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**FINC4101 Investment Analysis**

Instructor: Dr. Leng Ling Topic: Portfolio Theory I

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Learning objectives Compute different measures of investment performance: Holding-period return (HPR) Arithmetic average Geometric average Dollar-weighted return Compute the expected return, variance and standard deviation of a risky investment.

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**Concept Map FI4101 Theory Portfolio Pricing Asset Equity Fixed Income**

Efficiency Market Derivatives Exchange Foreign

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**Portfolio Theory I: Concept Map**

HPR $-weighted Geometric, Arithmetic, Expected return Variance

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**Investment return over 1 period: Holding period return (HPR)**

Rate of return over a given investment (holding) period. Has two components: Price change = ending price – beginning price Cash income HPR measures investment performance for a single asset or a portfolio of assets over a given period. Measures investment performance over a single period. To measure ‘average’ performance/return over many periods, use arithmetic, geometric, or dollar-weighted return. HPR can be expressed in percentages or in decimals.

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**Holding period return (HPR)**

Assumes that cash income is paid at the end of the holding period. If cash income is received earlier, reinvestment income is ignored. HPR can be used for different types of investments: stock, bond, mutual fund etc. For stock, cash income = dividend For bond, cash income = coupon If the cash income is received earlier, HPR ignores reinvestment income earned between receipt of income and the end of the holding period.

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**Holding period return (HPR)**

Stock

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Simple HPR example You are thinking of investing in ABC Inc’s stock. You intend to hold the stock for 1 year. ABC’s stock is currently selling at $50 and is expected to rise to $56 by the end of the year. The company is expected to pay a per share dividend of $0.60 during the year. Compute: HPR Capital gains yield Dividend yield. Sum up the capital gains yield and the dividend yield. Is that the same as the HPR? HPR = ( )/50 = (6.6)/50 = or 13.2% Capital gains yield = 6/50 = 0.12 Dividend yield = 0.6/50 = 0.012 Capital gains yield + Dividend yield = = = HPR. So, yes.

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**Investment return over many periods**

Three alternative ways of measuring average returns over multiple periods: Arithmetic average (arithmetic mean) Geometric average (geometric mean) Dollar-weighted return Use the following example to illustrate each return measure. Suppose you want the average return of an investment over longer periods of time. There are three alternative average return measures you can use.

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**Example: Table 5.1 Quarterly cash flows and HPRs of a mutual fund.**

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr Beg. Assets ($m) 1 1.2 2 0.8 HPR (%) 10 25 (20) Assets before flows ($m) 1.1 1.5 1.6 Net inflow ($m) 0.1 0.5 (0.8) 0.0 End. Assets ($m) Example is the mutual fund data in Table 5.1. What is the arithmetic average, geometric average and dollar-weighted return over the four quarters?

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**Arithmetic average = (r1 + r2 + r3 + ... rN) /N**

Suppose we hold an asset over N periods: 1, 2,…, and N. And we label the HPR in each period as: r1, r2, …, rN The arithmetic average is the sum of returns in each period divided by number of periods. Arithmetic average = (r1 + r2 + r rN) /N In the example, arithmetic average = ( )/4 = 10% It is standard in finance to use subscript to denote period and ‘r’ to denote HPR. Period can mean any length of time: year, quarter, month, day etc. In the example, there are 4 quarters, thus, 4 periods and n = 4. Arithmetic average ignores compounding and it does not represent an equivalent single quarterly rate for the year. The arithmetic average is useful because it is the best forecast of performance in future periods, using this particular sample of historic returns.

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**Geometric Average (time-weighted)**

Single per period return that gives the same cumulative performance as the sequence of actual returns. Geometric average = [(1+r1) x (1+r2) ... x (1+rN)]1/N - 1 In the example, geometric average = [(1.1) x (1.25) x (.8) x (1.25)] 1/4 - 1 = (1.375) 1/4 -1 = = 8.29% Important: When computing geometric average, you must always state the actual returns in decimals. Thus: 1+r1 = = 1.1, 1+r2 = = 1.25, 1+r3 = 1 + (-0.2) = 0.8, 1+r4 = = 1.25 The geometric average is also called the time-weighted average return because it ignores the period-by-period variation in dollars invested in the asset (funds under management in the example). When we want to account for variation in dollars invested, we use the dollar-weighted return, which we turn to next.

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**Dollar-weighted return (IRR)**

This is simply the internal rate of return (IRR) on an investment ! IRR: the interest rate that will make the PV of cash inflows equal to the PV of cash outflows. In other words, IRR is the discount rate such that the NPV is 0. This is simply a revision of FI3300. CIFt is the cash inflow for period t. COFt is the cash outflow for period t.

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**Dollar-weighted return (IRR)**

In the example, think in terms of capital budgeting. So, the mutual fund is a “project” from investor’s perspective. Initial Investment is an outflow Ending value is an inflow Additional investment is an outflow Reduced investment (withdraw money) is an inflow In this example, classify cash flows in the following way: Initial portfolio values ($1m) = initial investment, therefore COF Net cash inflows INTO the fund = additional investment, therefore also COF Net cash outflows FROM the fund = cash you receive from the fund (the ‘project’), therefore CIF. Ending portfolio value = final liquidation value, also CIF

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**Dollar-Weighted Return**

Time 1 2 3 4 Net cash flow ($m) -1 -0.1 -0.5 0.8 Using the definition of the IRR, Use financial calculator to solve for IRR. Using BA II Plus, C0 = -1, C1 = -0.1, F1=1, C2 = -0.5, F2=1, C3 = 0.8, F3=1, C4 = 1, F4 = 1, CPT, IRR = 4.17% The IRR is less than the time-weighted return of 8.29% because the portfolio returns were higher when less money was under management. To see this look at the beginning assets under management and then relate that to the HPR. There were less TNAs during Q2 and Q4 (which experience big returns) and more TNAs during Q3 which experience a big loss.

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**Quoting Rates of Return**

Annual percentage rate, APR = rate per period X n Where n = no. of compounding periods per year Effective annual interest rate, EAR We just seen several ways to compute average rates of return. There are also several ways to annualize periodic returns. Rate per period = interest rate per period, Periods per year = no. of compounding periods in a year. APR: APR annualizes per-period (periodic) rates using a simple interest approach, ignoring compound interest. APR is just the nominal stated interest rate introduced in FI3300. Returns on assets with regular cash flows, such as mortgages (with monthly payments) and bonds (with semiannual coupons), usually are quoted as annual percentage rates or APRs. EAR: Effective annual interest rate, also studied in FI3300. Accounts for compound interest when annualizing interest rates. In the EAR formula, APR must be in decimals. The formula for EAR assumes that you can earn the APR each period. Therefore, after one year (when n periods have passed), your cumulative return would be (1+APR/n)n . Note that one needs to know the holding period when given an APR in order to convert it to an EAR.

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**Quoting Rates of Return**

With continuous compounding, the relationship between EAR and APR becomes EAR = eAPR – 1 ‘e’ is the exponential function (that appears on your financial calculator as [ex]) Equivalently, APR = Ln(1 + EAR) Ln is the natural log function. Again, we covered this in FI3300. Remember that in the EAR formula, APR must be in decimals. Likewise, in the APR formula, EAR must be in decimals.

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HPR, APR, EAR problem Suppose you buy a bond of General Electric at a price of $990. The bond pays coupons semi-annually, has an annual coupon rate of 6%, a face value of $1,000 and will mature in six months’ time. You intend to hold the bond till it matures. What is the 6-month HPR? What is the APR of this investment? What is the EAR of this investment? This example serves two function: Allows student to compute HPR for Bonds Opportunity to use APR and EAR from a six-month rate. Hold this investment for 6-months, until maturity. Semiannual coupon = 0.06 x 1000 x 0.5 = 30 6-month HPR = (1000 – )/990 = 40/990 = or 4.04% for 6-months APR = 4.04 x 2 [because there are two 6-month periods in a year] = 8.08% EAR = ( )2 – 1 = = 8.24%

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Another example of HPR Suppose you bought a bond of General Electric at a price of $990 6 months ago. The bond pays coupons semi-annually, has an annual coupon rate of 6%, a face value of $1,000 and will mature in 12 months from today. Today it just paid the coupon and you intend to sell it immediately at current market price. The current YTM is 15%. What is the current market price? What will be your HPR? FV=1,000 PMT=1000x6%/2=30 N=2 I/Y=7.5 PV=?= HPR=( )/990=

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**Describing investment uncertainty: Scenario analysis**

Investment is risky simply because we don’t know what will happen in the future for certain. One way of quantifying risk is through scenario analysis. Scenario analysis: The process of devising a list of possible economic scenarios and specifying: - The likelihood (probability) of each scenario. The HPR that will be realized in each scenario. The list of possible HPRs with associated probabilities is called the probability distribution of HPRs. This is critical in helping us to evaluate risky investments. Scenario analysis builds the probability distribution of HPRs. The ideas/ concepts discussed here apply to a single risky asset or a portfolio of assets.

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**Probability distribution of HPR**

The probability distribution provides information for us to measure the reward and risk of an investment. Reward of the investment: Expected return Also known as ‘mean return’, ‘mean of the distribution of HPRs’. Risk of the investment: Variance Let’s start with a simple scenario analysis. The expected return can be thought of as the average HPR you would earn if you were to repeat an investment in the asset many times. Use the scenario analysis in Table 5.2 to illustrate the computation of expected return and variance.

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**Say you want to buy Google’s stock and hold it for a year.**

During this coming year, you think there are 3 possible economic scenarios: boom, normal growth, recession. State of the Economy Scenario Probability, p(s) HPR (%) Boom 1 0.25 44 Normal 2 0.50 14 Recession 3 -16 How to read the table. If you look at the first row, it says that one of the possible economic scenario is a boom. We label this as scenario 1. The likelihood of a boom occurring over the coming year is 0.25 or 25% or 1 chance in 4. If a boom occurs, the HPR for investing in Google’s stock will be 44%. Apply the same interpretation to the other three scenarios.

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Expected Return, E(r) The weighted average of returns in all possible scenarios, s = 1,2,…S, with weights equal to the probability of that particular scenario. p(s): probability of scenario s r(s): HPR in scenario s The equation simply uses symbols to express the idea contained in the definition. E.g., p(1) = probability of scenario 1, r(1) is the HPR in scenario 1. In general, p(s) means the probability of scenario s, and r(s) is the HPR in scenario s. The second line is just a compact way of stating the formula.

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**Expected Return, E(r) With the formula, Google’s expected return is:**

E(r) = (0.25 x 44) + (0.5 x 14) + (0.25 x -16) = 14% Probability in each scenario Numbers in blue are the probabilities. Numbers in red are the scenario HPRs. I used different colors and arrows to show how the general formula on the previous slide is applied to the simple example. HPR in each scenario

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Variance, Var(r) When we talk about risk, we often think of surprises or deviations from what we expect. Variance captures this idea. Variance: The expected value of squared deviation from the mean. Also known as σ2 (read as ‘sigma squared’). In the formula, the term [r(s) – E(r)] is the deviation from the mean, specifically, the deviation of scenario s’s HPR from the expected return. Variance is the sum of the probability-weighted squared deviations. We square the deviations because otherwise, negative deviations would offset positive deviations, with the result that the expected deviation from the mean return would necessarily be zero. Squared deviations are necessarily positive. Squaring exaggerates large (positive or negative) deviations and relatively deemphasizes small deviations.

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**Standard deviation, SD(r)**

Standard deviation: Square root of variance Returning to the Google example, Var(r) = 0.25(44 – 14) (14 – 14) ( )2 = 450 SD(r)= (450)1/2 = 21.21% Another result of squaring deviations is that the variance has a dimension of percent squared. To give the measure of risk the same dimension as expected return (%), we use the standard deviation, defined as the square root of the variance. A potential drawback to the use of variance and std dev as measures of risk is that they treat positive deviations and negative deviations from the expected return symmetrically. In practice investors welcome positive surprises, and a natural measure of risk would focus only on bad outcomes. However, if the distribution of returns is symmetric (meaning that the likelihood and magnitude of negative surprises are roughly equal to those of positive surprises), then standard deviation will approximate risk measures that concentrate solely on negative deviations. In the special case that the distribution of returns is approximately normal the standard deviation will be perfectly adequate to measure risk. The evidence shows that for fairly short holding periods, the returns of most diversified portfolios are well described by a normal distribution.

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**How to interpret E(r), Var(r) and SD(r)**

The bigger the expected return, the bigger the potential reward from the investment, vice versa. The bigger the variance, the bigger the risk of the investment, vice versa. The bigger the standard deviation, the bigger the risk of the investment, vice versa. In the previous slides, I only discuss the computation. Now I interpret the different measures. The same interpretation applies to the sample/historical moments. The interpretations will be very important when we compare different risky assets.

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**Describing investment performance in the past**

If we are interested in the rewards from investing in the past (using historical data), we can use (1) arithmetic average, (2) geometric average. To quantify risk, use ‘historical’ or ‘sample’ variance: Arithmetic average Expected returns, variance and standard deviation apply to possible outcomes in the future. If we are interested in the rewards from investing in the past, we can use the arithmetic average and the geometric average. So we simply applied these formulas directly. To quantify the risk from investing, we use the ‘historical’ or ‘sample’ variance. This is just the variance formula we saw earlier adapted to historical data. The modification involves: using the arithmetic average (sample average) return in place of the expected return, doing away with the scenario probabilities since we are looking at past data so these probabilities are irrelevant, dividing the sum of squared deviations from average return by (n – 1). The sample variance nevertheless captures the key idea of variance, i.e., incorporating deviations from the average return in the past. When we use the sample average return in place of the mean return, we must modify the average of the squared deviations for what statisticians call a “lost degree of freedom”. The modification is easy: multiply the average value of the squared deviations by n/(n-1). When you are using large samples and n is large, the modification is unimportant, since n/(n-1) is close to 1 and 1/(n-1) is close to 1/n. No. of periods

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**Example: S&P500 index, 1988-1992 Year HPR(%) 1988 16.9 1989 31.3 1990**

-3.2 1991 30.7 1992 7.7 Example 5.5 (Data from Table 5.3). To compute column 2, express the HPRs in column 1 in decimals and then add 1. So for 1988, 1 + HPR = 1 + (16.9/100) = 1.169 Arithmetic average = 83.4/5 = = 16.7 Geometric average = [(1.169)*(1.313)*(0.968)*(1.307)*(1.077)]1/5 – 1 = or 15.9% Compute the arithmetic average, geometric average, and variance.

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**Deviation from arithmetic average**

Example: S&P500 index, Year (1) HPR(%) (2) 1+HPR (3) Deviation from arithmetic average (4) Squared deviation 1988 16.9 1.169 16.9 – 16.7 = 0.2 0.04 1989 31.3 1.313 31.3 – 16.7 = 14.6 213.16 1990 -3.2 0.968 -3.2 – 16.7 = -19.9 396.01 1991 30.7 1.307 30.7 – 16.7 = 14 196 1992 7.7 1.077 7.7 – 16.7 = -9 81 Total 83.4 886.21 Example 5.5 (Data from Table 5.3). To compute column 2, express the HPRs in column 1 in decimals and then add 1. So for 1988, 1 + HPR = 1 + (16.9/100) = 1.169 Arithmetic average = 83.4/5 = = 16.7 Geometric average = [(1.169)*(1.313)*(0.968)*(1.307)*(1.077)]1/5 – 1 = or 15.9%

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**Example: S&P500 index, 1988-1992, cont’d**

Arithmetic average = 83.4/5 =16.7% Geometric average = [(1.169) x (1.313) x (0.968) x (1.307) x (1.077)]1/5 – 1 = or 15.9% Variance = /(5 – 1) = 221.6 Standard deviation = (221.6)1/2 =14.9%

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**Risk premiums & risk aversion**

If you don’t want to invest in a risky asset like a stock, what is the alternative? Risk-free assets like treasury bills. The return you get is the risk-free rate (rf). Risk-free rate = rate of return that can be earned with certainty. Risk premium: expected return in excess of the risk-free rate. Risk premium = E(r) – rf Why is the treasury bill considered a risk-free asset? Because we know the rate of return we will earn on T-bills at the beginning of the holding period. For risky assets, we can’t know the return we will earn until the end of the holding period.

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**Risk premium depends on risk aversion and variance**

Risk aversion: reluctance to accept risk. Risk premium of a portfolio, E(rp) – rf E(rp) – rf = A x var(rp) A = measures degree of investor’s risk aversion, Var(rp) = variance (risk) of the portfolio Risk premium increases if: Portfolio variance increases OR Risk-aversion, A, increases The term ½ is merely a scale factor and has no real bearing on the analysis. Investors are assumed to be risk-averse. The formula makes two very intuitive points: The higher the portfolio variance (the riskier the portfolio), the higher the risk premium demanded. The higher the risk-aversion (higher A), the higher the risk premium demanded. The equation requires that we put rates of return in decimal form. The formula applies to the investor’s overall portfolio, not to individual assets held in that portfolio.

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**Inferred Risk Aversion (price of risk)**

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Average A in market

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**Sharpe Ratio Measure the risk-return tradeoff**

is the standard deviation * It works for portfolio only, not individual security. * Sharpe ratio works for portfolio, not individual asset. It is because in a portfolio, idiosyncratic (individual) risk will be offset with each other and thus there should not be any return for those individual risk. Here, the standard deviation is from the systematic risk, not individual risk because it is not priced in the market.

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**Problem sets after Chapter 5, # 11.**

The expected cash flow is: (0.5 x $50,000) + (0.5 x $150,000) = $100,000 With a risk premium of 10%, the required rate of return is 15%. Therefore, if the value of the portfolio is X, then, in order to earn a 15% expected return: X(1.15) = $100,000 X = $86,957 If the portfolio is purchased at $86,957, and the expected payoff is $100,000, then the expected rate of return, E(r), is: = 0.15 = 15.0% The portfolio price is set to equate the expected return with the required rate of return. If the risk premium over T-bills is now 15%, then the required return is: 5% + 15% = 20% The value of the portfolio (X) must satisfy: X(1.20) = $100, 000 X = $83,333 For a given expected cash flow, portfolios that command greater risk premium must sell at lower prices. The extra discount from expected value is a penalty for risk.

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**Summary Single period: holding-period return (HPR)**

Many periods: arithmetic average, geometric average, and dollar-weighted return. Expected return measures the ‘reward’ from an investment. Variance (standard deviation) measures the ‘risk’ from an investment.

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**Practice 1 1.Chapter 5 problem sets : 5,6,7,8.**

2. Suppose you bought a bond of BT Co. at a price of $990 6 months ago. The bond pays coupons semi-annually, has an annual coupon rate of 6%, a face value of $1,000 and will mature in 24 months from today. Today it just paid the coupon. The current YTM is 15%. What is the current market price? What is your HPR for the last 6 months? If the interest rate on the market (YTM) does not change for the next 2 years, what will be your HPR for one year if you intend to sell the bond after 6 months receiving the 2nd coupon payment? (assume all coupons do not earn any investment returns)

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Homework 1 1. Suppose an investor’s risk aversion A=2 and the variance of the return of the portfolio he chose is The risk-free rate is 2%. The value of the portfolio has 0.3 probability to go to $2000 in one year, 0.5 probability to $1500 and 0.2 probability to $1000. What is the maximum price he would pay for this portfolio? 2. Suppose you bought a bond at a price of $ months ago. The bond pays coupons annually, has an annual coupon rate of 6%, a face value of $1,000 and will mature in 36 months. Today it just paid the coupon. The current YTM is 15%. Suppose the YTM will decrease to 10% after 12 months and then remains the same till expiration. What is the arithmetic average annual return for the next two years?

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