Introduction to Financial Markets PDF

Introduction to Financial Markets Unit 5 Risk and Return Prof. Dr. M. De Ceuster Prof. Dr. M. De Ceuster Introduction to Financial Markets 1 / 24  What is Risk?...

Introduction to Financial Markets Unit 5 Risk and Return Prof. Dr. M. De Ceuster Prof. Dr. M. De Ceuster Introduction to Financial Markets 1 / 24  What is Risk? Section 1 What is Risk? Prof. Dr. M. De Ceuster Introduction to Financial Markets 2 / 24  What is Risk? Risk In Italian riscare means ‘to dare’. In Chinese feng xian refers to danger and opportunity. IN BUSINESS, TAKING RISK IN AN INTELLIGENT WAY IS KEY. Prof. Dr. M. De Ceuster Introduction to Financial Markets 3 / 24  Distribution of the Returns of an Asset Section 2 Distribution of the Returns of an Asset Prof. Dr. M. De Ceuster Introduction to Financial Markets 4 / 24  Distribution of the Returns of an Asset Price Plot of Apple (AAPL) AAPL [2010−01−04/2023−02−02] Last 150.820007 150 100 50 0 Volume (millions): 1500 116,868,600 1000 500 0 Jan 04 2010 Jan 03 2012 Jan 02 2014 Jan 04 2016 Jan 02 2018 Jan 02 2020 Jan 03 2022 Prof. Dr. M. De Ceuster Introduction to Financial Markets 5 / 24  Distribution of the Returns of an Asset Observations This is a time series on a daily frequency. We observe a positive trend over time. Technically, this time series is non-stationary. We seean exponential shape which reflects the principle of compounding. The absolute price changes increase over time. An investor is indifferent between buying 1 stock of 100 or buying two shares of 50. What matters is the return s(he) can realize. From a statistical perspective, returns are stationary. Prof. Dr. M. De Ceuster Introduction to Financial Markets 6 / 24  Distribution of the Returns of an Asset Returns Simple returns (ordinary returns) are the percentual changes in the value of the asset (taking into account all cash flows). rt = P 1 - P 0 + C P0 where C is the cash distribution (dividend), P t the price of the asset at time t and rt the ordinary return at time t. Example P 0 = 100; P 1 = 105 and C = 1 gives an ordinary return of 105 - 100 + 1 = 6% 100 Prof. Dr. M. De Ceuster Introduction to Financial Markets 7 / 24  Distribution of the Returns of an Asset Relationship Prices and Returns Ordinary returns are multiplicative in nature. P 1 = P 0 ◊ (1 + r1) P T = P 0 x (1 + r1 ) x (1 + r2 ) x · · · x (1 + rT ) Example Suppose P 0 = 100, r1 = 25% and r2 = -20% Then P 1 = 100 x (1 + 25%) x (1 - 20%) = 100. Prof. Dr. M. De Ceuster Introduction to Financial Markets 8 / 24  Distribution of the Returns of an Asset Daily Returns r_d 2010−01−05/2023−02−02 0.10 0.10 0.05 0.05 0.00 0.00 −0.05 −0.05 −0.10 −0.10 Jan 05 2010 Jul 01 2011 Jan 02 2013 Jul 01 2014 Jan 04 2016 Jul 03 2017 Jan 02 2019 Jul 01 2020 Jan 03 2022 Prof. Dr. M. De Ceuster Introduction to Financial Markets 9 / 24  Distribution of the Returns of an Asset Monthly Returns 0. 2 0. 1 0. 0 −0. 1 Feb 2010 Jul 2011 Nov 2012 Apr 2014 Sep 2015 Feb 2017 Jul 2018 Nov 2019 Apr 2021 Sep 2022 Prof. Dr. M. De Ceuster Introduction to Financial Markets 10 / 24  Distribution of the Returns of an Asset Returns of Risky Assets are Random Variables A probability distribution assigns a probability to each possible out- come of the random variable. For the returns of speculative prices we use continuous random variables. The moments of aprobability distribution measurecertain characteris- tics of the probability distribution. The first two moments are the mean and the variance. Under simplifying assumptions the first moment i.e. the mean measures the expected return. the second moment i.e. the variance measures dispersion. Prof. Dr. M. De Ceuster Introduction to Financial Markets 11 / 24  Distribution of the Returns of an Asset The Histogram of Daily Returns Histogram of r_d 30 25 20 Density 15 10 5 0 −0.10 −0.05 0.00 0.05 0.10 r_d Prof. Dr. M. De Ceuster Introduction to Financial Markets 12 / 24  Distribution of the Returns of an Asset The Histogram of Monthly Returns Histogram of r_m 7 6 5 4 Density 3 2 1 0 −0.2 −0.1 0.0 0.1 0.2 r_m Prof. Dr. M. De Ceuster Introduction to Financial Markets 13 / 24  Portfolio Theory Section 3 Portfolio Theory Prof. Dr. M. De Ceuster Introduction to Financial Markets 14 / 24  Portfolio Theory Harry Markowitz (1952) Investors invest money based on the expected return and the risk. This has to be evaluated at portfolio level. Prof. Dr. M. De Ceuster Introduction to Financial Markets 15 / 24  Portfolio Theory An Estimator of Expected Returns (under a simplified model for the returns) ÿN E(r̃) = r̄ = ri · p(ri ) i= 1 ÿN 1 E(r̃) = r̄ = N ri i= 1 The mean monthly return on AAPL is ## 0.02402937 Prof. Dr. M. De Ceuster Introduction to Financial Markets 16 / 24  Portfolio Theory In Search for A Convenient Measure of “Risk” Range = maximum – minimum Interquartile range: IQR(r) = Q 0.75 - Q 0.25 Mean Absolute Deviation: MAD(r) = E |r - E(r)| Mean Squared Deviation or Variance: var(r) = E [r - E(r)]2 Standard deviation: std(r) = var(r) Prof. Dr. M. De Ceuster Introduction to Financial Markets 17 / 24  Portfolio Theory In Search for A Convenient Measure of “Risk” The standard deviation of the monthly returns on AAPL is ## 0.07967378 Prof. Dr. M. De Ceuster Introduction to Financial Markets 18 / 24  Portfolio Theory In Search for A Convenient Measure of “Risk” 1 Standard deviations are expressedin the sameunit of measurement asthe returns. 2 Assuming normality, wecan makeprobability statements basedon the mean and the standard deviation. The following rules of thumb are very useful. Prob(E(X̃ ) ≠ 1‡ < x < E(X̃ ) + 1‡ ) ¥ 2/3 Prob(E(X̃ ) ≠ 2‡ < x < E(X̃ ) + 2‡ ) ¥ 95% Prob(E(X̃ ) ≠ 3‡ < x < E(X̃ ) + 3‡ ) ¥ 99% Prof. Dr. M. De Ceuster Introduction to Financial Markets 19 / 24  Portfolio Theory Expected Return and Risk on Portfolio Level Expected returns are linear. The expected return of a portfolio is the weighted average of the expected returns on the individual assets. E(rA + B ) = wA E(rA ) + wB E(rB ) The variance of a portfolio is nonlinear. The co-movements between the assets enter into the equation. 2 2 2 2 ‡A2 + B = wA ‡A + wB ‡ B + 2wA wB cov(rA , rB ) Prof. Dr. M. De Ceuster Introduction to Financial Markets 20 / 24  Portfolio Theory Co-movement Co-movement can be measured with Covariances as cov(A, B ) = E(rA - rĀ )(rB - rB̄ ) Correlations (i.e. standardized covariances) as covA,B fl A,B = ‡A‡B Correlations are easier to interpret since ≠1 Æfl A,B Æ+1. Prof. Dr. M. De Ceuster Introduction to Financial Markets 21 / 24  Portfolio Theory Efficient Frontier “with two risky assets” Prof. Dr. M. De Ceuster Introduction to Financial Markets 22 / 24  Portfolio Theory Efficient Frontier How can weconstruct portfolios in such a way that wereduce risk but do not sacrifice return? Two step procedure 1 Build an efficient frontier Maximize expected return given risk Minimize risk given expected return 2 Choose one portfolio based on the investor’s risk aversion Prof. Dr. M. De Ceuster Introduction to Financial Markets 23 / 24  Portfolio Theory Diversification Prof. Dr. M. De Ceuster Introduction to Financial Markets 24 / 24