The Time Value of Money

Transcription

1 Study Sessio #, Readig # 5 The Time Value of Moey Revisio-, released o November 3, 00. Compoud Iterest or Iterest o Iterest Growth i the value of ivestmet icludes, iterest eared o: Origial pricipal. Previous period s iterest earigs. Time Lie Diagram of the cash flows associated with a TVM problem. Discoutig Movig CF to the begiig of a ivestmet period to calculate PV. Compoudig Movig cash flow to the ed of the ivestmet period to calculate FV. FV = PV ( +i) N (+i) N is FV factor Required iterest rate o security. = Nomial RFR. + Default risk + Liquidity + premium. premium. Maturity risk premium. Real RFR + Expected iflatio rate. It shows a icrease i purchasig power. Premium for the risk that borrower will ot make the promised paymets i a timely maer. Premium for receivig less tha fair value for a ivestmet if it must be sold quickly. Loger-term bods have more maturity risk, because their prices are more volatile. Loa Amortizatio Process of payig off a loa with a series of periodic loa paymets, whereby a portio of the outstadig loa amout is paid off, or amortized, with each paymet. Cash flow Additivity Priciple PV of ay stream of cash flows equals the sum of PV of each cash flow. Perpetuity Perpetual auity. Fixed paymet at set itervals over a ifiite time period. is the discoutig factor for perpetuity. PV of auity due. > PV of ordiary auity. Auity Due Cash flows occur at the begiig of each period. Auity Stream of equal cash flows accruig at equal itervals. Two types Ordiary Auity Cash flows occur at the ed of each period. Iterpretatios of Iterest Rate Required rate of retur. Discout rate. Opportuity cost. Effective Aual Rate (EAR) Rate of retur actually beig eared after adjustmets have bee made for differet compoudig periods. EAR = (+ periodic rate) m - Stated rate will be equal to the actual (effective) rate oly whe it is compouded aually.

2 Study Sessio #, Readig # 6 Discouted Cash Flow Applicatios D.R. = Discout Rate. NPV: PV of expected PV of expected cash iflows. cash outflows. NPV = N Σ t = 0 CFt ( + r) t D.R used is the market based opportuity cost of capital. NPV assumes reivestmet at D.R. IRR The D.R. at which NPV = 0. The rate of retur at which; PV iflows = PV outflows. It assumes reivestmet at IRR. Decisio rule IRR > r Accept IRR < r Reject For Sigle Project: IRR / NPV rules lead to exactly the same accept /reject decisio For Mutually Exclusive Project: Select the project with the greatest NPV. Decisio rule: NPV Decisio Impact +ve (IRR> r) Accept 0 - Icrease shareholder s wealth. Size of the compay rises but shareholder s wealth is ot affected. Problems i IRR For mutually exclusive projects, NPV & IRR may give coflictig project rakigs due to: Differet sizes of project s iitial cost Differeces i timigs of cash flows. -ve (IRR< r) Reject Decreases shareholder s wealth. Moey-Weighted Retur (MWR) IRR of a ivestmet. It is highly sesitive to the timig & amout of withdrawals from & additios to a portfolio. If oe has complete cotrol over the timigs of cash flows the it is more appropriate. Fuds cotributed prior to a period of relatively.. Time-Weighted Retur (TWR) It measures compoud growth (Geometric retur). It is ot affected by the timigs of cash flows. It is preferred over MWR. TWR ca t be calculated if we do t kow the period ed values of ivestmet. Poor returs High returs Holdig Period Yield or Holdig Period Retur (HPR) Total retur a ivestor ears betwee the purchase date & the sale or maturity date. MWR< TWR MWR> TWR Moey Market Yield (CD equivalet yield) r MM Aualized HPY. Assumes 360-day year. Does ot icorporate effects of compoudig. r MM = HPY 360 t Bak Discout Yield (r BD ) Dollar discout from the face (par) value as a fractio of the face value. Based o face value. Not based o market or purchase price. r BD = D 360 F t Effective Aual Yield (EAY) Aualized HPY. Assumes 365-day year. Icorporates effects of compoudig. EAY =(+HPY) 365/ t - Covertig r BD ito r MM Bod equivalet yield (BEY) BEY = (semi aual effective yield.) It is the actual retur a ivestor receives.

3 Study Sessio #, Readig # 7 Statistical Cocepts & Market Returs Populatio Set of all possible members. Sample Subset of populatio. Statistics Refers to data & methods used to aalyze data. Parameter Measures characteristics of populatio. Sample Statistic Measures characteristics of a sample. Descriptive Statistics Used to summarize & cosolidate large data sets ito useful iformatio. Two Categories Iferetial Statistics Forecastig, estimatig or makig judgmet about a large set based o a smaller set. Types of Measuremet Scales Nomial Scale Least accurate. No particular order or rak. Provides least ifo. Least refied. Ordial Scale Provides raks/orders. No equal differece b/w scale values. Iterval Scale Provides raks/orders. Differece b/w the scales are equal. Zero does ot mea total absece. Ratio Scale Provides raks/orders Equal differeces b/w scale. A true zero poit exists as the origi. Most refied. Costructig a Frequecy Distributio. Defie Itervals / Classes Iterval is set of values that a observatio may take o. Itervals must be, All-iclusive. No-overlappig. Mutually Exclusive. Importace of Number of Itervals Frequecy Distributio Tabular (summarized) presetatio of statistical data.. Tally the observatios Assigig observatios to their appropriate itervals. 3. Cout the observatios Cout actual umber of observatios i each iterval i.e., absolute frequecy of iterval. Cumulative Absolute Frequecy Sum of absolute frequecies startig with lowest iterval & progressig through the highest. Relative Frequecy %age of total observatios fallig i each iterval. Cumulative Relative Frequecy Sum of relative frequecies startig with the lowest iterval & progressig through highest. Too few itervals. Importat characteristics may be lost. Too may itervals. Data may ot be summarized well eough. Modal Iterval Iterval with highest frequecy. Histogram Bar chart of cotiuous data frequecy distributio. Helps i quickly idetifyig the modal iterval. X-axis: Class itervals. Y-axis: Absolute frequecies. Frequecy Polygo X-axis: Mid poits of each iterval. Y-axis: Absolute frequecies.

4 Study Sessio #, Readig # 7 Measures of Cetral Tedecy Idetify cetre of data set. Measure of reward. Used to represet typical or expected values i data set. Weighted Mea It recogizes the disproportioate ifluece of differet observatios o mea. Quatiles: Geometric Mea (GM) Calculatig multi periods retur. Measurig compoud growth rates. (applicable oly to o-egative values) +R G = (+R ) ( + R ) ( + R ) Mea Sum of all values divided by total umber of values. Populatio = Sample = Properties: Mea icludes all values of data set. Mea is uique for each data. All itervals & ratio data sets have a Mea. Sum of deviatios from Mea is always zero i.e., Mea is the best estimate of true mea & the value of ext observatio. Shortcomig: Mea is affected by extremely large & small values. Media Midpoit of a arraged data set. Divides data ito two equal halves. It is ot affected by extreme values; hece it is a better measure of cetral tedecy i the presece of extremely large or small values. Mode Most frequet value i the data set. No. of Modes Oe Two Three Names of Distributios Uimodal Bimodal Trimodal Harmoic Mea (H.M) H.M is used: Whe time is ivolved. Equal $ ivestmet at differet times. For values that are ot all equal Quartiles: Distributio divided ito 4 parts (quarters). Quitiles: Distributio divided ito 5 parts. Deciles: Distributio divided ito 0 parts. Percetiles: Distributio divided ito 00 parts (percets). H.M < GM < AM Measures Measures of Locatio of Cetral + Quatiles Tedecy Dispersio Variability aroud the cetral tedecy. Measure of risk. Mea Absolute Deviatio (MAD) Average of absolute deviatios from mea: Rage Max Mi Value Value Sample Variace Sample Stadard Deviatio Populatio Variace σ Averaged squared deviatios from mea. Usig - observatios Usig etire umber of observatios will systemically uderestimate the populatio parameter & cause the sample variace & S.D to be referred to as biased estimator. Populatio Stadard Deviatio (S.D) σ. Square root of populatio variace. I geeral S.D > MAD Relative Dispersio Amout of variability relative to a referece poit. CV= Coefficiet of Variatio i.e., risk per uit of expected retur. Helps make direct comparisos of dispersio across differet data sets. Sharpe Ratio Measures excess retur per uit of risk. Sharpe ratio = Higher sharpe ratios are preferred.

5 Study Sessio #, Readig # 7 3 Chebyshev s Iequality Gives the %age of observatios that lie withi k stadard deviatios of the mea is at least for all k>, regardless of the shape of the distributio. ±.5 s.d 36% obs. ±.5 s.d 56% obs ± s.d 75% obs. ± 3 s.d 89% obs. ± 4 s.d 94% obs. Symmetrical Distributio Idetical o both sides of the mea. Itervals of losses & gais exhibit the same frequecy. Mea = Media = Mode. Mea = Media = Mode. Negatively Skewed Loger tail towards left. More outliers i the lower regio. More ve deviatios. Mea < Media < Mode Skewess No Symmetrical. Sample Skewess Sum of cubed deviatios from mea divided by umber of observatios & cubed stadard deviatio. >0.5 is cosidered sigificat level of skewes. Positively Skewed Loger tail towards right. More outliers i the upper regio. More + ve deviatios. Mea > Media > Mode. Hit Media is always i the ceter. Mea is i the directio of skew. Kurtosis Measure of degree of more or less peaked tha a ormal distributio. Kurtosis of ormal distributio is 3. Excess kurtosis = sample kurtosis-3 Excess kurtosis value exceedig absolute is cosidered large. Distributio Excess Kurtosis Leptokurtic >0 Mesokurtic =0 (Normal) Platykurtic <0 Greater +ve Icreased kurtosis & Risk. more ve skew. I Leptokurtic distributio, there is a greater probability of both, very small & very large deviatio sfrom mea.

6 Study Sessio #, Readig # 8 Probability Cocepts Radom Variable Ucertai & uforcastable umber. Outcome A observed value of a radom variable. Evet A sigle outcome or a set of outcomes. Mutually Exclusive Evets Both ca t happe at the same time. P(A/B) = 0 & P(AB) = P(A/B). P(B) =0 Exhaustive Evets Iclude all possible outcomes. Two Defiig Properties of Probability Probability 0 P(E) i.e., Probability of a evet lies b/w 0 &. Odds for the evet Probability of occurrece divided by probability of o-occurrece. Probability i terms of Multiplicatio Rule (Joit Probability) Probability that both evets will occur. P(AB) = P(A/B). P(B) For mutually exclusive evets;. P(A/B) = 0, hece, P(AB) = 0. ΣP( E i ) = i.e., Total probability is equal to. Odds agaist the evet Probability of o-occurrece divided by probability of occurrece. Additio Rule Probability that at least oe evet will occur. P(A or B) = P(A) + P(B) - P(AB) For mutually exclusive evets. P(A or B) = P(A) + P(B). Total Probability Rule It highlights the relatioship b/w ucoditioal & coditioal probabilities of mutually exclusive & exhaustive evets. P(R) = P(RI) + P(RI c ) = P(R/I). P(I) + P(R/I c ). P(I c ) Empirical Probability Based o historical facts or data. No judgmets ivolved. Historical + o radom. A Priori Probability Based o sure logic or formula. Radom + historical. Objective Probability Subjective Probability A iformal guess. Ivolves persoal judgmet. Ucoditioal Probability Margial probability. Probability of occurrece of a evet- regardless of the past or future occurrece. Coditioal Probability; P(A/B) Probability of the occurrece of a evet is affectedly the occurrece of aother evet. It is also kow as likelihood of a occurrece. / deotes give or coditioal upo. P(A/B) = P (AB) P(B) Mutually exclusive evets P(A/B) = 0. For idepedet evets, P(A/B) = P(A) Idepedet Evets Evets for which occurrece of oe has o effect o occurrece of the other. P(A/B) = P(A) P(B/A) = P(B) Expected Value Weighted avg. value of a radom variable that results from multiple experimets. It is the best guess of the outcome of a radom variable. Coditioal Expected Value Calculated usig coditioal probabilities. Are cotiget upo the occurrece of some other evet.

7 Study Sessio #, Readig # 8 Covariace Measure of how two assets move together. It measures oly directio. - Cov(x, y) + (property). It is measured i squared uits. Cov(Ri,Rj) = E {[R i - E(R i )] [R j E(R j )]} = Σ P(S) [R i E(R i )] [R j E(R j ). Cov (R A,R A ) = variace (R A ) (property). Covariace Variables ted to + ve Move i same directio. - ve Move i opposite directio. 0 No liear relatioship. Correlatio Measures the directio as well as the magitude. It is a stadardized measure of co-movemet. It has o uits. - corr (Ri,Rj) +. Value Correlatio Variables ted to + Perfectly positive - Perfectly egative Move proportioally i the same directio. Move proportioally i the opposite directio. 0 Ucorrelated No liear relatioship. Corr (R i,r j ) = Cov (R i,r j ) σ (R i ) σ (R j ) Portfolio Baye s Formula Expected Value Variace Used to update a give set of prior probabilities i respose to the arrival of ew iformatio. Where w i = market value of ivestmet i asset i market value of the portfolio Updated probability prior Probability = of ew ifo. probability of ucoditioal evet. probability of ew ifo. Coutig Methods Labelig Formula Assigig each elemet of the etire group i oe of the three or more subgroups. ABCDE Factorial [!] Arragig a give set of items. No subgroups. There are! ways of arragig items. ABC ABC Permutatio [ P r ] Specific orderig of a group of objects ito oly two groups of predetermied size. ABCDE A C B D C Combiatio [ C r ] Choosig r items from a set of items whe order does ot matter. It applies to oly two groups of predetermied size. ABCDE Multiplicatio Rule Selectig oly oe item from each of the two or more groups. AB α β A α ABC DE

8 Study Sessio # 3, Readig # 9 Commo Probability Distributios Probability Distributio Describes the probabilities of all possible outcomes for a radom variable. Sum of probabilities of all possible outcomes is. Radom Variable Distributio Probability Desity Fuctio (PDF) It is used for cotiuous distributio. Deoted by f(x). Fids the probability of a outcome withi a particular rage (b/w two values). Probability of ay oe particular outcome is zero. Biomial Distributio Properties: Two outcomes (success & failure). umber of idepedet trials. With replacemet. Probability of success remais costat. E(x) = p. p(x) = Discrete Fiite (measurable) # of possible outcomes. P(x) ca t be 0 if x ca occur. We ca fid the probability of a specific poit i time. Probability Fuctio Probability of a radom variable beig equal to a specific value. Properties: 0 p(x) Σ p(x) = Cotiuous Ifiite (immeasurable) # of possible outcomes. P(x) ca be zero eve if x ca occur. We ca t fid the probability of a specific poit i time. Cumulative Distributio Fuctio (CDF) Calculates the probability of a radom variable x takig o the value less tha or equal to a specific value x. F(x) = P (X x) Biomial Tree Shows all possible combiatios of up & dow moves over a umber of successive periods. Node: Each of the possible values alog the tree. U is up-move factor. D is dow-move factor (/U). p is probability of up move. (-p) is probability of dow move. Discrete uiform radom variable Discrete Has a fiite umber of specified outcomes. P(x) k. K is the probability for k umber of possible outcomes i a rage. cdf: F(x ) =.p(x). Uivariate Distributio Probability distributio of a sigle radom variable. All outcomes have the same probability. Uiform Probability Distributio Cotiuous Defied over a rage with parameters b (upper limit) & a (lower limit). cdf: It is liear over the variable s rage. Properties: P(x < a or x > b) = 0 P (x X x ) = For all a x < x b. Multivariate Distributio Specifies the probabilities associated with a group of radom variables. Normal Distributio Completely described by µ & σ. Stated as X N (µ, σ ). Symmetric about its mea, Skewess =0. P(X µ) = 0.5 = P(X µ). Kurtosis = 3. Liear combiatio of ormally distributed radom variables is also ormally distributed. Large deviatios from mea are less likely tha small deviatios. Probability becomes smaller & smaller as we move away from mea, but ever becomes 0. Multivariate Normal Distributio It ca be completely described by three parameters meas. variaces. pair wise correlatios. Log-Normal Distributio Geerated by fuctio e x where x is ormally distributed. Skewed to the right. Bouded from below by 0. Stadard Normal Distributio Has µ = 0 & σ = i.e., N (0,). Stadardizatio: Process of covertig a observed value of a radom variable to its z-value.

9 Study Sessio # 3, Readig # 9 Cofidece Iterval Rage of values aroud the expected value withi which actual outcome is expected to be some specified percetage of time. Cofidece Iterval %age x ± s 68% x ±.65s 90% x ±.96s 95% x ± s 95.45% Roy s Safety First Criterio Optimal portfolio miimizes the probability that the retur of the portfolio falls below some miimum acceptable level. Miimize P(R P < R L ). SFR is the umber of stadard deviatios below the mea. SFRatio = Choose the portfolio with greatest SFRatio. Shortfall Risk Probability that portfolio value will fall below some miimum level at a future date. x ±.58s 99% x ± 3s 99.73% Discrete Daily, aually, weekly, mothly compoudig Compouded Rates of Returs Historical Simulatio Based o actual values & actual distributios of the factors i.e., based o historical data. Limitatio: History does ot repeat itself. Historical data does ot provide flexibility. Cotiuous l(s /S 0 ) = l(+hpr) These are additive for multiple periods. Effective aual rate based o cotiuous compoudig is give as: EAR = e Rcc - Specify prob. dist. of stock prices & relevat iterest rate as well as their parameters. Mote-Carlo Simulatio Repeated geeratio of oe or more factors (e.g. risk) that affect required value (e.g., stock price) i order to geerate a distributio of the values (stock price). We have the flexibility of providig the data. Simulatio Procedure for Stock Optio Valuatio Radomly geerate values of stock prices & iterest rates. Uses Valuig complex securities. Simulatig gais / losses from tradig strategy. Estimatig value at risk (VAR). Examiig variability of the differece b/w assets & liabilities of pesio fuds. Valuig portfolio with o - ormal retur distributio. Value the optios for each pair of risk factors. Calculate mea optio value performig may iteratios & use it as estimated optio value. Limitatios Complex procedure. Highly depedet o assumed distributios. Based o a statistical rather tha a aalytical method.

10 s.e = stadard error = rises Study Sessio # 3, Readig # 0 Samplig & Estimatio = approaches to d.f = degrees of freedom = sample size Sample A subgroup of populatio. Sample Statistic It describes the characteristic of a sample. Sample statistic itself is a radom variable. Data Time Variables Time series Cross- sectioal Observatios take over equally spaced time iterval Sigle Save time poit estimate Cetral Limit Theorem (CLT) For a radom sample of size with; populatio mea µ, fiite variace σ, the samplig distributio of sample mea x approaches a ormal probability distributio with mea µ & variace as becomes large. i.e, As ; x Properties of CLT For 30 samplig distributio of mea is approx. ormal. For 30 Not ormal. Mea of distributio of all possible samples = populatio mea µ. Variace o distributio = CLT applies oly whe samples is radom. Simple Radom Samplig Each item of the populatio uder study has equal probability of beig selected. There is o guaratee of selectio of items of a particular category. Sigle Differet variables Poit Estimate (P.E) Sigle (sample) value used to estimate populatio parameter. Estimator: Formula used to compute P.E. Ubiased Expected value of estimator equals parameter e.g., E( x ) = µ i.e, samplig error is zero. Methods of Samplig Stratified Radom Samplig Uses a classificatio system. Separates the populatio ito strata (small groups) based o oe or more distiguishig characteristics. Take radom sample for each stratum. It guaratees the selectio of items from a particular category. Systematic Samplig Select every th umber. It gives approximately radom samplig. Data Etity Characteristics Logitudial Pael Same Multiple Cofidece Iterval (CI) Estimates Results i a rage of values withi which actual parameter value will fall. P.E ±(reliability factor s.e) α= level of sigificace. - α= degree of cofidece. Desirable properties of a estimator Efficiet If var ( ) < x var( x ) the x is efficiet tha x Mea satisfies all 3 properties but media & mode do t. Multiple Save Same If sample is ot radom the CLT does ot apply. CI is ot ubiased. Estimates will ot have the desirable properties. Cosistet As, value of estimator approaches parameter & sample error approaches 0 e.g., As x µ & s.e. 0 Samplig error Sample Correspodig Statistic Populatio Parameter. Samplig Distributio Probability distributio of all possible sample statistics computed from a set of equal size samples radomly draw. Stadard Error (s.e) of Sample Mea Stadard deviatio of the distributio of sample meas. σ σ x = If σ is ot kow the; s s x = As ; x approaches µ ad s.e. Studet s T-Distributio Bell shaped. Shape is based o d.f. d.f. is based o. t-distributio depeds o (d.f) [ormal distributio does ot deped o (d.f)]. Symmetrical about it s mea. Less peaked tha ormal distributio. Has fatter tails. More probability i tails i.e., more observatio are away from the cetre of the distributio & more outliers. More difficult to reject H 0 usig t distributio. C.I for a r.v. usig t distributio must be wider whe d.f are less.

11 Study Sessio # 3, Readig # 0 Distributio Variace Sample Test Statistic Normal No ormal Kow Ukow Small (<30) Large ( 30) * * *The z-statistic is theoretically acceptable here, but use of the t-statistic is more coservative. t z Issues Regardig Selectio of Appropriate Sample Size As ; s.e. & hece C.I is arrow. Large sample may iclude observatios from differet populatio, hece: Precisio may ot icrease. Precisio may eve fall. Limitatios of Large Sample Size Cost may icrease more relative to a icrease i precisio. Biases Data Miig Bias Statistical sigificace of the patter is overestimated because the results were foud through data miig. Sample Selectio Bias Systematically excludig some data from aalysis. It makes the sample o-radom. Look ahead Bias Usig sample data that was t available o the test date. Time-period Bias Time period over which the data is gathered is either too short or too log. Data Miig Usig the same data to fid patters util the oe that works is discovered. Warig Sigs of Data Miig Survivorship Bias Most commo form of sample selectio bias. Excludig weak performaces. Survivig sample is ot radom. Evidece of testig may differet, mostly ureported variables. Lack of ecoomic theory cosistet with empirical results.

12 Study Sessio # 3, Readig # Hypothesis Testig = Level of sigificace t.s = Test statistics t.v = Table Value t.v provides the critical values called as rejectio poits s.s = Sample statistic c.v = Critical value s.e = Stadard error Hypothesis Statemet about parameter value developed for testig. Null Hypothesis H 0 Tested for possible rejectio. Always icludes = sig. Two Types Hypothesis Testig Procedure It is based o sample statistics & probability theory. It is used to determie whether a hypothesis is a reasoable statemet or ot. Alterative Hypothesis H a The oe that we wat to prove. (Source: Waye W. Daiel ad James C. Terrell, Busiess Statistics, Basic Cocepts ad Methodology, Houghto Miffli, Bosto, 997.) Oe Tailed Test Alterative hypothesis havig oe side. Upper Tail H 0 :µ µ 0 vs H a : µ > µ 0. Decisio rule Reject H 0 if t.s > t.v. Lower Tail H 0 :µ µ 0 vs H a : µ < µ 0. Decisio rule Reject H 0 if t.s < - t.v. Two Tailed Test Alterative hypothesis havig two sides. H 0 : µ = µ 0 vs H a µ µ 0. Reject H 0 if Test Statistics (t.s) Hypothesis testig ivolves two statistics: t.s calculated from sample data. critical values of t.s. t.s is a radom variable that follows some distributio. Type I Error Rejectig a true ull hypothesis. Two Types of Errors Type II Error Failig to reject a false ull hypothesis. Decisio Rule It is based o distributio of t.s. It is specific & quatitative. Sigificace Level (α ) Probability of makig a type I error. Deoted by Greek letter alpha (α ). Used to idetify critical values. Power of a Test P(type II error). Probability of correctly rejectig a false ull hypothesis. p- value Probability of obtaiig a critical value that would lead to a rejectio of a true ull hypothesis. Reject H 0 if p-value < α. Statistical Sigificace vs Ecoomical Sigificace Statistically sigificat results may ot ecessarily be ecoomically sigificat. A very large sample size may result i highly statistically sigificat results that may be quite small i absolute terms. Relatioship b/w Cofidece Itervals & Hypothesis Tests Related because of critical value. C.I [(s.s)- (c.v)(s.e)] parameter [(s.s) + (c.v)(s.e)]. It gives the rage withi which parameter value is believed to lie give a level of cofidece. Hypothesis Test -c.v t.s + c.v. rage withi which we fail to reject ull hypothesis of two tailed test give level of sigificace.

13 Study Sessio # 3, Readig # σ = populatio variace N.dist = Normally distributed N.N.dist = No Normally distributed = sample size 30 = large sample < 30 = small sample t.s. = Test statistics t.v = table value d.f = degree of freedom Testig Coditios Test Statistics Decisio Rule Populatio Mea σ kow x µ z = 0 N. dist. σ x µ 30 z = 0 or σ σ ukow *(more coservative) H o :µ µ 0 vs H a : µ >µ 0 Reject H 0 if t.s. > t.v H o :µ > µ 0 vs H a : µ <µ 0 Reject H 0 if t.s. < - t.v σ ukow <30 N. dist. t x µ 0 = ; d.f = - σ H o :µ = µ 0 vs H a : µ µ 0 Reject H 0 if t.s > t.v Equality of the Meas of Two Normally Distributed Populatios based o Idepedet Samples. Ukow variaces assumed equal. Uequal ukow variaces. t( + where; s P = ) ( ( x = d.f = + - ( x t = d. f = x ) ( µ µ ) s P + ) s + ( ) s + x ) ( µ µ ) s s s s + s + s + H o :µ - µ 0 vs Ha: µ -µ > 0 Reject H 0 if t.s > t.v H o :µ - µ > 0 vs Ha: µ -µ < 0 Reject H 0 if t.s < -t.v H o :µ - µ = 0 vs Ha: µ - µ 0 Reject H 0 if t.s > t.v Paired Comparisos Test T.S t (- ) = Testig Variace of a N.dist. Populatio T.s Testig Equality of Two Variaces from N.dist. Populatio T.s Parametric Test Specific to populatio parameter. Relies o assumptios regardig the distributio of the populatio. Decisio Rule Decisio Rule Reject H 0 if t.s > t.v Chi-Square Distributio Asymmetrical. Bouded from below by zero. Chi-Square values ca ever be ve. Decisio Rule Reject H 0 if t.s > t.v F- Distributio Right skewed. Bouded by zero. No-Parametric Test Do t cosider a particular populatio parameter. Or Have few assumptios regardig populatio. H 0: µ d µ d0 vs H a : µ d > µd 0 Reject H 0 if t.s > t.v. H 0: µ d µ d0 vs Ha:µ d < µ d0 Reject H 0 if t.s <-t.v. H 0 : µ d = µ d0 vs H a : µd µ d0 Reject H 0 if t.s > t.v.

14 Study Sessio # 3, Readig # TECHNICAL ANALYSIS T.A = Techical Aalysis F.S = Fiacial Statemets C.F = Cash Flows ROC = Rate of Chage RSI = Relative Stregth Idex. a Study of collective market setimet. Prices are determied by iteractio of supply & demad. Key assumptio of T.A is that EMH does ot hold. Usefuless is limited i illiquid & outside maipulatio markets. Compariso Techical Aalysis Fudametal Aalysis Share price & tradig volume Itrisic value Data is observable Use F.S & other iformatio Ca applied o assets without C.F S.D = Stadard Deviatio M.A = Movig Average M.V = Market Value MACD = Movig Average Covergece/divergece. b Charts of price & volume. Expoetial price chage charts o a log scale. Time iterval reflects horizo of iterest. Types of price charts Lie charts Bar charts Cadlestick charts Poit & figure charts Show closig price as cotiuous lie. Opeig & high & low prices. Use dash & vertical lies as symbols. Same data as bar charts. Use box for opeig & closig price. Patters easier to recogize. i directio of price. Horizotal axis o. of i directio ot time. Price icremet chose is box size. Relative stregth aalysis Volume Chart Assets closig price Bechmark values. i tred, asset is outperformig & vice versa. Usually displayed below price charts. Volume o vertical axis.. c Tred i prices Uptred Dowtred Reachig higher highs & retracig higher lows Show, demad is Tred lie coects icreasig lows Breakout from dowtred (sigificat price chage. Lower lows & retracig lower highs. Show, supply is. Tred lie coects decreasig highs. Breakdow from dowtred (sigificat price chage). Support Level Chage i polarity Resistace Level Buyig emerge. Prevet further price declie. Breached resistace levels become support levels & vice versa. Sellig emerge. Prevet price rise.

15 Study Sessio # 3, Readig #. d Reversal Patters Tred approach a rage of prices but fail to cotiue. Head & shoulder patters for uptred & iverse H&S for dowtreds. Aalyst use size of H&S patter to project price target. Dowtred is projected to cotiue (price after the right shoulder forms). Double top & triple top Cotiuatio Patters Idicate weakeig the buyig pressure. (Similar to H&S). Sellig pressure appears after resistace level. Double bottom & triple bottom for dow treds. Pause i tred rather reversal Triagles Rectagles Form whe prices reach lower highs & higher lows. Ca be symmetrical, ascedig or descedig. Suggest buyig & sellig pressure roughly equal. Size of triagle to set a price target. Form whe tradig temporarily form rage b/w support & resistace level. Suggest prevailig tred will resume. Flags & peats short term price charts, rectagles & triagles.. e Price based Idicators Movig avg. lies Mea of last closig prices. i, smoother the avg. lie. Uptred price is higher the movig avg. & vice versa. M.A for differet periods ca be used together. Short term avg. above log term (golde cross) buy sigal, (dead cross) if vice versa sell sigal. Bolliger bads Based o S.D of closig prices over last periods. Aalyst draw high & low bads above & below -period M.A Prices above upper Bolliger over bought, vice versa over sold. Cotraria strategy buy whe most traders are sellig. Oscillators Tool to idetify overbought or oversold market. Based o M.P but scaled so they oscillate aroud a or betwee two values. Charts used to idetify covergece or divergece of oscillator & M.P Covergece same patter as price, divergece vice versa. Examples of oscillators ROC or Mometum RSI MACD Stochastic oscillator 00 Diff. b/w closig price & closig price period earlier. Buy whe oscillator from to + & vice versa. Ca be aroud o or aroud 00. Ratio of = Total price Total price Oscillate b/w Value> 70 overbought. Value < 30 oversold. Use expoetially smoothed M.V Oscillate aroud 0 but ot bouded. MACD lie above sigal lie buy sigal & vice versa. Calculated from latest closig price & highest & lowest prices. Use two lies bouded 0 & 00. %k = diff. b/w latest price & recet low as % of diff. b/w recet high & lows. % D lie is 3-period avg. of the % k lie.

16 Study Sessio # 3, Readig # 3. e No-Price-Based Idicators Setimet idicators to gai isight ito treds. Bullish icreasig prices, bearish decreasig prices. Opiio polls measure ivestor setimet directly. a. Setimet Idicators. Put/Call Ratio Put/call ratio = Put volume Call volume Ratio egative price outlook. Viewed as cotraria idicator. Extremely high ratio bearish outlook & vice versa.. Volatility Idex (VIX) Measure volatility of optios o S&P 500 stock idex. High VIX fear declies i stock market. Techical aalysts iterpret VIX i cotraria way. b. Flow of fuds idicators Useful for observig chages i demad & supply of securities.. Arms idex or short-term tradig idex (TRIN) Measure of fuds flowig ito advacig & decliig stocks. No.of adv.issue /No.of decliig issue TRIN = Volume of adv.issue/ volume of decliig issue Idex value close to flowig evely to advacig & decliig stocks. Value > majority i decliig stocks, value < majority i advacig stocks.. Margi debt 3. Margi Debt i M.D, buyig, whe reach their limit, buyig, prices, ivestor sell securities to meet margi calls. M.D coicides with prices & M.D with prices. 4. Short Iterest Ratio Short iterest is o. of shares borrowed & sold short. Short iterest ratio = short iterest / Avg. daily tradig volume. Ratio, expect to i price & vice versa. Margi debt ivestor wats to buy more stocks & vice versa. 3. Mutual fud cash positio Ratio of = fud s cash/total assets. Uptred ratio & vice versa. T.A aalysts view as cotraria idicator. 4. New equity issuace IPO & secodary issues add to supply of stocks. Issuer teds to issue whe market peaks, so issuace coicide with high price.. f Cycle theory is study of processes that occur i cycles. Cycle Periods Presidetial Cycles Deceial Patters Kodratieff wave 4-year 0-year 8-year & 54 year. g Elliott wave theory. h Itermarket aalysis Fiacial market prices described by itercoected set of cycles. Waves chart patters, uptred 5 upward waves, 3 dow wards & vice versa i dowtred. Size of waves Fiboacci ratios. Fiboacci umbers states with 0 & ad add each of previous two to produce ext. Ratio of 0.68 ad. 68 used to project price targets. Aalysis of iterrelatioship amog M.V of asset classes (e.g. stocks, bods) Relative stregth ratio to idetify outperformer asset class, the assets withi class.