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1. YEOJU Technical Institute in Tashkent 1 Investment

2. Risk and Return: Past and Prologue 2 Week 5

3. Preview 3

4. RATES OF RETURN 4

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6. Measuring Investment Returns over Multiple Periods Arithmetic average The arithmetic average of the quarterly returns is just the sum of the quarterly returns divided by the number of quarters: (10 + 25 - 20 + 20)/4 = 8.75% Geometric average The geometric average of the quarterly returns is equal to the single per-period return that would give the same cumulative performance as the sequence of actual returns. 6

7. dollar-weighted return. To account for varying amounts under management, we treat the fund cash flows as we would a capital budgeting problem in corporate finance and compute the portfolio manager’s internal rate of return (IRR). 7

8. Conventions for Annualizing Rates of Return 8

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10. INFLATION AND THE REAL RATE OF INTEREST 10

11. Example 11

12. The Equilibrium Nominal Rate of Interest 12

13. RISK AND RISK PREMIUMS 13

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17. The Normal Distribution 17

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20. Two special properties of the normal distribution lead to critical simplifications of investment management when returns are normally distributed: 1. The return on a portfolio comprising two or more assets whose returns are normally distributed also will be normally distributed. 2. The normal distribution is completely described by its mean and standard deviation. No other statistic is needed to learn about the behavior of normally distributed returns. These two properties in turn imply this far-reaching conclusion: 3. The standard deviation is the appropriate measure of risk for a portfolio of assets with normally distributed returns. In this case, no other statistic can improve the risk assessment conveyed by the standard deviation of a portfolio. 20

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23. Normality over Time 23

24. Using Time Series of Returns 24

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26. Risk Premiums and Risk Aversion 26

27. The Sharpe Ratio 27

28. Asset allocation across risky and risk free assets Asset allocation means dividing your investment money among different asset classes — mainly: Stocks (risky, high return) Bonds (medium risk) Treasury bills / cash (low risk, low return) Instead of putting all money in one type of asset, investors mix them to balance risk and return . Why it matters History shows: Stocks have the highest average returns — but also the biggest ups and downs (risk). Bonds are safer than stocks, but not completely risk-free. Treasury bills (T-bills) are the safest — but their returns are the lowest. So, investors must choose how much risk they are willing to take by deciding how much to invest in each class. 28

29. Capital allocation When you invest, you decide: How much to put in risk-free assets (like T-bills) How much to put in a risky portfolio (like a mutual fund of stocks) That choice determines your overall portfolio risk and return . Complete portfolio The complete portfolio = your total investment = (risk-free asset part) + (risky asset portfolio part). Example: If you invest: 40% in T-bills 60% in a stock fund Your portfolio return and risk will depend on that 40/60 mix. 29

30. Idea of Portfolio Expected Return and Risk Investors can combine risky and risk-free assets to create a portfolio that fits their personal tolerance for risk. Risk-free asset (T-bill) has a guaranteed return. Risky portfolio (P) (like a stock fund) has a higher expected return but also some risk (measured by standard deviation). The complete portfolio (C) is a mix of these two. You decide how much to invest in the risky portfolio. That proportion is called y . y = 1 → all money in risky assets y = 0 → all money in risk-free asset 0 < y < 1 → a mix of both y > 1 → borrowing money to invest more in risky assets (leveraging) 30

31. Variable Meaning Value E(rₚ) Expected return of risky portfolio 15% σ ₚ Risk (standard deviation) of risky portfolio 22% r𝑓 Risk-free rate 7% Risk premium on risky asset E(rₚ) – r𝑓 = 15% – 7% = 8% 31

32. Two extreme cases All in risk-free asset (y = 0): Return = 7%, Risk = 0 All in risky portfolio (y = 1): Return = 15%, Risk = 22% Middle case (50/50 mix) If y = 0.5 , Expected return: E(r𝑐) = (0.5 × 15%) + (0.5 × 7%) = 11% Risk: σ𝑐 = 0.5 × 22% = 11% 👉 Both return and risk are reduced by half. 32

33. ✅ Expected return of complete portfolio: ✅ Standard deviation of complete portfolio: What this means If you increase y (invest more in risky assets), → risk and expected return both increase proportionally . If you decrease y , → both decrease . So, all combinations of (risk, return) fall on a straight line between the two extreme points: F (risk-free) and P (risky portfolio) This line is called the Capital Allocation Line (CAL) .   33

34. Sharpe Ratio — Slope of the Line The Sharpe ratio shows how much extra return (risk premium) you get per unit of risk: That means: 👉 For every 1% of risk, the investor gets 0.36% extra return above the risk-free rate.   34

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37. Step 1. Expected return of complete portfolio Step 2. Risk premium of complete portfolio Step 3. Standard deviation of complete portfolio Step 4. Ratio of risk premium to standard deviation   37

38. What the CAL shows You mix a risk-free asset (return ) with a risky portfolio (expected return , risk ). For any allocation (the fraction in ; the rest is in the risk-free asset), your complete portfolio has: Plotting as varies gives a straight line starting at the risk-free point and passing through . That line is the CAL .   38

39. The slope = Sharpe ratio The CAL’s slope (rise/run) is the reward-to-volatility (Sharpe) ratio . Every portfolio on the same CAL (any ) has the same Sharpe ratio :   39

40. Quick example (numbers from your page) , , . Sharpe of (and of any mix on this CAL): Sample mixes: : , , ratio . : , , ratio . (levered): , , ratio . Takeaway: By sliding along the CAL (changing ), you pick your preferred risk–return point . The efficiency (Sharpe) stays the same for that line; to improve it, you need a better risky portfolio (a higher-Sharpe ), which pivots the CAL upward.   40

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44. a) Expected end-of-year payoff = 0.5·50,000 + 0.5·150,000 = 100,000. Required return = risk-free 5% + required risk premium 10% = 15%. Price you’re willing to pay now: (≈ $86,957). b) If you buy at , the expected return is , which equals 5% + 10%. c) If you require a 15% risk premium, required return = 5% + 15% = 20%. (≈ $83,333). d) When the required risk premium (and thus required return) rises, the price you’re willing to pay falls. The relationship between required risk premium and the portfolio’s price is inverse.   44

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