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## Empirical Issues Portfolio Performance Evaluation

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**Content**• Simple Investment Return Measurement • Time-weighted VS Dollar-weighted Returns • Arithmetic VS Geometric Returns • Risk-adjusted Measures • Jensen’s • Treynor’s • Sharpe • Characteristics of Investment Portfolio • Style Box • Sector Weighting Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Motivation**Main question: how well does our investment portfolios do? • As trivial as this question, a scientific measurement is tricky to formulate. • Even average portfolio return is not as straightforward to measure • Adjusted for risk is even more problematic Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Rate of Return**ONE PERIOD Return (R) = Total Proceeds/Initial Investment Total Proceeds includes cash distributions and capital gains. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Trivial Example I**• Suppose you have invested $10,000 in a carefully-chosen investment portfolio. Exactly 1 year from the time of initial investment, the investment portfolio gives out $100 of cash dividends, and the investment portfolio has a market value of $11,000. R = (Dividends + Capital gains)/Initial Investment = [$100 + ($11,000 - $10,000)] /$10,000 = 11% Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Trivial Example II**• Suppose you have invested $10,000 in a carefully-chosen investment portfolio. Exactly 1/2 year from the time of initial investment, the investment portfolio gives out $100 of cash dividends, and exactly 1 year from the time of initial investment, the investment portfolio has a market value of $11,000. MULTI-Period Let r be the rate of return such that Initial Investment= Present Value of All cash flows from investment discounted at r Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Trivial Example II**• Suppose you have invested $10,000 in a carefully-chosen investment portfolio. Exactly 1/2 year from the time of initial investment, the investment portfolio gives out $100 of cash dividends, and exactly 1 year from the time of initial investment, the investment portfolio has a market value of $11,000. MULTI-Period Let r be the rate of return such that $10,000 = $100/[(1+r)1/2] + $11,000 /[(1+r)1] SOLVE for r, the rate of return of investment. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Trivial Example II**Initial Investment= Present Value of All cash flows from investment discounted at r This is extremely similar to the internal rate of return. I’ve talked about IRR having some problems in Lecture 2 when I compared IRR rule to NPV rule for project selection. Bottom Line: Calculating returns is really not that simple once we’re dealing with multi-periods. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Multi-periods, ΔCash ≠ 0**• When you add or withdraw cash from your investment portfolio, measuring the rate of return becomes more difficult. Trivial Example III • Continuing with our example, with $10,000 initial investment, $100 year-end cash dividend payout, and the portfolio has a market value of $11,000. At this point, you think the portfolio is doing great and decide to invest $11,000 more on this portfolio without changing the proportions of the content in it. By the end of year 2, the portfolio is worth $23,500 with no cash dividend during year 2. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Trivial Example III**Dollar-Weighted Return: • Calculate the internal rate of return: Present value of= Present Value of All cash Initial Investment flows from investment discounted at r $10,000 =$100/[(1+r)1] + $11,000/[(1+r)1] + $23,500/[(1+r)2] SOLVE for r, the dollar-weighted rate of return of investment. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Trivial Example III**Dollar-Weighted Return: • It is dollar-weighed because when you double the size of the portfolio, it has a greater influence on the average overall return than when you hold less of this portfolio in year 1. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Trivial Example III**Time-Weighted Return: • Alternative to the dollar-weighed returns • Ignores the number of shares of stock held in each period. • For the example, it ignores the changing size of your investment portfolio when you decided to double up the investment in year-end. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Trivial Example III**Time-Weighted Return: • 1st year return = 11% as calculated in Trivial Example I • 2nd year return = 6.82% • Because: at the beginning of year 2, the portfolio is worth $22,000. By the end of year 2, it is worth $23,500. • 6.82% = ($23,500 – $22,000)/$22,000 Time-weighted return = (11% + 6.82%)/2 = 8.91% Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Time VS Dollar**• Which one to use? Time-weighted or dollar-weighted? Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Time VS Dollar**• Which one to use? Time-weighted or dollar-weighted? • ANSWER: it depends (Typical answer from economists) Shopping for mutual funds • Time-weighted is better • Since the value and the composition of most mutual funds do change frequently Assessing your own portfolio for the past years • Dollar-weighted is better • Since more money you invest when the portfolio performs well, the more money you earn. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Arithmetic VS Geometric**Time-weighted return = (11% + 6.82%)/2 = 8.91% This is an arithmetic average. It ignores compounding. Geometric average return takes into account the effect of compounding. • If invest for 2 years, 1st year got 11%, 2nd year got 6.82%. • Compound growth rate = (1+11%) (1 + 6.82%) = 1.17502 • Geometric average return (rG) (1+rG) (1+rG) = (1+11%) (1 + 6.82%) => rG = 8.398% Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Rules**RULE 1: Arithmetic average return > Geometric average return => 8.91%> 8.398% RULE 2: “(Arithmetic average - Geometric average) ↑” as period-by-period returns are more volatile. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Rules**RULE 2: (Arithmetic average - Geometric average) ↑ as period-by-period returns are more volatile. In general, relationships between the two returns: rG = r – 0.5(σ2) Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Arithmetic VS Geometric**• Which one to use? Arithmetic avg. or Geometric avg.? Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Arithmetic VS Geometric**• Which one to use? Arithmetic avg. or Geometric avg.? • ANSWER: it depends (AGAIN!) Past returns • Use geometric average for looking at past returns • Geometric average represents the constant rate of return needed to earn in each year to match the actual performance over some past investment period. Thus, it serves its purpose as the right measurement of the past performance Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Arithmetic VS Geometric**• Which one to use? Arithmetic avg. or Geometric avg.? • ANSWER: it depends (AGAIN!) Future expected returns • Use arithmetic average for future expected returns • It is an unbiased estimate of the portfolio’s expected future return. In contrast, since geometric average is always lower than the arithmetic average, it gives a downward biased estimate. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Trivial Example IV**Why Unbiased? • Suppose your investment portfolio has the risk of 50% of the chance, it doubles in value; and another 50% of the chance, its value drops by half. Suppose it did double in value in the first year, but dropped by half in value in the second year. The geometric average is exactly equal to zero. • Arithmetic average return = [100% + (-50%)]/2 = 25% • True Expected return = 50%(100%) + 50%(-50%) = 25% (UNBIASED!!!) Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Risk-adjusted Measures**Why Risk-adjusted? • Does earning 11% return in year 1 means you are smart? Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Risk-adjusted Measures**Why Risk-adjusted? • Does earning 11% return in year 1 means you are smart? • ANSWER: It depends! • Case 1: Suppose for the same level of risk, on average other investors would get 20% in year 1. 11% is really low, and you are really not that smart. • Case 2: Suppose for the same level of risk, on average other investors would get 10% in year 1. 11% is good, and you are lucky. Bottom line: Returns must be adjusted for risk before they can be compared meaningfully. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Jensen’s Measures**Formula: αp = E(Rp) – {Rf + E(RM) – Rf]βp} • Also known as Portfolio’s Alpha. • Uses CAPM as benchmark. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Treynor’s Measures**Formula: Tp = [E(Rp)– Rf]/βp • Measures the slope of the line that connects the point of the portfolio in question to the y-intercept on the SML graph. • Also uses CAPM as benchmark. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Treynor’s Jensen’s**Tp = [E(Rp)– Rf]/βp αp = E(Rp) – {Rf + E(RM) – Rf]βp} e.g., 2 Portfolios: A ~ βA = 0.9, E(RA)– Rf = 0.11, αA = 0.02 B ~ βB = 1.6, E(RB)– Rf = 0.19, αB = 0.03 M ~ βM = 1.0, E(RM)– Rf = 0.10, αM = 0 E(Ri) Security Market Line E(RM) slope = [E(RM) - Rf] = Eqm. Price of risk = 0.1 Rf b = [COV(Ri, RM)/Var(RM)] bM= 1.0 Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Treynor’s Jensen’s**Tp = [E(Rp)– Rf]/βp αp = E(Rp) – {Rf + E(RM) – Rf]βp} e.g., 2 Portfolios: A ~ βA = 0.9, [E(RA)– Rf] = 0.11, αA = 0.02 B ~ βB = 1.6, [E(RB)– Rf] = 0.19, αB = 0.03 M ~ βM = 1.0, [E(RM)– Rf] = 0.10, αM = 0 E(Ri) Security Market Line M αA = 0.02 E(RM) slope = [E(RM) - Rf] = Eqm. Price of risk = 0.1 Rf b = [COV(Ri, RM)/Var(RM)] bM= 1.0 Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Treynor’s Jensen’s**Tp = [E(Rp)– Rf]/βp αp = E(Rp) – {Rf + E(RM) – Rf]βp} e.g., 2 Portfolios: A ~ βA = 0.9, [E(RA)– Rf] = 0.11, αA = 0.02 B ~ βB = 1.6, [E(RB)– Rf] = 0.19, αB = 0.03 M ~ βM = 1.0, [E(RM)– Rf] = 0.10, αM = 0 E(Ri) Slope = TA = 0.11/0.9 = 0.12222 Security Market Line M E(RM) slope = [E(RM) - Rf] = Eqm. Price of risk = 0.1 Rf b = [COV(Ri, RM)/Var(RM)] bM= 1.0 Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Treynor’s Jensen’s**Tp = [E(Rp)– Rf]/βp αp = E(Rp) – {Rf + E(RM) – Rf]βp} e.g., 2 Portfolios: A ~ βA = 0.9, [E(RA)– Rf] = 0.11, αA = 0.02 B ~ βB = 1.6, [E(RB)– Rf] = 0.19, αB = 0.03 M ~ βM = 1.0, [E(RM)– Rf] = 0.10, αM = 0 E(Ri) Security Market Line αB = 0.03 M E(RM) slope = [E(RM) - Rf] = Eqm. Price of risk = 0.1 Rf b = [COV(Ri, RM)/Var(RM)] bM= 1.0 Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Treynor’s Jensen’s**Tp = [E(Rp)– Rf]/βp αp = E(Rp) – {Rf + E(RM) – Rf]βp} e.g., 2 Portfolios: A ~ βA = 0.9, [E(RA)– Rf] = 0.11, αA = 0.02 B ~ βB = 1.6, [E(RB)– Rf] = 0.19, αB = 0.03 M ~ βM = 1.0, [E(RM)– Rf] = 0.10, αM = 0 E(Ri) Slope = TB = 0.19/1.6 = 0.11875 Security Market Line M E(RM) slope = [E(RM) - Rf] = Eqm. Price of risk = 0.1 Rf b = [COV(Ri, RM)/Var(RM)] bM= 1.0 Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Sharpe’s Measures**Formula: Sp = [E(Rp)– Rf]/σ(Rp) • Measures the slope of the line that connects the point of the portfolio in question to the y-intercept on the CML graph. • Also uses CAPM as benchmark, but built on the portfolio theory and the Capital Market line. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Sharpe’s Measures**E(Rp) Sp = [E(Rp)– Rf]/σ(Rp) Slope = Sp M Portfolio P E(RM) Rf σp σM Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Which Measure?**• Which one to use? • Answer: It depends (Our friend again!!!) • If the portfolio represents the entire investment for an individual, Sharpe’s Measure should be used. • If many alternatives are possible, use Jensen’s measure or the Treynor’s Measure because both are measures appropriately adjusted for risk. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Other Measures**• Morningstar’s Risk-adjusted rating • Widely used in the industry • Lots of research about mutual funds in Morningstar’s website. • Please check out the details from the website. Will not be tested in the exam, but I want you to know it. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Portfolio Characteristics**• Based on the idea that current make-up of a portfolio will be a good predictor for the next period’s returns. • Mainly uses classifications of different risky assets, into different types of assets or different sectors of assets. • 2 examples are shown as follows: Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Portfolio Characteristics**• Style Box Vertical Axis – Dividing stocks by market capitalization. Horizontal Axis – Dividing stocks by P/E ratios and Book-to-Price Ratio to determine whether a fund is classified as growth, blend or value. Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics**Portfolio Characteristics**• Sector Weighting • Display the percentage of stocks in the fund or portfolio that is invested in each sector.e.g., BMO Dividend Fund Return Measurement Time VS Dollar Arithmetic VS Geometric Risk-Adjusted Measure Portfolio Characteristics