the startig ad edig asset values. This sectio discusses ad presets three additioal examples of such objectives that lead to the selectio of discout rates. Let us cosider a simple modificatio of the fudig problem discussed earlier i the paper. Thik of a ivestor that has made a commitmet to accumulate $F at the ed of the period by meas of ivestig i a portfolio that geerates (ucertai) ivestmet retur R. To fud the commitmet, the ivestor wats to make a cotributio that is the subject to certai coditios. For coveiece, let us recall Objective itroduced i Sectio 2: Preset Values, Ivestmet Returs ad Discout Rates 3 05/02/203

14 Objective : To "coect" the startig ad edig asset values. As was demostrated i Sectio 2, the right discout rate for this objective is the geometric retur. Objective 2: To have a "safety cushio". Let us assume that the ivestor's objective is have more tha a 50% chace that ivestmet returs are greater tha the discout rate (the "safety cushio"). For example, if it is required to have a P% chace that the ivestmet retur is greater tha the discout rate, the the discout rate that delivers this safety level is the (00 - P)th percetile of the retur distributio. Objective 3: No expected gais/losses i the future. Let us assume that the ivestor's objective is to have either expected gais or losses at the ed of the period. If C f is the ivestor's cotributio at the preset, the this objective implies that the commitmet is the mea of the (ucertai) future value of C f : 0 E FV E C R F (5.) Equatio (5.) gives the followig formula for cotributio C f (subscript f i C f idicates that the objective is "o expected gais or losses i the future"): f C f F (5.2) E R Formula (5.2) shows that the objective "o expected gais or losses i the future" leads to cotributio C f calculated as the preset value of the commitmet usig the arithmetic mea retur. Hece, the right discout rate d for this objective is the arithmetic mea retur: f d f E R (5.3) As discussed i a prior sectio, there is a certai symmetry ad fudametal coectio betwee future ad preset values. I light of this discussio, the followig objective is a atural couterpart to Objective 3. Objective 4: No expected gais/losses at the preset. At first, this objective looks somewhat peculiar. Everythig is supposed to be kow at the preset, so what kid of gais/losses ca exist ow? But remember that that the asset value required to fud the commitmet the preset value of the commitmet is ucertai at the preset. Therefore, the objective "today's cotributio is the mea of the preset value of the Preset Values, Ivestmet Returs ad Discout Rates 4 05/02/203

15 commitmet" is as meaigful as the objective "the commitmet is the mea of the future value of today's cotributio" discussed i Objective 3. If C p is the cotributio the ivestor makes at the preset, the the objective "o expected gais/losses at the preset" implies the followig equatio. F E PV E C R 0 P (5.4) Equatio (5.4) gives the followig formula for cotributio C p (subscript p i C p idicates that the objective is "o expected gais or losses at the preset"): CP F E R (5.5) Formula (5.5) shows that the objective "o expected gais or losses at the preset" leads to cotributio C that is equal to the preset value of the commitmet usig discout rate d : P p C p F (5.6) d p where d p is calculated from (5.5) ad (5.6) as d p E R (5.7) Note that Jese iequality etails E R E R (5.8) Therefore, dp df. Uder the logormal retur factor assumptio, we ca tell more about discout rate R as d p. Defiig R V ER 2 (5.9) Preset Values, Ivestmet Returs ad Discout Rates 5 05/02/203

16 where V is the variace of retur R, it ca be show that the expected value of the reciprocal retur factor is R E R E R (5.0) Combiig (5.7) ad (5.0), we get d p E R (5.) R Furthermore, uder the logormal retur factor assumptio, there is a iterestig relatioship betwee the geometric mea retur G ad discout rates d ad d geerated by Objective 3 ad Objective 4: p f G d d (5.2) Thus, the geometric mea retur G is the "geometric mid-poit" betwee the discout rates geerated by the objectives of o expected gais/losses i the future ad at the preset. Example 5.. This example is a cotiuatio of Example 3. ad Example 4.. As i these examples, A 6.70% ad V 0.925%. The.0080 ad d 6.70% f d 5.85% p The 45th percetile of the retur distributio is R % (see Example 3.). Coclusio R The selectio of a discout rate is oe of the most importat assumptios for the calculatios of preset values. This paper presets the basic properties of the key measuremets of ivestmet returs ad the discout rates associated with these measuremets. The paper shows that the selectio of the discout rate depeds o the objective of the calculatio. The paper demostrates the selectio of discout rates for the followig four objectives. Objective : To "coect" the startig ad edig asset values. The correct discout rate for this objective is the geometric mea retur. p f Preset Values, Ivestmet Returs ad Discout Rates 6 05/02/203

17 Objective 2: To have a certai "safety cushio". The correct discout rate for this objective is the (00 - P)th percetile of the retur distributio if it is required to have a P% chace that the ivestmet retur is greater tha the discout rate. Objective 3: No expected gais/losses i the future. The correct discout rate for this objective is the arithmetic mea retur. Objective 4: No expected gais/losses at the preset. The correct discout rate for this objective is give i formula (5.7). It is worth remidig that the mai purpose of a discout rate is to calculate a determiistic preset value. Yet, preset values associated with vital fudig problems are iheretly stochastic. As a result, the presece of a discout rate assumptio has sigificat pros ad cos. The primary advatage of a discout rate is the simplicity of calculatios. The mai disadvatage is a discout rate based determiistic preset value caot adequately describe the preset value of a ucertai fiacial commitmet fuded via ivestig i risky assets. This author believes that the direct aalysis of preset values ad their stochastic properties is the most appropriate approach to the process of fudig fiacial commitmets, but this subject is outside of the scope of this paper. This author hopes that the paper would be useful to practitioers specializig i the area of fudig fiacial commitmets. REFERENCES DeFusco R. A., McLeavey D. W., Pito J. E., Rukle, D. E. [2007]. Quatitative Ivestmet Aalysis, Wiley, 2 d Ed., Siegel, J. J. [2008]. Stocks for the Log Ru, McGraw-Hill, 4 th Ed., Bodie, Z., Kae, A., Marcus, A.J. [999]. Ivestmets, McGraw-Hill, 4 th Ed., 999. Jorda, B. D., Miller T. W. [2008]. Fudametals of Ivestmets, McGraw-Hill, 4 th Ed., Midli, D., [2009]. The Case for Stochastic Preset Values, CDI Advisors Research, CDI Advisors LLC, 2009, Midli, D., [200]. O the Relatioship betwee Arithmetic ad Geometric Returs, CDI Advisors Research, CDI Advisors LLC, 200, Pito, J. E., Hery, E., Robiso, T. R., Stowe, J. D. [200]. Equity Asset Valuatio, Wiley, 2 d Ed., 200. Preset Values, Ivestmet Returs ad Discout Rates 7 05/02/203

18 Edotes There are exceptios, e.g. a iflatio-adjusted cash flow with a matchig TIPS portfolio. 2 See Midli [2009] for more details. 3 That is as log as the returs i the series are ot the same. 4 For the purposes of this paper, the cocers that the sample variace as defied i (2.6) is ot a ubiased estimate are set aside. 5 This fact is a corollary of the Jece s iequality. 6 For example, see Bodie [999], p. 75, Jorda [2008], p. 25, Pito [200], p. 49., Siegel [2008], p. 22., DeFusco [2007], p 28, That is, obviously, as log as the returs i the series are ot the same ad V > 0. 8 Midli [200] cotais a simple example for which (R) > (R2). 9 See Midli [200] for more details. 0 The presece of discout rate is critical for these observatios. I geeral, the media of the preset value distributio calculated usig the full rage of returs (ad without discout rates) is ot equal to the preset value calculated usig the geometric mea (except whe the cash flow cotais just oe paymet). I other words, the media of preset value is ot the same as the preset value at the media retur. See Midli [2009] for more details regardig stochastic preset values. Importat Iformatio This material is iteded for the exclusive use of the perso to whom it is provided. It may ot be modified, sold or otherwise provided, i whole or i part, to ay other perso or etity. The iformatio cotaied herei has bee obtaied from sources believed to be reliable. CDI Advisors LLC gives o represetatios or warraties as to the accuracy of such iformatio, ad accepts o resposibility or liability (icludig for idirect, cosequetial or icidetal damages) for ay error, omissio or iaccuracy i such iformatio ad for results obtaied from its use. Iformatio ad opiios are as of the date idicated, ad are subject to chage without otice. This material is iteded for iformatioal purposes oly ad should ot be costrued as legal, accoutig, tax, ivestmet, or other professioal advice. Copyright 20, CDI Advisors LLC. All rights reserved. No part of this publicatio may be reproduced or trasmitted i ay form or by ay meas, electroic or mechaical, icludig photocopyig, recordig, or by ay iformatio storage ad retrieval system, without permissio i writig from CDI Advisors LLC. Preset Values, Ivestmet Returs ad Discout Rates 8 05/02/203