Real vs Nominal Rates Fisher effect: Approximation R = r + i or r = R - i Example: r = 3%, i = 6% R = 9% = 3%+6% or r = 3% = 9%-6% Fisher effect: Exact R −i r = ; or 1+i 0.09 − 0.06 Numerically: r = 2.83% = + 0.06 1+R 1+r = 1+i Rates of Return: Single Period HPR = P − P0 + D1 P HPR = Holding Period Return P0 = Beginning price P1 = Ending price D1 = Dividend during period one Rates of Return: Single Period Example Ending Price = 48 Beginning Price = 40 Dividend = 48 − 40 + HPR = = 25% 40 Characteristics of Probability Distributions 1) Mean: most likely value 2) Variance or standard deviation 3) Skewness * If a distribution is approximately normal, the distribution is described by characteristics and Normal Distribution s.d s.d r Symmetric distribution Measuring Mean: Scenario or Subjective Returns Subjective returns E(r) = s ∑p i =1 i ⋅ ri ‘s’ = number of scenarios considered pi = probability that scenario ‘i’ will occur ri = return if scenario ‘i’ occurs Numerical example: Scenario Distributions Scenario Probability 0.1 0.2 0.4 Return -5% 5% 15% 0.2 0.1 25% 35% E(r) = (.1)(-.05)+(.2)(.05) +(.1)(.35) E(r) = 15 = 15% Measuring Variance or Dispersion of Returns Subjective or Scenario Distributions Variance = σ = s ∑ p(i) ⋅ [r(i) − E(r)] i =1 Standard deviation = [variance]1/2 = σ Using Our Example: σ2=[(.1)(-.05-.15)2+(.2)(.05- 15)2+…] =.01199 σ = [ 01199]1/2 = 1095 = 10.95% Risk - Uncertain Outcomes p = W = 100 1-p = W1 = 150; Profit = 50 W2 = 80; Profit = -20 E(W) = pW1 + (1-p)W2 = 122 σ2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2 σ2 = 1,176 and σ = 34.29% Risky Investments with Risk-Free Investment p = W1 = 150 Profit = 50 1-p = W2 = 80 Profit = -20 Risky Investment 100 Risk Free T-bills Risk Premium = 22-5 = 17 Profit = Portfolio Selection & Risk Aversion E(r) U’’’ U’’ U’ P Q More risk-averse investor S Efficient frontier of risky assets Less risk-averse investor σ Efficient Frontier with Lending & Borrowing E(r) CAL B Q P A rf F s Capital Asset Pricing Model (CAPM)    Equilibrium model that underlies all modern financial theory Derived using principles of diversification with simplified assumptions Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development Assumptions     Individual investors are price takers Single-period investment horizon Investments are limited to traded financial assets No taxes, and transaction costs Assumptions (cont’d)    Information is costless and available to all investors Investors are rational mean-variance optimizers There are homogeneous expectations Resulting Equilibrium Conditions    All investors will hold the same portfolio of risky assets – market portfolio Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value The market portfolio is on the efficient frontier and, moreover, it is the tangency portfolio Resulting Equilibrium Conditions (cont’d)   Risk premium on the market depends on the average risk aversion of all market participants Risk premium on an individual security is a function of its covariance with the market Capital Market Line E(r) CML E(rM) M rf σm σ Slope and Market Risk Premium M rf E(rM) - rf = = = = E(rM) − rf σM The market portfolio Risk free rate Market risk premium Slope of the CML Expected Return and Risk on Individual Securities   The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio Individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio Security Market Line E(r) SML E(rM) rf ß M = 1.0 ß SML Relationships β Slope SML = = Cov(ri,rm) / σm E(rm) - rf = market risk premium E(r)SML = rf + β[E(rm) - rf] BetaM = Cov (rM,rM) / σΜ2 2 = σM / σM = Sample Calculations for SML E(rm) - rf = 08 a) rf = 03 βx = 1.25 E(rx) = 03 + 1.25(.08) = 13 or 13% b) βy = E(ry) = 03 + 6(.08) = 078 or 7.8% Graph of Sample Calculations E(r) SML Rx=13% 08 Rm=11% Ry=7.8% 3% ß 1.0 1.25 ß ß y m x ß Disequilibrium Example E(r) SML 15% Rm=11% rf=3% 1.0 1.25 ß [...]... larger proportions of the risk free rate Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations CAL with Risk Preferences E(r) P Borrower 7% Lender σp = 22% σ CAL with Higher Borrowing Rate E(r) P 9% 7% ) S = 27 ) S = 36 σp = 22% σ Risk Reduction with Diversification... rates of return of each asset comprising the portfolio, with the portfolio proportions as weights rp = w1r1 + w2r2 Portfolio Mathematics: Risk with Risk-Free Asset Rule 4: When a risky asset is combined with a risk-free asset, the portfolio standard deviation equals the risky asset’s standard deviation multiplied by the portfolio proportion invested in the risky asset σ p = wrisky asset × σrisky asset... Allocation Line   Borrow at the Risk-Free Rate and invest in stock Using 50% Leverage rc = (-.5) (.07) + (1.5) (.15) = 19 σc = (1.5) (.22) = 33 Indifference Curves and Risk Aversion E(r) E(rp)=15% Certainty equivalent of portfolio P’s expected return for two different investors A=4 A=2 P rf=7% σp = 22% σ Risk Aversion and Allocation    Greater levels of risk aversion lead to larger proportions of the... Deviation Portfolio Mathematics: Assets’ Expected Return Rule 1 : The return for an asset is the probability weighted average return in all scenarios E(r) = s ∑p i =1 i ⋅ ri Portfolio Mathematics: Assets’ Variance of Return Rule 2: The variance of an asset’s return is the expected value of the squared deviations from the expected return 2 σ = s ∑p i =1 i 2 ⋅ [ri − E(r)] Portfolio Mathematics: Return on a... S = 36 σp = 22% σ Risk Reduction with Diversification St Deviation Unique Risk Market Risk Number of Securities Two-Security Portfolio: Return rP = w1 ⋅ r1 + w2 ⋅ r 2 w1 w2 r1 r2 = proportion of funds in Security 1 = proportion of funds in Security 2 = expected return on Security 1 = expected return on Security 2 n ∑w i i =1 =1 ... investors Portfolios of One Risky Asset and One Risk-Free Asset  Assume a risky portfolio P defined by : E(rp) = 15% and σp = 22%  The available risk-free asset has: rf = 7% and σrf = 0%  And the proportions invested: y% in P and (1-y)% in rf Expected Returns for Combinations E(rc) = yE(rp) + (1 - y)rf rc = complete or combined portfolio If, for example, y = 75 E(rc) = 75(.15) + 25(.07) = 13 or... variance is given by: σ p 2 = w12 σ12 + w22 σ22 + 2w1w2Cov(r1, r2) Allocating Capital Between Risky & Risk Free Assets    Possible to split investment funds between safe and risky assets Risk free asset: proxy; T-bills Risky asset: stock (or a portfolio) Allocating Capital Between Risky & Risk Free Assets   Examine risk/return tradeoff Demonstrate how different degrees of risk aversion will affect allocations ... efficient frontier These portfolios are dominant The Minimum-Variance Frontier of Risky Assets E(r) Efficient frontier Global minimum variance portfolio Individual assets Minimum variance frontier... 150; Profit = 50 W2 = 80; Profit = -20 E(W) = pW1 + (1-p)W2 = 122 σ2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2 σ2 = 1,176 and σ = 34.29% Risky Investments with Risk-Free Investment p = W1 = 150 Profit... HPR = = 25% 40 Characteristics of Probability Distributions 1) Mean: most likely value 2) Variance or standard deviation 3) Skewness * If a distribution is approximately normal, the distribution