Unformatted text preview: Chapter 5 Risk & Return 5.1 Rates of Return Measuring Ex
Post (Past) Returns One
period investment: regardless of the length of the period. Holding period return (HPR): HPR = (PS+ CF
PB)/PB where PS = Sale price (or P1) PB = Buy price ($ you put up) (or P0) CF = Cash ﬂow during holding period • HPR is percentage return. No size
of
investment concerns. • CF realized by the end of the period. Annualizing HPRs Q: Why would you want to annualize returns? 1. Annualizing HPRs for holding periods of greater than one year: – Without compounding (Simple or APR): HPRann = HPR/n – With compounding: EAR HPRann = [(1+HPR)1/n]
1 where n = number of years held Example: Annualize HPR Suppose you buy one share of a stock today for $45 and you hold it for 2 years and sell it for $52. You also received $8 in dividends at the end of the two years. • • • • PB=$45, PS=52, CF=8, n=2 HPR=(52+8
45)/45=33.33% HPRann= 33.33%/2=16.66% simple rate Assuming annual compouding, – HPRann=(1+0.3333)1/2
1 = 15.47% Example: Annualize HPR An example when the HP is < 1 year: Suppose you have a 5% HPR on a 3 month investment. What is the annual rate of return with and without compounding? • n=3/12= 0.25 year • Without compounding (Simple): – HPRann = HPR/n = 5%/0.25= 20% • With compounding: – HPRann = [(1+HPR)1/n]
1 = [(1+5%)4]
1 = 21.55% ArithmeZc Average Xme weighted average return Finding the average HPR for a Zme series of returns: • a. Without compounding (AAR or ArithmeZc Average Return): HPR T HPR avg = ∑ n T =1 • n = number of Zme periods €
n ArithmeZc Average HPR T HPR avg = ∑ n T =1
HPR avg = (.2156 + .4463 + .2335 + .2098 + .0311 + .3446 + .1762) = 17.51% 7 n €
€ This DOES NOT correspond to any single
period return! Geometric Average Xme weighted average return • b. With compounding (geometric average or GAR: Geometric Average Return): Ⱥ n Ⱥ HPR avg = Ⱥ∏ (1 + HPR T ) Ⱥ Ⱥ T =1 Ⱥ
1/ n −1 HPR avg = (0.7844 × 1.4463 × 1.2335 × 1.2098 × 1.0311 × 1.3446 × 1.1762)1/7 − 1 = 15.61% € How to interpret GAR? What is Zme weighted average return? Por_olio’s Ex
Post (Past) Returns • Finding the average HPR for a por_olio of assets for a given Zme period: Ⱥ VI Ⱥ HPR avg = ∑ȺHPR I × Ⱥ Ⱥ TV Ⱥ I=1
J • where VI = $ amount invested in asset I, • J = Total # of securiZes € • and TV = total $ amount invested; • thus VI/TV = percentage of total investment invested in asset I Example: Por[olio returns Suppose you have $1000 invested in a stock por[olio in September. You have $200 invested in Stock A, $300 in Stock B and $500 in Stock C. The HPR for the month of September for Stock A was 2%, for Stock B the HPR was 4% and for Stock C the HPR was
5%. • The average HPR for the month of September for this por[olio is: HPR avg Ⱥ VI Ⱥ = ∑ȺHPR I × Ⱥ Ⱥ TV Ⱥ I=1
J HPR avg = (.02 × (200/1000)) + (.04 × (300/1000)) + (.05 × (500/1000)) = −0.9% €
€ Example: Por[olio returns • Measuring returns when there are investment changes (buying or selling) or other cash ﬂows within the period. • An example: – Today you buy one share of stock cosZng $50. The stock pays a $2 dividend one year from now. – Also one year from now you purchase a second share of stock for $53. – Two years from now you collect a $2 per share dividend and sell both shares of stock for $54 a share • Q: What was your average (annual) return? Dollar
Weighted Return I. Dollar
weighted return procedure (DWR): Find the internal rate of return for the cash ﬂows (i.e. ﬁnd the discount rate that makes the NPV of the net cash ﬂows equal zero.) Tips on CalculaZng Dollar Weighted Returns This measure of return considers both changes in investment and security performance • IniXal Investment is an ou[low • Ending value is considered as an inﬂow • AddiXonal investment is an ou[low • Security sales are an inﬂow Por_olio Returns . Dollar
weighted return procedure (DWR): Find the internal rate of return for the cash ﬂows (i.e. ﬁnd the discount rate that makes the NPV of the net cash ﬂows equal zero.) • NPV =
$50/(1+IRR)0
$51/(1+IRR)1 + $112/(1+IRR)2 set to zero • Solve for IRR: • IRR = 7.117% average annual dollar weighted return The DWR gives you an average return based on the stock’s performance and the dollar amount invested each period. Time
Weighted Returns II. Time
weighted returns (TWR): TWRs assume you buy one share of the stock at the beginning of each interim period and sell one share at the end of each interim period. TWRs reﬂect the security performance NOT the investment in
and out
ﬂow change. To calculate TWRs: • Calculate the return for each Xme period, typically a year. • Then calculate either an arithmeXc (AAR) or a geometric average (GAR) of the returns. Time
Weighted Returns • Same example: iniZally buy one share at $50, in one year collect a $2 dividend, and you buy another share at $53. In two years you sell the stock for $54, amer collecZng another $2 dividend per share. TWRs assume you buy one share of the stock at the beginning of each period and sell it at the end of each period aber collecXng any cash ﬂow. Ex
post returns • HPR for year 1: – [$53 + $2
$50] / $50 = 10% • HPR for year 2: – [$54
$53 +$2] / $53 = 5.66% • Calculate the arithmeZc average TW return – AAR = [0.10 + 0.0566] / 2 = 7.83% Ex
post returns • HPR1 = 10% • HPR2 = 5.66% • Calculate the geometric average TW return (GAR): Ⱥ Ⱥ
n 1/ n HPR avg = Ⱥ∏ (1 + HPR T ) Ⱥ Ⱥ T =1 Ⱥ
1/2 −1 HPR avg = (1.10 × 1.0566) − 1 = 7.81% = GAR € € GAR vs AAR A1: When you are evalua>ng PAST RESULTS (ex
post): – Use the AAR (average without compounding) if you ARE NOT reinvesXng any cash ﬂows received before the end of the period. – Use the GAR (average with compounding) if you ARE reinvesXng any cash ﬂows received before the end of the period. A2: When you are trying to es>mate an expected return (ex
ante return):  Use the ARR 5.2 Risk and Risk Premiums Measuring expected returns: Scenario analysis • SubjecZve or Scenario expected returns E ( r) = ∑ p(s)r(s)
s=1 M • E(r) = Expected Return • p(s) = likelihood (probability) of a state • r(s) = return if a state occurs € • s=1, 2, …, M states Measuring Variance or Dispersion of Returns • SubjecZve or Scenario Variance σ = Var( r) = ∑ p( s)[ r( s) − E ( r)]
2 s=1 M 2 € • • • • • E(r) = Expected Return p(s) = likelihood (probability) of a state r(s) = return if a state occurs s=1, 2, …, M states σ is typically called volaXlity Numerical Example: SubjecZve or Scenario DistribuZons State 1 2 3 Probability of a State 0.2 0.5 0.3 Returns
0.05 0.05 0.15 E(r)= (.2)(
0.05) + (.5)(0.05) + (.3)(0.15) = 6% σ2 = [(.2)(
0.05
0.06)2 + (.5)(0.05
0.06)2 + (.3)(0.15
0.06)2 σ = [ 0.0049]1/2 = .07 or 7% Ex
post Expected Return & σ • Ex
post return and volaXlity are sample esXmates, i.e. they are computed from data/ realizaXon. 1n Expost Variance : σ 2 = ( ri − r ) 2 ∑ n − 1 i=1 • Annualize: rannual = rperiod × # periods
ˆ ˆ σ annual = σ period x # periods Using Ex
Post Returns to esZmate Expected HPR • EsZmaZng Expected HPR (E[r]) from ex
post data. – Use the arithmeXc average of past returns as a forecast of expected future returns as we did and, – Perhaps apply some (usually ad
hoc) adjustment to past returns • Which historical Xme period? – Problems? • Have to adjust for current economic situaXon CharacterisZcs of Probability DistribuZons Normal DistribuZon E[r] = 10% σ = 20% Average = Median Skewed DistribuZon: Large NegaZve Returns Possible Median NegaZve PosiZve ImplicaXon? • Skewed to the leb • σ is an incomplete risk measure r € Skewed DistribuZon: Large PosiZve Returns Possible Median NegaZve PosiZve ImplicaXon? • Skewed to the right r € Leptokurtosis Value at Risk (VaR) Value at Risk aqempts to answer the following quesZon: • How many dollars can I expect to lose on my por_olio in a given Zme 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 deviaZons below the mean represents a 5% probability: – From Excel: =Norminv (0.05,0,1) =
1.64485 standard deviaZons From the standard deviaZon we can ﬁnd the corresponding level of the por_olio return: VaR = E[r]
1.64485σ For Example: A $500,000 stock por_olio has an annual expected return of 12% and a volaZlity of 35%. What is the por_olio VaR at a 5% probability level? VaR = 0.12 + (
1.64485 * 0.35) VaR =
45.57% (rounded slightly) VaR$ = $500,000 x (
.4557) =
$227,850 What does this number mean? Value at Risk (VaR) Value at Risk (VaR) VaR versus standard deviaZon: • For normally distributed returns VaR is equivalent to standard deviaZon (although VaR is typically reported in dollars rather than in % returns) • VaR adds value as a risk measure when return distribuZons are not normally distributed. – Actual 5% probability level will diﬀer from 1.68445 standard deviaZons from the mean due to kurtosis and skewness. 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. 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. Sharpe RaZo • A measure that is omen used to rank por_olio performance • Sharpe RaZo = [E(rp)
rf ]/σp 5.3 The Historical Record Frequency distribuZons of annual HPRs, 1926
2008 Rates of return on stocks, bonds and bills, 1926
2008 Annual Holding Period Returns StaZsZcs 1926
2008 Geom. Series World Stk US Lg. Stk Sm. Stk World Bnd LT Bond Mean% 9.20 9.34 11.43 5.56 5.31 Arith. Mean% 11.00 11.43 17.26 5.92 5.60 Excess Return% 7.25 7.68 13.51 2.17 1.85 Kurt. 1.03 0.10 1.60 1.10 0.80 Skew. 0.16 0.26 0.81 0.77 0.51 • Devia>on from normality • Geometric mean: Best measure of compound historical return • Arithme>c Mean: Expected return DeviaZons from Normality: Another Measure Portfolio World Stock Arithmetic Average Geometric Average Difference ½ Historical Variance .1100 .0920 .0180 .0186 US Small Stock .1726 .1143 .0483 .0694 US Large Stock .1143 .0934 .0209 .0214 • If returns are normally distributed then the following relaZonship holds: ArithmeZc Average – Geometric Average = ½ σ2 •The comparisons above indicate that US Small Stocks may have deviaZons from normality and therefore VaR may be an important risk measure for this class. Actual vs. TheoreZcal VaR 1926
2008 Series World Stk US Lg. Stk US Sm. Stk World Bnd US LT Bond Actual VaR% 21.89 29.79 46.25 6.54 7.61 VaR% if Normal 21.07 22.92 44.93 8.69 7.25 These comparisons indicate that the U.S. Large Stock por_olio, the US small stock por_olio and the World Bond por_olio may exhibit diﬀerences from normality. Size
decile por_olios of the NYSE/AMEX/ NASDAQ of annual returns, 1927
2008 5.4 InﬂaZon and Real Rates of Return InﬂaZon, Taxes and Returns • InﬂaZon rate, rate at which prices are rising, is measured as the percentage change in the CPI (consumer price index). • The average inﬂaZon rate from 1966 to 2005 was 4.29%. This reduces the purchasing power of investment income • Taxes are paid on nominal investment income. This reduces real investment income even further. You earn a 6% nominal, pre
tax rate of return and you are in a 15% tax bracket and face a 4.29% inﬂaZon rate. What is your real amer tax rate of return? rreal ≈ [6% x (1
0.15)] – 4.29% ≈ 0.81%; taxed on nominal Real vs. Nominal Rates • rreal = real interest rate • rnom = nominal interest rate • i = expected inﬂaZon rate Fisher (1930): : rnom = rreal + E(i) ApproximaZon: rreal ≈ rnom
i Example rnom = 9%, i = 6%, so rreal ≈ 3% Exact RelaZon: 1+ rreal = [(1 + rnom) / (1 + i)] or rreal = (rnom
i) / (1 + i) rreal = (9%
6%) / (1.06) = 2.83% The exact real rate is less than the approximate real rate. Nominal and Real interest rates and InﬂaZon Historical Real Returns & Sharpe RaZos Series World Stk US Lg. Stk Sm. Stk World Bnd LT Bond Real Returns% 6.00 6.13 8.17 2.46 2.22 Sharpe Ratio 0.37 0.37 0.36 0.24 0.24 • Real returns have been much higher for stocks than for bonds • Sharpe raZos measure the excess return to standard deviaZon. • The higher the Sharpe raZo the beqer, (all else equal). • Stocks have had much higher Sharpe raZos than bonds. 5.5 Asset AllocaZon Across Risky and Risk Free Por_olios Asset AllocaZon Possible to split investment funds between safe and risky assets Risk free asset rf : proxy; T
bills or money market fund Risky asset or por_olio rp Example. Your total wealth is $10,000. You put $2,500 in risk free T
Bills and $7,500 in a stock por_olio invested as follows: – Stock A you put $25,000 – Stock B you put $30,000 – Stock C you put $20,000 AllocaZng Capital Between Risky & Risk
Free Assets In the complete por_olio: Weights in rf, y=25% Weights in rp, 1
y=75% – WA = 25% – WB = 30% – WC = 20% Note: RelaZve weights in risky por_olio are ﬁxed (y may vary) A; $2,500 / $7,500 = 33.33% B: $3,000 / $7,500 = 40.00% C: $2,000 / $7,500 = 26.67% 100.00% AllocaZng Capital Between Risky & Risk
Free Assets • Issues in se}ng weights – Examine Risk and Return Tradeoﬀ – Demonstrate how diﬀerent degrees of risk aversion will aﬀect allocaZons between risky and risk free assets Example • rf = 5% σrf = 0% • E(rp) = 14% σrp = 22% • y = % in rp (1
y) = % in rf Expected Returns for CombinaZons • E(rc) = yE(rp) + (1
y)rf • σc = yσrp + (1
y)σrf • E(rC) is return for complete por_olio • For example, let y = 75% • E(rC) = (.75 x .14) + (.25 x .05)=11.75% • σC = (0.75 x 0.22) + (0.25 x 0) = 16.5% Complete por_olio • E(rc) = yE(rp) + (1
y)rf • σc = yσrp + (1
y)σrf • Varying y results in E[rC] and σC that are linear combiniZon of E[rp] and rf and σrp and σrf respecZvely. This is NOT generally the case for the σ of combinaZons of two or more risky assets. Possible CombinaZon E(rp)= 14% E(rp)= 11.75% y = .75 rf = 5% F y = 0 P y = 1 0 16.5% 22% σ Using Leverage with Capital AllocaZon Line Borrow at the Risk
Free Rate and invest in stock Using 50% Leverage y=1.5 E(rc)=(1.5) (.14) + (
.5) (.05) = 0.185 = 18.5% σc = (1.5) (.22) = 0.33 or 33% Is this on the capital allocaXon line?? Risk Aversion and AllocaZon • Greater levels of risk aversion lead investors to choose larger proporZons of the risk free rate; Lower levels of risk aversion lead investors to choose larger proporZons of the por_olio of risky assets • Willingness to accept high levels of risk for high levels of returns would result in leveraged combinaZons QuanZfying Risk Aversion E ( rp ) − rf = 0.5 × A × σ p
E(rp) = Expected return on por_olio p rf = the risk free rate 0.5 = Scale factor A x σp2 = ProporZonal risk premium The larger A is, the larger will be the investor’s added return required to bear risk 2 QuanZfying Risk Aversion Rearranging the equaZon and solving for A A= E ( rp ) − rf 0.5 × σ p
2 Many studies have concluded that investors’ average risk aversion is between 2 and 4. € 5.6 Passive Strategies and the Capital Market Line A Passive Strategy • InvesZng in a broad stock index and a risk free investment is an example of a passive strategy. – The investor makes no aqempt to acZvely ﬁnd undervalued strategies nor acZvely switch their asset allocaZons. – The CAL that employs the market (or an index that mimics overall market performance) is called the Capital Market Line or CML. Excess Returns and Sharpe RaZos implied by the CML Excess Return or Risk Premium Time Period 19262008 19261955 19561984 19852008 Average 7.86 11.67 5.01 5.95 σ 20.88 25.40 17.58 18.23 Sharpe Ratio 0.37 0.46 0.28 0.33 The average risk premium implied by the CML for large common stocks over the enZre Zme period is 7.86%. • How much conﬁdence do we have that this historical data can be used to predict the risk premium now? AcZve versus Passive Strategies • AcZve strategies entail more trading costs than passive strategies. • Passive investor “free
rides” in a compeZZve investment environment. • Passive involves investment in two passive por_olios – Short
term T
bills – Fund of common stocks that mimics a broad market index – Vary combinaZons according to investor’s risk aversion. ...
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