Present Values, Investment Returns and Discount Rates


 Michael Bell
 1 years ago
 Views:
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 forwardlookig (prospective) calculatio of the edofperiod 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 forwardlookig 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 rewrite 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 suboptimal 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 otrivial 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 midpoit" 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, McGrawHill, 4 th Ed., Bodie, Z., Kae, A., Marcus, A.J. [999]. Ivestmets, McGrawHill, 4 th Ed., 999. Jorda, B. D., Miller T. W. [2008]. Fudametals of Ivestmets, McGrawHill, 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 iflatioadjusted 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
I. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
More informationOn the Relationship between Arithmetic and Geometric Returns
O the Relatioship betwee Arithmetic ad Geometric Returs Dimitry Midli, ASA, MAAA, PhD Presidet CDI Advisors LLC dmidli@cdiadvisors.com August 14, 011 Copyright 011, CDI Advisors LLC Arithmetic ad geometric
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationKey Ideas Section 81: Overview hypothesis testing Hypothesis Hypothesis Test Section 82: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, Pvalue Type I Error, Type II Error, Sigificace Level, Power Sectio 81: Overview Cofidece Itervals (Chapter 7) are
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationI. Why is there a time value to money (TVM)?
Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationBENEFITCOST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEITCST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal  Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationStandard Errors and Confidence Intervals
Stadard Errors ad Cofidece Itervals Itroductio I the documet Data Descriptio, Populatios ad the Normal Distributio a sample had bee obtaied from the populatio of heights of 5yearold boys. If we assume
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
More informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationHow to read A Mutual Fund shareholder report
Ivestor BulletI How to read A Mutual Fud shareholder report The SEC s Office of Ivestor Educatio ad Advocacy is issuig this Ivestor Bulleti to educate idividual ivestors about mutual fud shareholder reports.
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationINVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology
Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology
More informationSubject CT5 Contingencies Core Technical Syllabus
Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationBASIC STATISTICS. Discrete. Mass Probability Function: P(X=x i ) Only one finite set of values is considered {x 1, x 2,...} Prob. t = 1.
BASIC STATISTICS 1.) Basic Cocepts: Statistics: is a sciece that aalyzes iformatio variables (for istace, populatio age, height of a basketball team, the temperatures of summer moths, etc.) ad attempts
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationQuadrat Sampling in Population Ecology
Quadrat Samplig i Populatio Ecology Backgroud Estimatig the abudace of orgaisms. Ecology is ofte referred to as the "study of distributio ad abudace". This beig true, we would ofte like to kow how may
More informationBond Valuation I. What is a bond? Cash Flows of A Typical Bond. Bond Valuation. Coupon Rate and Current Yield. Cash Flows of A Typical Bond
What is a bod? Bod Valuatio I Bod is a I.O.U. Bod is a borrowig agreemet Bod issuers borrow moey from bod holders Bod is a fixedicome security that typically pays periodic coupo paymets, ad a pricipal
More information9.8: THE POWER OF A TEST
9.8: The Power of a Test CD91 9.8: THE POWER OF A TEST I the iitial discussio of statistical hypothesis testig, the two types of risks that are take whe decisios are made about populatio parameters based
More informationSimple Annuities Present Value.
Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX9850GB PLUS to efficietly compute values associated with preset value auities.
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationUnit 20 Hypotheses Testing
Uit 2 Hypotheses Testig Objectives: To uderstad how to formulate a ull hypothesis ad a alterative hypothesis about a populatio proportio, ad how to choose a sigificace level To uderstad how to collect
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationTerminology for Bonds and Loans
³ ² ± Termiology for Bods ad Loas Pricipal give to borrower whe loa is made Simple loa: pricipal plus iterest repaid at oe date Fixedpaymet loa: series of (ofte equal) repaymets Bod is issued at some
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationEstimating the Mean and Variance of a Normal Distribution
Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers
More informationLearning objectives. Duc K. Nguyen  Corporate Finance 21/10/2014
1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the timevalue
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationCDs Bought at a Bank verses CD s Bought from a Brokerage. Floyd Vest
CDs Bought at a Bak verses CD s Bought from a Brokerage Floyd Vest CDs bought at a bak. CD stads for Certificate of Deposit with the CD origiatig i a FDIC isured bak so that the CD is isured by the Uited
More informationTO: Users of the ACTEX Review Seminar on DVD for SOA Exam MLC
TO: Users of the ACTEX Review Semiar o DVD for SOA Eam MLC FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Eam M, Life Cotigecies
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationThe Arithmetic of Investment Expenses
Fiacial Aalysts Joural Volume 69 Number 2 2013 CFA Istitute The Arithmetic of Ivestmet Expeses William F. Sharpe Recet regulatory chages have brought a reewed focus o the impact of ivestmet expeses o ivestors
More informationGrade 7. Strand: Number Specific Learning Outcomes It is expected that students will:
Strad: Number Specific Learig Outcomes It is expected that studets will: 7.N.1. Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] [C]
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More information15.075 Exam 3. Instructor: Cynthia Rudin TA: Dimitrios Bisias. November 22, 2011
15.075 Exam 3 Istructor: Cythia Rudi TA: Dimitrios Bisias November 22, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 A compay makes highdefiitio
More informationTIEE Teaching Issues and Experiments in Ecology  Volume 1, January 2004
TIEE Teachig Issues ad Experimets i Ecology  Volume 1, Jauary 2004 EXPERIMENTS Evirometal Correlates of Leaf Stomata Desity Bruce W. Grat ad Itzick Vatick Biology, Wideer Uiversity, Chester PA, 19013
More informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More informationOverview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals
Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationAmendments to employer debt Regulations
March 2008 Pesios Legal Alert Amedmets to employer debt Regulatios The Govermet has at last issued Regulatios which will amed the law as to employer debts uder s75 Pesios Act 1995. The amedig Regulatios
More informationA Guide to the Pricing Conventions of SFE Interest Rate Products
A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More informationTime Value of Money. First some technical stuff. HP10B II users
Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributiofree statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More informationUsing Excel to Construct Confidence Intervals
OPIM 303 Statistics Ja Stallaert Usig Excel to Costruct Cofidece Itervals This hadout explais how to costruct cofidece itervals i Excel for the followig cases: 1. Cofidece Itervals for the mea of a populatio
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
More informationPage 1. Real Options for Engineering Systems. What are we up to? Today s agenda. J1: Real Options for Engineering Systems. Richard de Neufville
Real Optios for Egieerig Systems J: Real Optios for Egieerig Systems By (MIT) Stefa Scholtes (CU) Course website: http://msl.mit.edu/cmi/ardet_2002 Stefa Scholtes Judge Istitute of Maagemet, CU Slide What
More informationThe Stable Marriage Problem
The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,
More informationBaan Service Master Data Management
Baa Service Master Data Maagemet Module Procedure UP069A US Documetiformatio Documet Documet code : UP069A US Documet group : User Documetatio Documet title : Master Data Maagemet Applicatio/Package :
More informationChapter 7  Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7  Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationSavings and Retirement Benefits
60 Baltimore Couty Public Schools offers you several ways to begi savig moey through payroll deductios. Defied Beefit Pesio Pla Tax Sheltered Auities ad Custodial Accouts Defied Beefit Pesio Pla Did you
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chisquare (χ ) distributio.
More informationInformation about Bankruptcy
Iformatio about Bakruptcy Isolvecy Service of Irelad Seirbhís Dócmhaieachta a héirea Isolvecy Service of Irelad Seirbhís Dócmhaieachta a héirea What is the? The Isolvecy Service of Irelad () is a idepedet
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationTO: Users of the ACTEX Review Seminar on DVD for SOA Exam FM/CAS Exam 2
TO: Users of the ACTEX Review Semiar o DVD for SOA Exam FM/CAS Exam FROM: Richard L. (Dick) Lodo, FSA Dear Studets, Thak you for purchasig the DVD recordig of the ACTEX Review Semiar for SOA Exam FM (CAS
More informationCHAPTER 3: FINANCIAL ANALYSIS WITH INFLATION
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
More informationGregory Carey, 1998 Linear Transformations & Composites  1. Linear Transformations and Linear Composites
Gregory Carey, 1998 Liear Trasformatios & Composites  1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio
More informationA Resource for Freestanding Mathematics Qualifications Working with %
Ca you aswer these questios? A savigs accout gives % iterest per aum.. If 000 is ivested i this accout, how much will be i the accout at the ed of years? A ew car costs 16 000 ad its value falls by 1%
More informationRainbow options. A rainbow is an option on a basket that pays in its most common form, a nonequally
Raibow optios INRODUCION A raibow is a optio o a basket that pays i its most commo form, a oequally weighted average of the assets of the basket accordig to their performace. he umber of assets is called
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationCovariance and correlation
Covariace ad correlatio The mea ad sd help us summarize a buch of umbers which are measuremets of just oe thig. A fudametal ad totally differet questio is how oe thig relates to aother. Stat 0: Quatitative
More informationTrading the randomness  Designing an optimal trading strategy under a drifted random walk price model
Tradig the radomess  Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationA Mathematical Perspective on Gambling
A Mathematical Perspective o Gamblig Molly Maxwell Abstract. This paper presets some basic topics i probability ad statistics, icludig sample spaces, probabilistic evets, expectatios, the biomial ad ormal
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by
More information