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32 lacible | Octobre 2021 FEATURE ARTICLE standard deviation is also the same before and after taxes. In the standard deviation formula, just use the after-tax returns already calculated and you will get the same outcome as if you had used the gross return. b) Standard Deviation in Investment Accounts Unlike the income in deferred plans and TFSAs, the income in a non-registered account is taxed annually. How earnings from a non-registered account are taxed depends on the type of income generated (interest, dividends, capital gains). The principle for after-tax standard deviation in a non-registered account is similar to a registered account, but you have to apply the same tax rate that is applied to the income. In other words, the standard deviation also depends on the type of income generated. This gives us the following formula: Take, for example, a portfolio comprised of only Canadian equities that earns a return of 6.10%, of which 4.60% is a capital gain and 1.50% is a dividend. With a maximum marginal tax rate, we can conclude that the investor will receive 70% of the gross return 1 but also bear about 70% of the risk. 2 c) Modern Portfolio Theory In modern portfolio theory, better known as the Markowitz model, standard deviation is especially important for creating an efficient portfolio, that is, a portfolio with the optimal combination of assets, which will generate the highest expected return based on a given standard deviation. But the Markowitz model is also affected by the latent income taxes on an investment account. In the following example, we will illustrate various combinations in a TFSA and an investment account. Assuming a neutral correlation between the accounts and an unadjusted standard deviation of 13% for equities and 8% for fixed income, we get the orange line, known as the efficient frontier, which represents the various combinations in a TFSA. The blue line represents the efficient frontier in a non-registered account. 1 ((4.60% (1 – 53.31% x 50%) + 1.50% (1 – 40.10%)) × 6.10%) 2 The investor receives 73% of the return on the 4.60% from the capital gain, plus 59.9% of the 1.50% from the dividend, for a total after-tax return of 4.2726%. The investor receives 4.2726% × 6.10%, or 70% of the gross return. 3 Assuming that the return does not include any dividend component. 4 Assuming that the return does not include any dividend component. You can see that when a hypothetical tax rate is applied to the expected returns, the portfolio's return and standard deviation become lower. 2. Beta Beta is another measure that can be used to assess the risk of a portfolio and that must also be calculated after taxes. There are several ways to calculate a security's beta, but the most common one uses market model regression. Briefly, this method regresses the returns on individual securities against the market's returns. The beta therefore represents the slope of the line of regression and compares the risk of a security to a benchmark index. In practice, the beta variable is used with the capital asset pricing model (CAPM) to estimate the rate of return expected by the market for a security based on its systemic risk. Like standard deviation, beta is the same before and after income tax in deferred plans and TFSAs but must be adjusted based on the type of income for investment accounts. Here is the CAPM formula for investment accounts: Let's take the example of a Canadian equity with an initial beta of 0.8, a market risk premium of 4.80% and a risk-free rate of 1.50%. Using the basic CAPM, we get an expected return of 5.3%, taxable as a capital gain. 3 By adjusting the beta to reflect the investment account, we get the orange line in the graph. The starting point is always the risk-free rate, taxed as if it were interest, and the risk premium is taxed as if it were a capital gain. 4 We can see that the

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