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Risk and Return: Past and Prologue

Department of Banking and Finance. SPRING 200 7 -0 8. Risk and Return: Past and Prologue. by Asst. Prof. Sami Fethi. Portfolio Theory.

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Risk and Return: Past and Prologue

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  1. Department of Banking and Finance SPRING 2007-08 Risk and Return: Past and Prologue by Asst. Prof. Sami Fethi

  2. Portfolio Theory • The 19th century Scottish/ American millionaire Andrew Carnegie said that “ The way to make a fortune is to put all your eggs into one basket and then watch the basket very carefully”. • In fact, this seems a high-risk strategy. If you choose the right one, you can make a fortune, but choosing the wrong one, it could turn out to be the way to lose. • A more logical approach to invest is to diversify and thus spread the risk.

  3. Portfolio Theory • If you are facing a future uncertain, as most of us do, you will be vulnerable to negative shocks as you rely on a single source of income. It is less risky to have diverse sources of income or put it another way, to hold a portfolio of assets or investments. • The bottom line is that “ Do not put all your eggs in one basket”.

  4. Portfolio Theory • Here, the main aim is to examine the extent of risk reduction when an investor switches from complete commitment to one asset, for example, shares in a company to the position where resources are split between two or more assets. • Briefly, we will focus on the use of portfolio theory, particularly in the context of investment in financial securities such as shares in companies.

  5. Portfolio Theory • The basis of portfolio theory was first developed by Harry Markowitz in 1952. The main theme behind the theory is to analyse the risk-reducing effect in spreading investment across a range of asset- • In a portfolio, you may have unexpected bad news whereas the other may have unexpected good news that can compensate each other. So Markowitz gives us the tool for identifying portfolios which give the highest return for a particular level of risk.

  6. HPR = Holding Period Return • Holding Period Return of a share of stock depends on the increase or decrease in the price of the share over the investment period as well as on any dividend income the share has provided. This can be formulated in the following page.

  7. Rates of Return: Single Period P P D - + HPR = 1 0 1 P 0 HPR = Holding Period Return P1 = Ending price P0 = Beginning price D1 = Dividend during period one

  8. Rates of Return: Single Period Example Ending Price = 24 Beginning Price = 20 Dividend = 1 HPR = ( 24 - 20 + 1 )/ ( 20) = 25%

  9. Rates of Return: Single Period Example • If a share was bought for $ 2, and paid a dividend after one year of 10c and the share was sold for $ 2.20 after one year. • a) what was the HPR? • b) if another share produced a HPR, say, 10 percent over a six month period. what was the HPR?

  10. Rates of Return: Single Period Solution • HPR = ( 2.20- 2 + 0.10 )/ (2 )= 15% • One year and six month return are related through the formula: • (1+s)2 = 1+R s=semi-annual rate, R=annual. • (1+0.1)2 = 1+R R= (1+0.1)2 –1 • R=21%

  11. Returns Using Arithmetic, Geometric and Dollar Weighted Averaging • Arithmetic Average: The sum of returns in each period divided by the number of periods. • Geometric Average: The single per period return that gives the same cumulative performance as the sequence of actual returns (called time weighted av return). • Dollar weighted Average returns : is the internal rate of return on an investment.

  12. Example 1 1 2 3 4 Assets(Beg.) 1.0 1.2 2.0 .8 HPR .10 .25 (.20) .25 TA (Before Net Flows 1.1 1.5 1.6 1.0 Net Flows 0.1 0.5 (0.8) 0.0 End Assets 1.2 2.0 .8 1.0

  13. Returns Using Arithmetic and Geometric Averaging Arithmetic ra = (r1 + r2 + r3 + ... rn) / n ra = (.10 + .25 - .20 + .25) / 4 = .10 or 10% Geometric rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1 rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1 = (1.5150) 1/4 -1 = .0829 = 8.29%

  14. Dollar Weighted Returns Internal Rate of Return (IRR) - the discount rate that results present value of the future cash flows being equal to the investment amount • Considers changes in investment • Initial Investment is an outflow • Ending value is considered as an inflow • Additional investment is a negative flow • Reduced investment is a positive flow

  15. Dollar Weighted Average Using Example 1 Net CFs $ (mil) 1 2 3 4 - .1 - .5 .8 1.0 The present value of cash flow is $ 1 million. Solving for IRR 1.0 = -.1/(1+r)1 + -.5/(1+r)2 + .8/(1+r)3 + 1.0/(1+r)4 r = .0417 or 4.17%

  16. Quoting Conventions APR = annual percentage rate (periods in year) X (rate for period) EAR = effective annual rate ( 1+ rate for period)Periods per yr - 1 Example: monthly return of 1% APR = 1% X 12 = 12% EAR = (1.01)12 - 1 = 12.68%

  17. Example 2 A FUND begins with $10mn Month 1 2 3 Net IFs $ (mil) 3 5 0 HPR (%) 2 8 (4) Compute the arithmetic, time-weighted and dollar-weighted average returns

  18. Answer • The arithmetic av. (2+8-4)/3=2%. • The time-weighted av. • [1+0.02) (1+0.08) (1-0.04)]1/3-1=1.88% • The dollar-weighted as follow: Solving for IRR 10 = -3.0/(1+irr)1 + -5.0/(1+irr)2 + 18.486/(1+irr)3 r = 1.17%

  19. Answer Month 1 2 3 Asset beginning 10 13.2 19.256 HPR (%) x assets 0.2 1.05 (0.77) Net IFs $ (mil) 3 5 0 Asset end 13.2 19.256 18.486

  20. Answer Time 0 1 2 3 Net CFs $ (mil) - 10 - 3.0 -5.0 18.486 Solving for IRR 10 = -3.0/(1+irr)1 + -5.0/(1+irr)2 + 18.486/(1+irr)3 r = 1.17%

  21. Characteristics of Probability Distributions 1) Mean: most likely value 2) Variance or standard deviation 3) Skewness * If a distribution is approximately normal, the distribution is described by characteristics 1 and 2

  22. Probability Distribution, Expected Return, Variance and Standard Deviation • Probability Distribution: List of the possible outcomes with associated probabilities. • Expected Return: The mean value of the distribution of holding period returns. • Variance: The expected value of the square deviation from the mean. • Standard Deviation: The square root of the variance.

  23. Normal Distribution s.d. s.d. r Symmetric distribution

  24. Skewed Distribution: Large Negative Returns Possible Median Negative Positive r

  25. Skewed Distribution: Large Positive Returns Possible Median Negative r Positive

  26. S E ( r ) = p ( s ) r ( s ) s Measuring Mean: Scenario or Subjective Returns Subjective returns p(s) = probability of a state r(s) = return if a state occurs 1 to s states

  27. Numerical Example 3: Subjective or Scenario Distributions StateProb. of State rin State 1 .1 -.05 2 .2 .05 3 .4 .15 4 .2 .25 5 .1 .35 E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35) E(r) = .15

  28. S 2 Variance = p ( s ) [ r - E ( r )] s s Measuring Variance or Dispersion of Returns Subjective or Scenario Standard deviation = [variance]1/2 Using Our Example: Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2] Var= .01199 S.D.= [ .01199] 1/2 = .1095

  29. Example 4 • A share of stock of company B is now selling for $ 23.50 and three possible scenarios as follows: Busi conscenario. Prob end-Y p A.Div High growth 1 0.35 $35 $4.4 Normal growth 2 0.30 $27 $4.0 No growth 3 0.35 $15 $4.0 Calculate HPR, ex HPR and Sdev of HPR.

  30. Solution • High growth HPR = (4.40+35-23.50)/23.50 • =67.66% • NG HPR = (4.0+27-23.50)/23.50 • =31.91% • NoG HPR = (4.0+15-23.50)/23.50 • =-19.15% • Er=0.35x67.66+0.30x31.91+0.35x(-19.15) • 26.55% • Sd= Sqroot of 0.35x(67.66-26.55)2…… • 36.5% and 1331 was found as variance.

  31. Example • The current price of a share is 100c and three possible scenarios as follows: • Eco. conscenario. Prob end-Y p A.Div • Boom 1 0.30 128c 7c • steady growth 2 0.40 117c 3c • slump 3 0.30 105c 0c • Calculate return on investment, exReturn, variance and Sdev of returns.

  32. Solution • Boom R = (128+7-100)/100 • =35% • SSG R = (117+3-100)/100 • =20% • slump R = (105+0-100)/100 • =5% • Er=0.30x35+0.40x20+0.30x(5) • 20% • Sd= Sqroot of 0.30x(35-20)2…… • 11.61% and 135 was found as variance.

  33. Annual Holding Period ReturnsExample 5 Geom. Arith. Stan. Series Mean% Mean% Dev.% Lg. Stk 10.23 12.25 20.50 Sm. Stk 11.80 18.43 38.11 LT Gov 5.10 5.64 8.19 T-Bills 3.71 3.79 3.18 Inflation 2.98 3.12 4.35

  34. Risk Premium and Real Return • First of all, we need to ask how much of an expected reward is offered to compensate for the risk involved in investing money in stocks. We measure the reward as the difference the expected HPR and the risk free rate- the rate of return that can be earned with certainty. • The rate you can earn by leaving money in risk-free assets such as Treasury bills, money market funds or banks.This difference is referred to as risk premium- an expected return in excess of that on risk free securities.

  35. Annual Holding Period Risk Premiums and Real Returns-solution Excess Real Series Returns%Returns% Lg. Stk 8.46 9.13 Sm. Stk 14.64 15.31 LT Gov 1.85 2.52 T-Bills --- 0.19 Inflation --- ---

  36. Excess and Real Returns • Excess return: Rate of return in excess of the Treasury-bill rate. • Real return: Rate of return in excess of the inflation rate. Lg. Stk 8.46 (Ex rt.)9.13 (RR) Sm. Stk 14.64 15.31 • The above figures from Table 5.3 (pp 135/6) can be computed as follows: • 12.25-3.79=8.46 12.25-3.12=9.13 • 18.43-3.79=14.64 18.43-3.15=15.31

  37. Risk Premium • The risk premium can be formulated as follows: • E (rP) – rf = ½ A P2 • A= E (rP) – rf / (½ P2 ) • Where E (rP) – rf= risk premium of portfolio E(rP) is expected return, rf is risk free-rate, A is risk aversion and P2 is the volatility of returns-risk of portfolio. In general, the equation above shows an investor demands of a portfolio as a function of risk.

  38. Risk Premium- Example 6 • Suppose the return of S&P 500 index portfolio over the coming year is 10%. The one year T-bill rate is 5%. The relevant index suggest that sd of returns is 18%. • What is the degree of risk aversion of average investor base on an assumption that the av. Portfolio resembles the S&P 500? • A= E (rP) – rf / ½ P2 A= (0.10-0.05)/ ½ (0.18)2 A= 3.09

  39. Inflation rate, Nominal and Real Interest rate • Inflation Rate: The rate at which prices are rising measured as the rate of increase of the CPI (consumer price index) or the percentage change in CPI. • NIR: The interest rate in terms of nominal (not adjusted for purchasing power) dollars. • RIR: The excess of the interest rate over the inflation rate or the growth rate of purchasing power derived from an investment (i.e., RIR=NIR-Inflation Rate).

  40. Real vs. Nominal Rates Fisher effect: Approximation nominal rate = real rate + inflation premium R = r + i or r = R - i Example r = 3%, i = 6% R = 9% = 3% + 6% or 3% = 9% - 6% Fisher effect: Exact R=r+E(i) 1+ r = (1+R ) / (1 + i) Growth factor of your purchasing power equals the growth factor of your money divided by the new price level. r = (R - i) / (1 + i) 2.83% = (9%-6%) / (1.06)

  41. Real vs. Nominal Rates • Suppose the real interest rate is 3% per year and the expected inflation rate is 8%. • A) what is the nominal interest rate? • B) Compute the nominal interest rate when the expected inflation rate rises to 10%. Answers: • A) 1+ R= (1.03) / (1.08)- R=0.1124 or 11.24% • B) 1+ R= (1.03) / (1.10)- R=0.133 or 13.3%

  42. Allocating Capital Between Risky & Risk-Free Assets • Asset allocation: Portfolio choice among broad investment classes • Possible to split investment funds between safe and risky assets • Risk free asset: T-bills • Risky asset: stock (or a portfolio) • Complete portfolio: the entire portfolio including risky and risk-free assets.

  43. Allocating Capital Between Risky & Risk-Free Assets (cont.) • Issues • Examine risk/ return tradeoff • Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets • When we shift wealth from the risky portfolio to the risk –free asset, we do not change the relative proportions of the various risky assets within the risky portfolio. Rather, we reduce the relative weight of the risky portfolio as a whole in favor of risk-free assets.

  44. Example 7 • Assume that the total market value of an investor’s portfolio is $300,000. A risk free asset is $90,000. The remaining $ 210,000 is in risky securities, say $113,400 A type S&P 500 index fund and $96,600 in B type investment grade bond fund. • The percentage weighted risky portfolio are : WA=113,400/210,000=0.54 WB=96,000/210,000=0.46

  45. Example (cont.) • Complete portfolio is denoted by (y) and the weight of the money fund is (1-y); • Y=210,000/300,000=0.7 (risky asset, P) • 1-Y= 90,000/300,000=0.3 (risk free asset) • The weights of the individual assets in the com.port. • A type 113,400/300,000=0.378 • B type 96,600/300,000=0.322 • Portfolio P 210,000/300,000=0.7 • Ready Assets F 90,000/300,000=0.3 • Portfolio C 300,000/300,000=1

  46. Expected Returns for Combinations E(rc) = yE(rp) + (1 - y)rf rc = complete or combined portfolio This refers the entire portfolio including risky and Risk-free assets. Suppose: y = .75 based onprevious example E(rc) = .75(.15) + .25(.07) = .13 or 13%

  47. The capital Allocation line • CAL: plot of risk-return combinations available by varying portfolio allocation between a risk-free asset and a risky portfolio. • Reward- to- variability ratio: Ratio of risk premium to standard deviation. • Note: the RTVR is the same for risky portfolio P and the complete portfolio that is formed by mixing P and the risk –free asset in equal proportions. While the risk-return combination differ, RTVR to risk is constant.

  48. Possible Combinations-capital allocation line Case 1: If you invest all of your funds in the risky asset, that is, y=1.0, the expected return on your complete portfolio will be 15% and standard deviation will be 22%- this combination of risk and return is plotted as point p in the relevant figure. Case 2: If you invest all of your funds in the risk-free asset, that is, y=0, the expected return on your complete portfolio will be 7% and standard deviation will be 0- point F. Case 3: If you allocate equal amounts of your overall or complete portfolio, C, (risky and risk-free assets, that is, choosing y=0.5, the expected return will be an av. of the expected return on portfolios F and P. E(rc)= 0.5(7%)+0.5(15%)=11% so the risk premium of complete portfolio is 11-7=4%. The slope of CAL is (15-7)/22=0.36. E(r) CAL (Capital Allocation Line) CAL E(rp) = 15% y = 1.25 P y = 0.50 E(rp) –rf = 8% s = 8/22 rf = 7% F 0 s 22%

  49. Variance on the Possible Combined Portfolios • Case 1: If you invest all of your funds in the risky asset, that is, y=1.0, the expected return on your complete portfolio will be 15% and standard deviation will be 22%- this combination of risk and return is plotted as point p in the relevant figure. • Case 2: If you invest all of your funds in the risk-free asset, that is, y=0, the expected return on your complete portfolio will be 7% and standard deviation will be 0- point F.

  50. Variance on the Possible Combined Portfolios • Case 3: If you allocate equal amounts of your overall or complete portfolio, C, (risky and risk-free assets, that is, choosing y=0.5, the expected return will be an av. of the expected return on portfolios F and P. • E(rc)= 0.5(7%)+0.5(15%)=11% so the risk premium of complete portfolio is 11-7=4%. • The slope of CAL is (15-7)/22=0.36.

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