MFIN1151 Lecture Notes - Topic 2 - Risk and Return - Complete PDF

Topic 2. Risk and Return Chapter 5 of the textbook Complete Version © McGraw Hill, LLC 1  Topics Covered Part One: Average return over multiple time periods Part Two: Scenario analysis Part Th...

Topic 2. Risk and Return Chapter 5 of the textbook Complete Version © McGraw Hill, LLC 1  Topics Covered Part One: Average return over multiple time periods Part Two: Scenario analysis Part Three: Risk aversion and risk premium Part Four: Asset allocation problem with one risk-free asset and one risky asset Part Five: Capital allocation line Part Six: Capital market line Reading: textbook Chapter 5 © McGraw Hill, LLC 2  1. Average return over multiple time periods © McGraw Hill, LLC 3  Rate of return over single period Holding-period return (HPR): capital gain plus dividend (or coupon) divided by initial investment Ending price − Beginning price + dividend (or coupon) HPR = Beginning price The HPR is the sum of capital gain yield plus dividend yield (or coupon rate) Example: What is the HPR for a share of stock that was purchase for $25, sold for $27 and distributed $1.25 in dividends? $27.00–$25.00+$1.25 𝐻𝑃𝑅 = = 0.13 = 13.00% $25.00 © McGraw Hill, LLC 4  Rate of return over multiple periods Table 5.1 Annual Cash Flows & Rates of Return of a Mutual Fund Example: A fund starts with $1 million assets under management. Investors can buy additional shares or withdraw shares. It receives additional investment from investors and also redeems shares of investors shareholders at the end of each year. Hence, the net investment (cash inflow) can be positive or negative. What is the average return for four years? 1st 2nd 3rd 4th Assets at the beginning of year ($million) 1 1.2 2 0.8 HPR(%) 10% 25% -20% 20% Assets before net inflows 1.1 1.5 1.6 0.96 Net inflow 0.1 0.5 -0.8 0.6 Assets at the end of year ($million) 1.2 2 0.8 1.56 © McGraw Hill, LLC 5  Rate of return over multiple periods © McGraw Hill, LLC 6  Rate of return over multiple periods Calculating the average annual return for multiple years: 1. Arithmetic average approach (𝑟𝐴 ) 2. Geometric average approach(𝑟𝐺 ) 3. Dollar-weighted return approach (IRR) © McGraw Hill, LLC 7  Rate of return over multiple periods 1. Arithmetic average (𝑟𝐴 ): the sum of return in each period divided by the number of periods 𝐻𝑃𝑅1 + 𝐻𝑃𝑅2 + 𝐻𝑃𝑅3 + 𝐻𝑃𝑅4 10% + 25% − 20% + 20% 𝑟𝐴 = = = 8.75% 4 4 - Ignore compounding - Arithmetic average return is related to APR (annualized percentage rate) 1st 2nd 3rd 4th Assets at the beginning of year ($million) 1 1.2 2 0.8 HPR(%) 10% 25% -20% 20% Assets before net inflows 1.1 1.5 1.6 0.96 Net inflow 0.1 0.5 -0.8 0.6 Assets at the end of year ($million) 1.2 2 0.8 1.56 © McGraw Hill, LLC 8  Rate of return over multiple periods 2. Geometric average (𝑟𝐺 ): the single per-period return that gives the same cumulative (compounding) performance as the sequence of actual returns (1+𝑟𝐺 )4=(1+10%)(1+25%)(1-20%)(1+20%)=1.32 4 → 𝑟𝐺 = 1.32 - 1 = 1.321/4 - 1 = 1.320.25 - 1 = 1.0719 – 1 = 7.19% Alternatively, 𝑟𝐺 = [(1 + 𝐻𝑃𝑅1 ) × (1 + 𝐻𝑃𝑅2 ) ×...× (1 + 𝐻𝑃𝑅𝑛 )]1/𝑛 − 1 = 1.10 × 1.25 ×.80 × 1.20 1/4 − 1 = 0.0719 = 7.19% - Take compounding into account. - Geometric average return is related to EAR (effective annual rate). © McGraw Hill, LLC 9  Rate of return over multiple periods 3. Dollar-weighted return: the internal rate of return (IRR) on an investment, which makes the net present value (NPV) equal to zero 𝐶𝐹1 𝐶𝐹2 NPV = 𝐶𝐹0 + + +... (1 + 𝑟)1 (1 + 𝑟)2 Invest (purchase shares) is negative cash flows (for investors). Withdraw (sell shares) is positive cash flows (for investors). −0.1 −0.5 0.8 −0.6 1.56 0 = −1.0 + + + + + 1 + IRR (1 + IRR)2 (1 + IRR)3 (1 + IRR)4 (1 + IRR)4 −0.1 −0.5 0.8 −0.6 + 1.56 = −1.0 + + + + 1 + IRR (1 + IRR)2 (1 + IRR)3 (1 + IRR)4 −0.1 −0.5 0.8 0.96 = −1.0 + + + + 1 + IRR (1 + IRR)2 (1 + IRR)3 (1 + IRR)4 (Use a financial calculator to solve it) CF0 = -1, CF1 = -0.1, CF2 = -0.5, CF3 = 0.8, CF4 = 0.96 → IRR = 3.38% Question: why is the dollar-weighted return very different from than geometric average and arithmetic average? © McGraw Hill, LLC 10  Rate of return over multiple periods Question: why is the dollar-weighted return very different from geometric average and arithmetic average? Arithmetic average and geometric average have similar returns because they only consider HPR. Dollar-weighted return has a different return than the above two returns because it considers additional investments and withdraw (i.e., cash inflows and outflows). © McGraw Hill, LLC 11  Annualizing rate of return Annualizing rate of return is expressing periodical rate of return (e.g., weekly, monthly, quarterly) as an annual rate of return. Question: What is the difference between calculating the average annual return for multiple years and annualizing rate of return? Average annual return: calculated based on the yearly returns over multiple years, e.g., a yearly return of 5% for year 1, a yearly return of 5% for year 2, and a yearly return of 5% for year 3 Annualizing: calculated based on a periodic return, e.g., a monthly return of 1%, Three main approaches of average annual return for multiple years: 1. Arithmetic average approach (𝑟𝐴 ) 2. Geometric average approach(𝑟𝐺 ) 3. Dollar-weighted return approach (IRR) Two main approaches of annualizing rate of return: 1. APR (Annual Percentage Rate) 2. EAR (Effective Annual Rate) © McGraw Hill, LLC 12  Annualizing rate of return APR (Annual Percentage Rate) uses a simple interest, ignoring compound interest. APR does not equal the rate at which your invested funds actually grow. Returns on assets with regular intra-year cash flows, such as mortgages (with monthly payments) and bonds (with semiannual coupons), usually are quoted as APRs APR (Annual Percentage Rate) = Periodical rate × Periods per year Example: Given a monthly rate of 1%, What is the APR? 𝐴𝑃𝑅 = 0.01 × 12 = 0.12 or 12.00%. © McGraw Hill, LLC 13  Annualizing rate of return EAR (Effective Annual Rate) uses a compound interest and equals the rate at which your invested funds actually grow. Effective Annual Rate (EAR) = (1 + 𝑃𝑒𝑟𝑖𝑜𝑑𝑖𝑐 𝑅𝑎𝑡𝑒)𝑛 − 1 For example, n = 12 for monthly payment mortgages and n = 2 for bonds making payments semiannually Example. Given a monthly rate of 1%, what is the EAR? 𝐸𝐴𝑅 = (1 + 0.01)12 − 1 = 0.1268 or 12.68% © McGraw Hill, LLC 14  Annualizing rate of return APR and EAR can be converted into each other. When there are n compounding periods in the year, we first recover the periodical rate as APR/n and then compound that rate for the number of periods in a year. For n periods per year of compounding: 𝐴𝑃𝑅 𝑛 𝐸𝐴𝑅 = (1 + ) −1 𝑛 1 𝐴𝑃𝑅 = [(𝐸𝐴𝑅 + 1)𝑛 − 1] × 𝑛 For monthly interest rate (MR), n = 12 𝐴𝑃𝑅 = 𝑀𝑅 × 12 𝐸𝐴𝑅 = (1 + 𝑀𝑅)12 − 1 © McGraw Hill, LLC 15  Annualizing rate of return Example. Given a 12% APR, what is the Effective Annual Rate, given monthly compounding? With a 12% APR, the monthly rate is 1% (= 12% / 12) 𝐸𝐴𝑅 = (1 + 𝑀𝑅)12 − 1 = (1 + 0.01)12 − 1 = 1.1268 − 1 = 0.1268 We can also calculate this using the financial calculator’s time value keys. Let us assume the PV is 1 for simplicity. There are 12 months, and the monthly interest rate is 1%. [N] [I/Y] [PV] [PMT] [CPT] [FV] The calculator should display FV = -1.1268 This means $1 PV is compounded into $1.1268 FV in one year. So, the EAR is ($1.1268 - $1) / $1 = 0.1268 = 12.68%. © McGraw Hill, LLC 16  Nominal Interest and Real Interest Nominal interest rate (denoted as Rnom) is the interest rate in terms of nominal (not adjusted for purchasing power) dollars (quoted by banks) Real interest rate (denoted as Rreal) is the rate at which the goods you can buy with your funds grows, i.e. the growth rate of purchasing power derived from an investment. Inflation rate (denoted as i) is the rate at which prices are rising, measured as the rate of increase of the CPI (Consumer Price Index). Rreal ≈ Rnom − i 1 + Rreal = (1 + Rnom) / (1+ i) Example: What is the real return on an investment that earns a nominal 10% return during a period of 5% inflation? 1 + 0.10 1 + Rreal = = 1.048 1 + 0.05 Rreal = 0.048 or 4.8% © McGraw Hill, LLC 17  2. Scenario Analysis © McGraw Hill, LLC 18  Scenario Analysis How to evaluate an uncertain return and quantify risk?—Scenario analysis It shows a list of possible economic scenarios and specifies the probability of each scenario and its holding period return (HPR). Expected return (mean return): the weighted average of returns in all possible scenarios, with weights equal to the probability of that scenario 𝐸(𝑟) = σ𝑠 𝑝(𝑠)𝑟(𝑠) 𝑝(𝑠): Probability of a scenario; 𝑟(𝑠): Return if a scenario occurs; There are 1 to s scenarios © McGraw Hill, LLC 19  Scenario Analysis Variance of return: the expected value of the squared deviation from the mean 𝑉𝑎𝑟(𝑟) = 𝜎 2 = σ𝑠 𝑝(𝑠)[𝑟(𝑠) − 𝐸(𝑟)]2 Standard deviation of return: the square root of the variance 𝑆𝑑 𝑟 = 𝜎 = 𝑉𝑎𝑟(𝑟) Question: Why do variance and standard deviation measure risk? They reflect the volatility of returns. A large variance or standard deviation indicates that the returns have a significant deviation from the mean, making them unstable and risky. Conversely, a small variance or standard deviation suggests that the returns have minimal deviation from the mean, indicating stability. See the concrete example on the next page. © McGraw Hill, LLC 20  Scenario Analysis Example: A financial analyst summarized the next year’s HPR of a stock portfolio using scenario analysis. What are the expected HPR and the standard deviation of HPR? © McGraw Hill, LLC 21  Scenario Analysis To find the expected return, we compute the weighted average of returns in all possible scenarios, with weights equal to the probability of that scenario 𝐸 𝑟 = 0.05 × −37 + 0.25 × −11 + 0.4 × 14 + 0.3 × 30 = 10 Expected rate of return on the stock is 10% To find the variance, we take the difference between the holding period return in each scenario and the mean return, then we square that difference, and finally we multiply by the probability of each scenario. The sum of the probability-weighted squared deviations is the variance, and its square root is standard deviation. 𝜎 2 = 0.05 × −37 − 10 2 + 0.25 × −11 − 10 2 + 0.4 × 14 − 10 2 + 0.3 × 30 − 10 2 = 347.1 𝜎= 𝜎 2 = 347.1 = 18.63 Standard deviation of return on the stock is 18.63% © McGraw Hill, LLC 22  3. Risk Aversion and Risk Premium © McGraw Hill, LLC 23  Risk Aversion Risk Aversion: how much you dislike risk Example: Which one of the two choices you prefer? A fair gamble with the following possible payoffs: 50% chance to get $200 and 50% change to get $0 You get $100 for sure Choice 1 has an expected return of $100 and has risk. Choice 2 has an expected return of $100 and has no risk. © McGraw Hill, LLC 24  Risk Aversion Risk Neutral: They only care about expected return (payoff), do not care about risk. For two investments with same expected return, they have no preference (Demand no risk premium as they invest in risky security.) Risk Averse: For two investments with same expected return (payoff), they prefer the one with smaller risk. (Demand a positive risk premium as they invest in risky security.) Risk loving: They would like to pay a price for taking risk. (Willing to accept a negative risk premium as they invest in risky security.) Assumption: Investors are generally risk averse People generally prefer more return to less, and don’t like risk People demand higher expected return for the investment with higher risk. © McGraw Hill, LLC 25  Risk Aversion Table 5.3: Historical Return and Risk. Stocks have the highest risk and highest expected return The standard deviation of the return on stocks over this period, 20.05% was nearly double that of T-bonds, 11.59%, and more than six times that of T-Bills. © McGraw Hill, LLC 26  Risk Premium Risk Premium or excess return is an expected return of portfolio in excess of that on risk-free securities Risk premium or excess return = 𝐸(𝑟𝑝 ) − 𝑟𝑓 = 𝐴𝜎𝑝2 The size of the risk premium on a risky asset depends on the degree of risk aversion of the investor (A) and the riskiness of the portfolio return measured by variance (𝜎𝑝2 ). © McGraw Hill, LLC 27  Sharpe Ratio Sharpe (Reward-to-Volatility) Ratio: ratio of portfolio risk premium to standard deviation Portfolio Risk Premium 𝐸 𝑟𝑝 − 𝑟𝑓 𝑆𝑃 = = Standard Deviation of Excess Returns 𝜎𝑃 𝐸 𝑟𝑝 = Expected Return of the portfolio 𝑟𝑓 = Risk Free rate of return 𝜎𝑃 = Standard Deviation of portfolio excess return (in excess of risk-free rate) © McGraw Hill, LLC 28  4. Asset allocation across a risky asset and a risk-free asset © McGraw Hill, LLC 29  Asset allocation How to split investment between safe and risky assets to form a complete portfolio (C)? The most basic asset allocation problem: How to build a portfolio of a risk-free asset (such as 3-month treasury bills) and a risky asset (such as a single stock or a stock portfolio targeting the S&P500 index)? © McGraw Hill, LLC 30  Asset allocation Risk free asset: T-bills has a return of 𝑟𝑓 and standard deviation of 0 Risky asset: A stock or a stock portfolio’s expected return and standard deviation are 𝐸(𝑟𝑝 ), 𝜎𝑝 Complete portfolio’s expected return and standard deviation are 𝐸(𝑟𝑐 ), 𝜎𝑐 The weight on risky asset, y, which could be less than, equal to, or greater than one. 𝐸(𝑟𝑐 ) = (1 − 𝑦)𝑟𝑓 + 𝑦𝐸(𝑟𝑝 ) = 𝑟𝑓 + 𝑦[𝐸(𝑟𝑝 ) − 𝑟𝑓 ] → 𝑬 𝒓𝒄 − 𝒓𝒇 = 𝒚[𝑬(𝒓𝒑 ) − 𝒓𝒇 ] 𝝈𝒄 = 𝒚𝝈𝒑 © McGraw Hill, LLC 31  Asset allocation The excess return (risk premium) of the complete portfolio is y times that of risky asset. The standard deviation (risk) of the complete portfolio is y times that of risky asst. The excess return of the complete portfolio is the weighted average of the excess return of the risky asset and risk-free asset. So, the excess return (risk premium) of the complete portfolio is y times that of risky asset. 𝐸 𝑟𝑐 − 𝑟𝑓 = 1 − 𝑦 𝑟𝑓 − 𝑟𝑓 + 𝑦 𝐸 𝑟𝑝 − 𝑟𝑓 = 0 + 𝑦 𝐸 𝑟𝑝 − 𝑟𝑓 = 𝑦[𝐸(𝑟𝑝 ) − 𝑟𝑓 ] Similarly, the standard deviation of the complete portfolio is the weighted average of the standard deviation of risky asset and risk-free asset. The standard deviation of risk-free asset is zero, so the standard deviation (risk) of the complete portfolio is y times that of risky asst. © McGraw Hill, LLC 32  Asset allocation Example: Suppose 𝑟𝑓 = 7%, 𝐸(𝑟𝑝 ) = 15% and 𝜎𝑝 = 22%, and assume that investors can borrow at the risk-free rate. Both the risk premium and the standard deviation of the complete portfolio increase in proportion to the investment in the risky asset. E.g., when weight on risky asset increases from 0.5 to 1 (doubled), risk premium increases from 4% to 8% (doubled), and standard deviation increases from 11% to 22% (doubled). 1-𝑦 𝑦 𝐸 𝑟𝑐 𝐸 𝑟𝑐 − 𝑟𝑓 𝜎𝐶 Sharpe ratio Weight on Weight on Expected return Risk premium Standard deviation risk free asset risky asset 1 0 1*7%+0*15% =7% 7% - 7% = 0% 0*22% = 0% 0.5 0.5 0.5*7%+0.5*15% =11% 11% - 7% = 4% 0.5*22% =11% 4%/11% = 0.36 0 1 0*7%+1*15% =15% 15% - 7% = 8% 1*22% = 22% 8%/22% = 0.36 -0.25 1.25 -0.25*7%+1.25*15%=17% 17% - 7% = 10% 1.25*22% = 27.5% 10%/27.5% = 0.36 (borrow) © McGraw Hill, LLC 33  Sharpe Ratio 𝐸(𝑟𝑝 )−𝑟𝑓 Reward-to-variability ratio (or Sharpe ratio) = 𝜎𝑝 Sharpe ratio measures the additional risk premium (or excess return) a risky asset offers for each unit of additional standard deviation (total risk), defined as the ratio of risk premium to standard deviation 4% 8% 10% The Sharpe ratio stays constant: = 0.36, 22% = 0.36, and 27.5% = 0.36 11% © McGraw Hill, LLC 34  5. Capital Allocation Line © McGraw Hill, LLC 35  Capital allocation line (CAL) Capital allocation line (CAL) is the line depicts the risk-return combinations available by choosing different values of y, that is, by varying capital allocation. The points that describe the risk and return of the complete portfolio for various capital allocations, y, all plot on the straight-line connecting F (100% in risk free asset) and P (100% in risky asset), with an intercept of 𝑟𝑓 and slope equal to the Sharpe ratio © McGraw Hill, LLC 36  Capital allocation line (CAL) 𝐸(𝑟𝑝 )−𝑟𝑓 Slope of the CAL: = (15% - 7%) / 22% = 0.36 𝜎𝑝 You can construct complete portfolios to the right of point P by borrowing, that is, by choosing y > 1. This means that you borrow a proportion of 1- y and invest both the borrowed funds and your own wealth in the risky portfolio P. © McGraw Hill, LLC 37  Capital allocation line (CAL) If you can borrow at the risk-free rate, 𝑟𝑓 = 7%, the Sharpe ratio stays constant on the CAL, i.e. the slope of CAL does not change. The rate of return of this complete portfolio will be 𝐸(𝑟𝑐 ) = (1 − 𝑦)𝑟𝑓 + 𝑦𝐸(𝑟𝑝 ) = 𝑟𝑓 + 𝑦[𝐸(𝑟𝑝 ) − 𝑟𝑓 ] → 𝐸 𝑟𝑐 − 𝑟𝑓 = 𝑦[𝐸(𝑟𝑝 ) − 𝑟𝑓 ] The risk premium of complete portfolio 𝐸 𝑟𝑐 − 𝑟𝑓 is equal to 𝑦[𝐸(𝑟𝑝 ) − 𝑟𝑓 ] The standard deviation of complete portfolio 𝜎𝑐 is equal to y𝜎𝑝. 𝐸 𝑟𝑐 − 𝑟𝑓 𝑦[𝐸(𝑟𝑝 ) − 𝑟𝑓 ] 𝐸(𝑟𝑝 ) − 𝑟𝑓 𝑆ℎ𝑎𝑟𝑝𝑒 = = = 𝜎𝑐 𝑦𝜎𝑝 𝜎𝑝 The Sharpe ratios of complete portfolio and risky portfolio are the same when you can borrow at the risk-free rate 𝑟𝑓. © McGraw Hill, LLC 38  Capital allocation line (CAL) Example: Suppose 𝑟𝑓 = 7%, 𝐸(𝑟𝑝 ) = 15% and 𝜎𝑝 = 22% (i.e. the same condition as the previous exam), and investors can borrow at the risk-free rate of 7%. Suppose the investment budget is $300,000, and an investor borrows an additional $120,000, investing the $420,000 in the risky asset. This is a levered position in the risky asset, which is financed in part by borrowing. What is Sharpe ratio of the complete portfolio? 420,000 𝑦 = 300,000 = 1.4, and 1 − 𝑦 = 1 − 1.4 = −0.4 With weights of –0.4 in the risk-free asset and 1.4 in the risky portfolio, the complete portfolio’s rate of return and standard deviation are 𝐸 𝑟𝑐 = 1 − 𝑦 𝑟𝑓 + 𝑦𝐸 𝑟𝑝 = −0.4 × 7% + 1.4 × 15% = 18.2% 𝜎𝑐 = 𝑦𝜎𝑝 = 1.4 × 22% = 30.8% 𝐸 𝑟𝑐 − 𝑟𝑓 18.2% − 7% 11.2% 𝑆ℎ𝑎𝑟𝑝𝑒 = = = = 0.36 𝜎𝑐 30.8% 30.8% © McGraw Hill, LLC 39  Capital allocation line (CAL) The levered complete portfolio still exhibits the same reward-to-volatility or Sharpe ratio. The levered complete portfolio has both a higher expected return and a higher standard deviation than an unlevered position. Compare with (see the Table earlier) 100% investment in risky portfolio, whose 𝐸 𝑟𝑐 = 15%, 𝜎𝑐 = 22%, and Sharpe = 0.36 50% investment in risky portfolio, whose 𝐸 𝑟𝑐 = 11%, 𝜎𝑐 = 11%, and Sharpe = 0.36 © McGraw Hill, LLC 40  Capital allocation line (CAL) There is no absolute optimal asset allocation point along the straight CAL, because they all have the same Sharpe ratio. Greater level of risk aversion leads to larger proportions of the risk-free asset Willingness to accept high levels of risk for high level of returns would result in greater proportion of risky asset © McGraw Hill, LLC 41  Capital allocation line (CAL) If an investor borrows at a rate higher than the risk-free rate, which is more realistic, (say 𝑟𝑏 =9%), how would that change the CAL? For points to the right of P, we use y = 1.4 as the example. With weights of –0.4 in the risk-free asset and 1.4 in the risky portfolio, the complete portfolio’s rate of return and standard deviation are 𝐸 𝑟𝑐 = 1 − 𝑦 𝑟𝑏 + 𝑦𝐸 𝑟𝑝 = −0.4 × 9% + 1.4 × 15% = 17.4% 𝜎𝑐 = 𝑦𝜎𝑝 = 1.4 × 22% = 30.8% 𝐸 𝑟𝑐 − 𝑟𝑓 17.4% − 7% 10.7% 𝑆ℎ𝑎𝑟𝑝𝑒 = = = = 0.35 𝜎𝑐 30.8% 30.8% © McGraw Hill, LLC 42  Capital allocation line (CAL) © McGraw Hill, LLC 43  Capital allocation line (CAL) When the investor borrows at an interest rate equal to the risk-free rate, the Sharpe ratio (slope) is constant, and the CAL is a straight line. When an investor borrows at a rate higher than the risk-free rate, the borrowing cost increases, leading to a lower expected return on the complete portfolio. Consequently, the Sharpe ratio (slope) decreases, and the CAL becomes kinked, with a flatter slope on the leveraged portion. Normally, the investor cannot borrow at an interest rate lower than the risk-free rate © McGraw Hill, LLC 44  6. Capital Market Line © McGraw Hill, LLC 45  Capital market line (CML) A passive investment strategy is often entails having a market index portfolio. The rate of return on the portfolio then replicates the return on the index. Capital market line (CML): A special capital allocation line (CAL) that goes through the risk-free asset and a market portfolio M. There are many possible risky portfolio P, and M is one of them. CML is one of the many possible CAL. CML represents the investment opportunity of the risky portfolio for a passive investment strategy (indexing). © McGraw Hill, LLC 46  Capital market line (CML) Example: Assume that you manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 27%. The T-bill rate is 7%. Your client chooses to invest $7,000 in your fund and $3,000 in a T-bill money market fund. What is the reward-to-variability ratio (Sharpe ratio) of your client’s overall portfolio? 7,000 𝑦 = 7,000+3,000 = 0.7 𝐸 𝑟𝑐 = 1 − 𝑦 𝑟𝑓 + 𝑦𝐸 𝑟𝑝 = 0.3 × 7% + 0.7 × 17% = 14% 𝜎𝑐 = 𝑦𝜎𝑝 = 0.7 × 27% = 18.9% 𝐸 𝑟𝑐 −𝑟𝑓 14%−7% 7% 𝑆ℎ𝑎𝑟𝑝𝑒 = = = 18.9% = 0.37 𝜎𝑐 18.9% © McGraw Hill, LLC 47  Capital market line (CML) Example: Suppose this client decides to invest a proportion (y) of his total investment budget ($10,000) in your risky portfolio, so that his overall portfolio will have an expected rate of return of 15%. What is the proportion y? What is the standard deviation of the rate of return on your client’s portfolio? 𝐸 𝑟𝑐 = 1 − 𝑦 𝑟𝑓 + 𝑦𝐸 𝑟𝑝 = 1 − 𝑦 × 7% + 𝑦 × 17% = 15% 7% + 𝑦 × 17% − 7% = 15% → 𝑦 = 0.8 𝜎𝑐 = 𝑦𝜎𝑝 = 0.8 × 27% = 21.6% © McGraw Hill, LLC 48