Investment Analysis and Portfolio Management Instructor Attila Odabasi
Investment Analysis and Portfolio Management Instructor: Attila Odabasi Introduction to Risk, Return, and the Historical Record on T-bills and US equity 1
Learning Objectives • Time Value of Money – Review • How to calculate the return on an investment using different methods. • The historical returns on various important types of investments. • The historical risks of various important types of investments. • The relationship between risk and return. 1 -2
Rates of Return on Zero Bonds • $FV: Par value • $P(T): Price of the zero-bond with maturity T years • P(T) = FV / (1+r(T)) • Gross total return is: (1+r(T)) = FV / P(T) • Net total Return is: r(T)) = [FV / P(T)] - 1 5 -3
Annualized Rates of Return Horizon, T Price, P(T) [100/P(T)]-1 Net Return for Given Horizon 0. 5 year $97. 36 100/97. 36 – 1 = 0. 0271 r(0. 5) = 2. 71% 1 year $95. 52 100/95. 52 – 1 = 0. 0469 r(1) = 4. 69% 10 years $60 100/60 – 1 = 1. 67 r(10) = 167% Bonds come with different maturities. How to compare their returns? We typically Express each total return as a rate of return for a common period. The common period is usually a year. 4
How to compare returns? • We convert each total return to an effective annual rate (EAR) as follows: • (EAR) Compound annual return = 5 -5
Annual Effective Rates of Return Horizon, T Price, P(T) [100/P(T)]-1 Net Return for Given Horizon 0. 5 year $97. 36 100/97. 36 – 1 = 0. 0271 r(0. 5) = 2. 71% 1 year $95. 52 100/95. 52 – 1 = 0. 0469 r(1) = 4. 69% 10 years $60 100/60 – 1 = 0. 67 r(10) = 67% Let us compute EAR’s: 6
Annual Percentage Rates • However, annualized rates on interest rates are reported (quoted) using simple interest. These are called annual percentage rates (APR). • For example, the APR of the 6 -month bond, with a 6 -month rate of 2. 71% is 2 x 2. 71 = 5. 42%. • Here the annualization period is m = 1/T = 2 7
Quoting Rates of Return • Therefore, the relationship among the EAR and the APR is: • • Note that APR for a given EAR: 5 -8
Compounding occurs m times per year The difference between EAR and APR grows with compounding: Continuous compounding: 5 -9
Continuous Compounding Example: The effective annual rate associated with continuously compounded (m=∞) APR= 10% is determined by: EAR = e 0. 1 – 1 = 5 -10
T-Bills and Inflation Approximation: nominal risk-free rate = real rate + inflation rate rreal rnom - iexp Example rnom = 9%, iexp = 6% rreal 3% Exact: Fisher effect rreal = [(1 + rnom) / (1 + i)] – 1 or rreal = (rnom - i) / (1 + i) rreal = (9% - 6%) / (1. 06) = 2. 83% The approximate rule overstates the real rate by the factor 1 + i. 11
Nominal and Real interest rates and Inflation Correlation between nominal rate and inflation is low. Fisher hypo is not very thight. 12
Measuring Equity Returns One period investment: • Holding Period Percentage Return (HPR): • HPR= r 1 = P 1 – P 0 + D 1 P 0 • P 0 = Beginning price (or, PV) • P 1 = Ending price (or, FV) • D 1 = Dividend (cash flow) during period • Q: Why use % returns at all? • Q: What are we assuming about the cash flows in the HPR calculation? 13
Measuring Equity Returns • HPR defined as: • r 1 = P 1 – P 0 + D 1 = (P 1+D 1) - 1 P 0 • Then we can define ‘gross return’: (1 + r 1) = (P 1+D 1) / P 0 14
Multi-period vs One-period Returns 5 -15
Ex: Multi-period vs One-period Returns • Suppose the price of Msoft stock in month t-2 is $80, $85 in t-1, and $90 at t. No dividends paid between t-2 and t. 5 -16
Average Performance of Multi-period Returns 17
Arithmetic vs Geometric Average Returns • Arithmetic and geometric average returns will give different values for the returns over the same evaluation period. • Arithmetic average return: the amount invested is assumed to be maintained at the initial market value. • The geometric average return: it is a return on an investment that varies in size because of the assumption that all proceeds are reinvested. 18
Continuously Compounded (cc) Returns • Let rt = simple monthly (or daily) return on an investment • Continuously compounded monthly return (rcc) that corresponds to the simple return: 5 -19
Example: Compute cc return • Let Pt-1 = 85, Pt = 90 then simple return is 0. 0588. r t= • The cc monthly return can be computed as: • rcc = ln(1. 0588) = 0. 0571 • rcc = ln(90) – ln(85) = 4. 4998 – 4. 4427 = 0. 0571 • Notice that cc return is slightly smaller than simple return as expected. 5 -20
Multi-period cc Returns This is the main difference between simple and cc returns. 5 -21
Ex: Multi and one-period cc returns • Suppose Pt-2= 80, Pt-1= 85, Pt= 90. The cc twomonth return can be computed in two equivalent ways: 1) Take difference in log prices: 2) Sum the two cc one-month returns: 5 -22
Ex-Ante Return Estimation • Some asset classes are called risky. They offer a risk premium over risk-free assets. • These assets involve some degree of uncertainty about future holding-period returns. • We have to estimate these future holding period returns and related uncertainty. • How? 23
Expected Return & Std; Scenario Analysis Purchase Price 100 T-bill Rate 0. 04 State of the market Prob Cash Y-E Price Dividend HPR Deviations from the mean Squared Deviations Excess from Mean Returns Sqrd Deviations from Mean Excellent 0. 25 126. 50 4. 50 0. 3100 0. 2124 0. 0451 0. 2700 0. 0729 Good 0. 45 110. 00 4. 00 0. 1400 0. 0424 0. 0018 0. 1000 0. 0100 Poor 0. 25 89. 75 3. 50 -0. 0675 -0. 1651 0. 0273 -0. 1075 0. 0116 Crash 0. 05 46. 00 2. 00 -0. 5200 -0. 6176 0. 3815 -0. 5600 0. 3136 Expected Value 0. 0976 <-- {=SUMPRODUCT(B 5: B 8, E 5: E 8)} Variance of HPR 0. 0380 <-- {=SUMPRODUCT(B 5: B 8, G 5: G 8)} Std of HPR 0. 1949 <-- =SQRT(E 11) Risk Premium 0. 0576 <-- {=SUMPRODUCT(B 5: B 8, H 5: H 8)} Std of Excess Returns 0. 2032 <-- {=SQRT(SUMPRODUCT(B 5: B 8, I 5: I 8))} 5 -24
Time Series Analysis of Past Returns Period Implicitly Assumed Prob = 1/5 2000 HPR (decimal) Squared Deviation Gross HPR= 1 + HPR Wealth Index 100. 000 2001 0. 2 88. 110 -0. 1189 0. 0196 0. 8811 88. 110 2002 0. 2 68. 638 -0. 2210 0. 0586 0. 7790 68. 638 2003 0. 2 88. 330 0. 2869 0. 0707 1. 2869 88. 330 2004 0. 2 97. 940 0. 1088 0. 0077 1. 1088 97. 940 2005 0. 2 102. 749 0. 0491 0. 0008 1. 0491 102. 749 Arithmetic Average 0. 0210 <-- =AVERAGE(D 4: D 8) Expected Value 0. 0210 <-- =SUMPRODUCT(B 4: B 8, D 4: D 8) Standard Deviation 0. 1774 <-- =SUMPRODUCT(B 4: B 8, E 4: E 8)^0. 5 Standard Deviation 0. 1774 <-- =STDEV. P(D 4: D 8) Geometric Average Return 0. 0054 <-- =GEOMEAN(F 4: F 8)-1 Geometric Average Return 0. 0054 <-- =((G 8/G 3)^0. 2)-1 5 -25
Using Ex-Post Returns to estimate Expected HPR Estimating Expected HPR (E[r]) from ex-post data. Use the arithmetic average of past returns as a forecast of expected future returns and, Perhaps apply some (usually ad-hoc) adjustment to past returns Problems? • Which historical time period? • Have to adjust for current economic situation? 26
The Normal Distribution • We assume that (past) returns are distributed normally! • Normal distribution: A symmetric, bell-shaped frequency distribution that can be described with only an average and a standard deviation. • Mean: Average return • Variance is a common measure of return dispersion. • Standard deviation is the square root of the variance. • Standard Deviation is handy because it is in the same "units" as the average. • Standard Deviation is a good measure of risk when returns are symmetric around the mean. • If security returns are symmetric, portfolio returns will be, too. 1 -27
Figure 5. 4 The Normal Distribution 5 -28
Given Historical Returns • Expected Return: 1 -29
Return Variability: The Statistical Tools • The formula for return variance is ("n" is the number of returns): • Sometimes, it is useful to use the standard deviation, which is related to variance like this: 1 -30
Skew and Kurtosis Skew. Kurtosis. 5 -31
Skewed Distribution: Large Negative Returns (Left Skewed) Implication? r = average is an incomplete risk measure Median Negative r Positive 32
Skewed Distribution: Large Positive Returns (Right Skewed) r = average Median Negative r Positive 33
Implication? is an incomplete risk measure Leptokurtosis 34
Value at Risk (Va. R) Value at Risk attempts to answer the following question: • How many dollars can I expect to lose on my portfolio in a given time period at a given level of probability? • The typical probability used is 5%. • We need to know what HPR corresponds to a 5% probability. • If returns are normally distributed then we can use a standard normal table or Excel to determine how many standard deviations below the mean represents a 5% probability: • From Excel: =Norminv (0. 05, 0, 1) = -1. 64485 standard deviations 35
Value at Risk (Va. R) From the standard deviation we can find the corresponding level of the portfolio return: Va. R = E[r] + -1. 64485 For Example: A $500, 000 stock portfolio has an annual expected return of 12% and a standard deviation of 35%. What is the portfolio Va. R at a 5% probability level? Va. R = 0. 12 + (-1. 64485 * 0. 35) Va. R = -45. 57% (rounded slightly) Va. R$ = $500, 000 x -. 4557 = -$227, 850 What does this number mean? 36
Value at Risk (Va. R) Va. R versus standard deviation: • For normally distributed returns Va. R is equivalent to standard deviation (although Va. R is typically reported in dollars rather than in % returns) • Va. R adds value as a risk measure when return distributions are not normally distributed. • Actual 5% probability level will differ from 1. 68445 standard deviations from the mean due to kurtosis and skewness. 37
Value at Risk (Va. R) • Simple approach: • Assume we have 100 HPR observations not necessrily normally distributed. • To obtain an estimate of Va. R of this sample of 100 observations, rank the returns from highest to lowest. • Find the 5 th percentile: • . . . -25% / -26% -30% -33% -35% -40% 38
Expected Shortfall (ES) • Also called conditional tail expectation (CTE) • More conservative measure of downside risk than Va. R • Va. R takes the highest return from the worst cases • ES takes an average return of the worst cases • (26% + 30% + 33% +35% +40%)/5= 32. 8% 39
Risk Premium & Risk Aversion • The risk free rate is the rate of return that can be earned with certainty. • The risk premium is the difference between the expected return of a risky asset and the risk-free rate. Excess Return or Risk Premiumasset = E[rasset] – rf Risk aversion is an investor’s reluctance to accept risk. How is the aversion to accept risk overcome? By offering investors a higher risk premium. 40
Frequency distributions of annual HPRs, 1926 -2008, Historical Records 41
Rates of return on stocks, bonds and bills, 1926 -2008 42
Annual Holding Period Returns Statistics 1926 -2008 Series Geom. Arith. Excess Mean% Return% Sharpe SD Kurt. Skew. Ratio World Stk 9. 20 11. 00 7. 25 18. 28 1. 03 -0. 16 0. 396 US Lg. Stk 9. 34 11. 43 7. 68 20. 67 -0. 10 -0. 26 0. 371 11. 43 17. 26 13. 51 37. 26 1. 60 0. 81 0. 362 World Bnd 5. 56 5. 92 2. 17 9. 05 1. 10 0. 77 0. 239 LT Bond T-Bill 5. 31 5. 60 3. 75 1. 85 8. 01 0. 80 0. 51 0. 231 Sm. Stk • Geometric mean: Best measure of compound historical return over the period • Arithmetic Mean: Expected return, best estimate for next year’s single-period return 3. 08 • Deviations from normality? 43
Deviations from Normality: Another Measure Portfolio World Stock US Small Stock US Large Stock Arithmetic Average . 1100 . 1726 . 1143 Geometric Average . 0920 . 1143 . 0934 Difference . 0180 . 0483 . 0209 ½ Historical Variance . 0186 . 0694 . 0214 If returns are normally distributed then the following relationship among geometric and arithmetic averages holds: Arithmetic Average – Geometric Average = ½ 2 • The comparisons above indicate that US Small Stocks may have deviations from normality and therefore Va. R may be an important risk measure for this class. 44
Sharpe Ratio (Reward-to-volatility) • Risk aversion implies that investors will accept a higher return in exchange of a higher risk as measured by the std of returns. • A statistic commonly used to rank assets in terms of risk-return trade-off is the Sharpe Measure: • The higher the Sharpe ratio the better. 5 -45
Historic Returns on Risky Portfolios • Observations: • Returns appear normally distributed • Except small stocks • Overall, no serious deviations from normality observed.
Lesson: Risk and Return • The First Lesson: There is a reward, on average, for bearing risk. • That is if we are willing to bear risk, then we can expect to earn a risk premium, at least on average. • Second Lesson: Further, the more risk we are willing to bear, the greater the expected risk premium. 1 -47
Adjusting for Inflation • The computation of real returns on an asset: • Deflate the nominal price Pt of the asset by an index of the general price level CPIt • Compute returns in the usual way using the deflated prices 5 -48
Ex: Adjusting for Inflation • Consider a one-month investment in Msoft stock. Suppose the CPI in months t-1 and t is 1 and 1. 01, respectively. The real prices of the stock are: 5 -49
Ex: Adjusting for Inflation • 5 -50
Ex: Adjusting for Inflation • Suppose you buy a 0 -coupon T-Bond maturing in 20 years, priced to yield 12% Price = $1000/(1. 12)20 = $103. 67 • If the CPI is 1. 00 today and 2. 65 in 20 years, what is your Real rate of return? 51
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