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LM05 Portfolio Mathematics 2025 Level I Notes © IFT. All rights reserved 1 LM05 Portfolio Mathematics 1. Introduction...........................................................................................................................................................2 2. Portfolio Expected Return and Variance of Return................................................................................2 3. Forecasting Correlation of Returns: Covariance Given a Joint Probability Function................6 4. Portfolio Risk Measures: Applications of The Normal Distribution ................................................7 Summary......................................................................................................................................................................9 Required disclaimer: IFT is a CFA Institute Prep Provider. Only CFA Institute Prep Providers are permitted to make use of CFA Institute copyrighted materials which are the building blocks of the exam. We are also required to create / use updated materials every year and this is validated by CFA Institute. Our products and services substantially cover the relevant curriculum and exam and this is validated by CFA Institute. In our advertising, any statement about the numbers of questions in our products and services relates to unique, original, proprietary questions. CFA Institute Prep Providers are forbidden from including CFA Institute official mock exam questions or any questions other than the end of reading questions within their products and services. CFA Institute does not endorse, promote, review or warrant the accuracy or quality of the product and services offered by IFT. CFA Institute®, CFA® and “Chartered Financial Analyst®” are trademarks owned by CFA Institute. © Copyright CFA Institute Version 1.0
LM05 Portfolio Mathematics 2025 Level I Notes © IFT. All rights reserved 2 1. Introduction This learning module covers: • Calculating the expected return and variance of a portfolio • Calculating covariance and correlation of portfolio returns using a joint probability function • Portfolio risk measures: Roy’s safety-first ratio 2. Portfolio Expected Return and Variance of Return Expected Return A portfolio’s expected return can be calculated as: E(RP) = w1E(R1) + w2E(R2) + ... + wnE(Rn) where: wn = portfolio weight of nth security in the portfolio Rn = expected return of nth security in the portfolio n = number of securities in the portfolio We will discuss portfolio expected return and variance of return using a two-stock portfolio. Example 40% of the portfolio is invested in Stock A and 60% is invested in Stock B. As shown in the table below, the expected return of each stock depends on the economic scenario. Scenario P(Scenario) Expected returns of A Expected returns of B Recession 0.25 2% 4% Normal 0.50 8% 10% Boom 0.25 12% 16% Calculate the expected return of A and B. Solution: Given the data presented above: The expected return of A is: 0.25 x 2 + 0.50 x 8 + 0.25 x 12 = 7.5%. The expected return of B is: 0.25 x 4 + 0.50 x 10 + 0.25 x 16 = 10%. Expected return of the portfolio = weight of A in the portfolio x expected return of A + weight of stock B in the portfolio x expected return of B = 0.4 x 7.5 + 0.6 x 10 = 9% The expected portfolio return is 9%. As the term implies, this is the expected return. The actual return will vary around 9%. The amount of variability is measured by the variance. In order to determine the variance of return, we must first calculate the covariance.

LM05 Portfolio Mathematics 2025 Level I Notes © IFT. All rights reserved 4 It is computed as: ρ(Ri , Rj) = Cov(Ri , Rj)/σ(Ri) σ(Rj) We will now apply this formula to calculate the correlation between the returns of A and B from our example. We have already shown that the covariance of returns is 0.0015. In order to calculate the correlation, we need the standard deviation of A and B. Using a financial calculator, we can determine that the standard deviation of A is 0.0357 and the standard deviation of B is 0.0424. The correlation, ρ(A, B) = Cov(A,B) σ(A)σ(B) = 0.0015/(0.0357 x 0.0424) = 0.99. Instructor’s Note: The keystrokes for calculating the standard deviation of A are shown below. Keystrokes Explanation Display [2nd] [DATA] Enters data entry mode [2nd] [CLR WRK] Clears data register X01 0.02 [ENTER] 1st possible value of random variable X01 = 0.02 [↓] 25 [ENTER] Probability of 25% for X01 Y01 = 25 [↓] 0.08 [ENTER] 2nd possible value of random variable X02 = 0.08 [↓] 50 [ENTER] Probability of 50% for X02 Y02 = 50 [↓] 0.12 [ENTER] 3rd possible value of random variable X03 = 0.12 [↓] 25 [ENTER] Probability of 25% for X03 Y03 = 25 [2nd] [STAT] Puts calculator into stats mode [2nd] [SET] Press repeatedly till you see → 1-V [↓] Total number of entries N = 100 [↓] Expected value of random variable X = 0.075 [↓] Sample standard deviation Sx = 0.0359 [↓] Population standard deviation σx = 0.0357 The correlation of 0.99 (almost 1) implies a very strong positive relationship between the returns of A and B. This is more meaningful than the covariance number of 0.0015 which tells us that there is a positive relationship between the returns of A and B but does not give a sense for the strength of the relationship. Variance of returns Once we know the covariance, we can calculate the variance of a portfolio using this formula: σ 2 (RP ) = w1 2σ1 2 (R1 ) + w2 2σ2 2 (R2 ) + 2w1w2Cov (R1R2 )