Investment Analysis and Portfolio Management Instructor: Attila Odabasi Introduction to Risk, Return, and the Historical Record on T-bills and US equity

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.

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

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.

How to compare returns? We convert each total return to an effective annual rate (EAR) as follows: (EAR) Compound annual return =

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:

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

Quoting Rates of Return Therefore, the relationship among the EAR and the APR is: Note that APR for a given EAR:

Compounding occurs m times per year The difference between EAR and APR grows with compounding: Continuous compounding:

Continuous Compounding Example: The effective annual rate associated with continuously compounded (m=∞) APR= 10% is determined by: EAR = e0.1 – 1 =

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 = or rreal = The approximate rule overstates the real rate by the factor 1 + i. [(1 + rnom) / (1 + i)] – 1 (rnom - i) / (1 + i) (9% - 6%) / (1.06) = 2.83% One issue we have ignored until now is the effect of inflation on our investments. The nominal rate is the rate we have been calculating. Not adjusted for inflation. To adjust for inflation we must calculate the real rate of return. At first glance you might think we just need to subtract the inflation rate from the nominal rate, but this is not correct. To get correct answer we must use the Fisher effect calculation. 1+r = 1+R/1+i which can be converted to r=(R-i)/(1+i)

Nominal and Real interest rates and Inflation Correlation between nominal rate and inflation is low. Fisher hypo is not very thight.

Measuring Equity Returns One period investment: Holding Period Percentage Return (HPR): HPR= r1 = P1 – P0 + D1 P0 P0 = Beginning price (or, PV) P1 = Ending price (or, FV) D1 = Dividend (cash flow) during period Q: Why use % returns at all? Q: What are we assuming about the cash flows in the HPR calculation?

Measuring Equity Returns HPR defined as: r1 = P1 – P0 + D1 = (P1+D1) - 1 P0 P0 Then we can define ‘gross return’: (1 + r1) = (P1+D1) / P0

Multi-period vs One-period Returns

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.

Average Performance of Multi-period Returns

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.

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:

Example: Compute cc return Let Pt-1 = 85, Pt = 90 then simple return is rt= 0.0588. 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.

Multi-period cc Returns This is the main difference between simple and cc returns.

Ex: Multi and one-period cc returns Suppose Pt-2= 80, Pt-1= 85, Pt= 90. The cc two-month return can be computed in two equivalent ways: 1) Take difference in log prices: 2) Sum the two cc one-month returns:

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?

Expected Return & Std; Scenario Analysis Purchase Price 100 T-bill Rate 0.04 State of the market Prob Y-E Price Cash Dividend HPR Deviations from the mean Squared Deviations from Mean Excess 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 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(B5:B8,E5:E8)} Variance of HPR 0.0380 <-- {=SUMPRODUCT(B5:B8,G5:G8)} Std of HPR 0.1949 <-- =SQRT(E11) Risk Premium 0.0576 <-- {=SUMPRODUCT(B5:B8,H5:H8)} Std of Excess Returns 0.2032 <-- {=SQRT(SUMPRODUCT(B5:B8,I5:I8))}

Time Series Analysis of Past Returns Period Implicitly Assumed Prob = 1/5   HPR (decimal) Squared Deviation Gross HPR= 1 + HPR Wealth Index 2000 100.000 2001 0.2 88.110 -0.1189 0.0196 0.8811 2002 68.638 -0.2210 0.0586 0.7790 2003 88.330 0.2869 0.0707 1.2869 2004 97.940 0.1088 0.0077 1.1088 2005 102.749 0.0491 0.0008 1.0491 Arithmetic Average 0.0210 <-- =AVERAGE(D4:D8) Expected Value <-- =SUMPRODUCT(B4:B8,D4:D8) Standard Deviation 0.1774 <-- =SUMPRODUCT(B4:B8,E4:E8)^0.5 <-- =STDEV.P(D4:D8) Geometric Average Return 0.0054 <-- =GEOMEAN(F4:F8)-1 <-- =((G8/G3)^0.2)-1

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? How much past data? How far back? How stationary is the data? Which historical time period? Have to adjust for current economic situation?

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.

Figure 5.4 The Normal Distribution

Given Historical Returns Expected Return: You might have subtracted 1926 from 2005. Of course, you got 79, which excludes the 1926 return. To include 1926, add one. This is because the 1926 return uses year-end 1925 prices. So, 2005 – 1925 = 80.

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:

Skew and Kurtosis Skew Kurtosis . .

Skewed Distribution: Large Negative Returns (Left Skewed) Implication?  is an incomplete risk measure r = average Skew = (E{r – E[r]}3) / 3 Here the distribution is skewed to the left. Most of the observations are slightly above the mean but there are a few large negative return outcomes that are pulling the mean down. Example 0,8,9,10,10,11,12. The average or mean is 8.57, but the median is 10. Median r Negative Positive

Skewed Distribution: Large Positive Returns (Right Skewed) = average This skewness isn’t really so bad. Most of the outcomes are slightly below the mean, but there are a few outcomes where the return is really high. Example 7,7,8,8,9,9,and 12 the mean is again 8.57 however this time the median is 8. Notice the E[r] in the sense of the most likely return is now the median. The implication is that investors are willing to accept a slightly lower E[r] for the small chance of a large positive gain. Lottery principle. Median Negative r Positive

Leptokurtosis Implication?  is an incomplete risk measure Kurtosis = (E{r – E[r]}4) / 4 - 3; Kurtosis for a normal distribution is 3. Thus Kurtosis above zero indicates fat tails. Note the  underestimates risk if stock returns are leptokurtotic.

Value at Risk (VaR) 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 in the Norminv function in Excel the 0.05 is the 5% probability, 0 is the mean and 1 is the standard deviation of a standard normal variate.

Value at Risk (VaR) From the standard deviation we can find the corresponding level of the portfolio return: VaR = E[r] + -1.64485 For Example: A $500,000 stock portfolio has an annual expected return of 12% and a standard deviation of 35%. What is the portfolio VaR at a 5% probability level? VaR = 0.12 + (-1.64485 * 0.35) VaR = -45.57% (rounded slightly) VaR$ = $500,000 x -.4557 = -$227,850 What does this number mean? in the Norminv function in Excel the 0.05 is the 5% probability, 0 is the mean and 1 is the standard deviation of a standard normal variate. Interpretation: The greatest annual expected loss 95% of the time is $227,85.

Value at Risk (VaR) VaR versus standard deviation: For normally distributed returns VaR is equivalent to standard deviation (although VaR is typically reported in dollars rather than in % returns) VaR 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.

Value at Risk (VaR) Simple approach: Assume we have 100 HPR observations not necessrily normally distributed. To obtain an estimate of VaR of this sample of 100 observations, rank the returns from highest to lowest. Find the 5th percentile: ........... -25% / -26% -30% -33% -35% -40%

Expected Shortfall (ES) Also called conditional tail expectation (CTE) More conservative measure of downside risk than VaR VaR 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%

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 = 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. E[rasset] – rf

Frequency distributions of annual HPRs, 1926-2008, Historical Records Forward looking purpose: Use Arithmetic Mean. Note relationship between means and standard deviations and the standard deviation of a large stock portfolio. Ask, which is the most risky and which is he least risky? Note apparent deviations from normality in small stocks particularly.

Rates of return on stocks, bonds and bills, 1926-2008

Annual Holding Period Returns Statistics 1926-2008 Geom. Arith. Excess Sharpe Series Mean% Return% 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 Sm. Stk 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 5.31 5.60 1.85 8.01 0.80 0.51 0.231 T-Bill 3.75 3.08 Notice the greater divergence of the GAR and AAR for small stocks. This is because of the high variance and the higher proportion of negative returns in the small stock portfolio. These are the historical returns between 1926-1998 listed in the text. Remember the arithmetic mean would be the typical expectation for any given year, while the geometric mean is the return needed for every year to get the same cumulative returns as the sequence of historical returns. Notice that generally the higher the returns the higher the variance or SD. Geometric mean: Best measure of compound historical return over the period Arithmetic Mean: Expected return, best estimate for next year’s single-period return Deviations from normality?

Deviations from Normality: Another Measure Portfolio World Stock US Small Stock US Large Stock Arithmetic Average .1100 .1726 .1143 Geometric Average .0920 .0934 Difference .0180 .0483 .0209 ½ Historical Variance .0186 .0694 .0214 Uses data from Table 5.3 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 VaR may be an important risk measure for this class.

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.

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.

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

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:

Ex: Adjusting for Inflation

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?