The Normal Distribution

Transcription

1 Overview Refresher on risk and return (refresher) Chapter 5, we have covered much already Risk aversion (refresher) Chapter 6 Optimal risky portfolios (refresher) Chapter 7 Index models Index models bridges CAPM and Optimal Risky Chapter 8 Portfolios. The Normal Distribution Why is it useful for financial analysis? Why do some stock returns sometimes not follow a normal pattern? We use the normal distribution to model certain continuous random variables such as the return on Google stock. How do we read the distribution? We are not picking exact values, we cannot say what the probability of an exact occurrence is. What we can say is: if I pull out a sample the probability that it is some value x is the area under the curve from the left up to that point x (the area of the left tail). We can only say what is the probability that something is greater than or less than something or between two ranges. Symmetrical about the FNCE30: Investments Lecture 6 Page 1

2 mean. We will use this along with the concept of "value of risk" to measure risk. Often used by banks to measure portfolios. Why do Normal Distributions Appear Naturally? This distribution was generated by measuring the return on JP Morgan stock on a monthly basis for 60 months. Then said, for a particular month, if the stock price goes up they see a positive return. Over that 60 month period, every time the price went up what was the average value? The result was approximately 0.7%. So on an average up month, based on the last 5 years, the price of JP Morgan stock went up by (.7%). Didn't measure the average down value, instead just decreased the stock value by.7%: UP 1.007Price Down Price So we use this UP value and this DOWN value and run a simulation 1000 times and plot the results and it is a normal distribution. Not perfectly but would be more so if we do a larger number of simulation runs. If we draw a tree using the up/down arrows results of many runs the natural thing we get out is a normal distribution. Value at Risk So the product of UP*DOWN=1. If a particular period goes up and then it goes down you come back to the same starting point. If you grow by the up amount when you come down you divide out that growth (?). Anyway, you come back to the same value. Go by the same amount you come down. o ASSUMING RETURNS ON QQQQ ARE NORMALLY DISTRIBUTED o use last 500 days of historical returns to measure the 5% VAR for a $1 million position QQQQ is the ETF (exchange traded fund) that trades and tracks the NASDAQ index, it is a stock market traded fund which is put together with the goal of mimicking the NASDAQ. We extract the last 500 days of historical returns. Use this data to estimate the risk I have in a daily position in a $1 million investment in the QQQQ. $1 million invested in QQQQ, on a bad day how much could I potentially lose based on historical data? FNCE30: Investments Lecture 6 Page

3 VAR data (not all of it) Date Adjusted Close Days Return Daily Return /7/ % 0.499% Mean 0.07% /6/ % 0.387% Std Dev 1.09% /5/ % 0.168% // % 0.368% /1/ % % /0/ % 0.457% /19/ % % /15/ % % /14/ % % Feb 7, 008 is the last closing price of that index fund, on that day the ETF closing price was $44.9. On the previous day it closed at $ price today - price yesterday price today Pt Weekday Return 1 1 price yesterday price yesterday A weekend or holiday involves several days, this requires us to spread the return over each of these days to keep our analysis homogenous: Closing price on Friday goes through Saturday, Sunday, and Monday. We are interested in daily returns so we have to adjust. We adjust the DAILY RETURN column based on the number of days the return represents. For weekdays the daily return equals the return. Weekend and Holiday returns are adjusted according to the above formula. Pt 1 P Weekend & Holiday Retur ( ) 1 1 t n n Pt 1 Now we calculate the parameters of the normal distribution of the daily returns by using the Excel functions 'AVERAGE' and 'STDEV'. QQQQ= N( =0.0007, =0.0109) FNCE30: Investments Lecture 6 Page 3

4 We want to calculate the 1 in 0 bad day meaning the amount I have at risk for what constitutes a bad day with only a 1/0 = 5% (or worst) chance of occurring. This is a pretty standard evaluation of exposure. A 5% bad day is not really bad, a 1 in 100 (1%) bad day would be much worst. Now we want to measure our Value at Risk The 5%-Tail of a normal distribution occurs standard deviations below the mean (we know our std dev = 1.09%, solve for position on x axis) When evaluating VAR it is common to set the mean at zero. This works because the mean over so many days will be very small compared to the standard deviation. For our example, the 5%-Tail begins at: x 1.09% = 1.69% = Therefore, for a $1 million position DEAR is: 1.69% x $1 million $17,000 DEAR: DAILY EARNINGS AT RISK We find that a 5% tail represents a loss of 1.69% which, on a million dollar investment is $17,000. So on a 1 in 0 bad day we will lose AT LEAST $17,000. The analysis could be taken a little further. We just picked off the 5% tail from one curve but a more accurate way would be to run about a hundred simulations, find the 5% point on each, and average them all together. What we have done above is measure the "top half" of the 5% bad day which is meant to give us relative exposure. Example, out of 1000 days 50 would be 1 in 5 bad days and they would be scattered along a certain range of the curve. So we would take an average of all those results. That is the most accurate way. When doing a TOP END analysis we just calculate based on one. Also, REMEMBER TO SET THE MEAN EQUAL TO 0. We are measuring the top of the tail. FNCE30: Investments Lecture 6 Page 4

5 So we have calculated that our 5% DEAR is approximately $17,000. Now we want to calculate a 5% 5-Day Value At Risk. We are asking: Over a 5 day period what is my potential 1 in 0 bad "5 day" exposure? Of course it's not just 5 times the 1 in 0 bad day risk, that would be the probability of 5 1-in-0 bad days in a row! Usually, because markets are at least weak form efficient, there is no correspondence between a gain and a loss over two given days. Each day is independent of the previous day. Without going into the reasons why the correct procedure would be: 1 in 0 worst day number of days worst 1 in 0 bad day over 5 days THIS IS THE WORST CASE 1-IN-0 BAD DAY 5-DAY SCENERIO Reasoning is based on covariance between these events being 0 therefore the standard deviation of the 5 independent variables is 5 times the individual standard deviations. The 5 day period calculation is typical but can be used for more or less days. This is an approximation of an approximation but comes out really close. Why Use the Normal Distribution The normal distribution is symmetric The probability of any positive deviation above the mean is equal to that of a negative deviation of the same magnitude The normal distribution is stable When assets with normally distributed returns are mixed to construct a portfolio, the portfolio return also is normally distributed (most distributions are not like this) Scenario analysis is greatly simplified (eg Monty Carlo) Only two parameters (mean and SD) need to be estimated to obtain the probabilities of future scenarios (modeling is easier) While asset returns are pretty much normally distributed they tend to have FAT TAILS. This means that the extremes of real asset returns tend to be pretty close to normal but do have a higher probability (the extremes) than if they were actually normal. EXTREME EVENTS HAPPEN MORE THAN THEY SHOULD! This is one of the big problems with modeling asset returns as normally distributed, this feature causes the estimates of really good and really bad days to be low (underestimated). FNCE30: Investments Lecture 6 Page 5

6 Normal & Fat Tails Distributions Mean =.1 StDev =. For asset returns the normal curve will underestimate extreme events (tail). Asset returns tend to have fat tails. NASDAQ Daily Close-Up of Tails The really bad events and the really good events happen with greater probability with the fat tail distribution. Research also shows that the low side tails are higher than the high side, so low side is skewed. FNCE30: Investments Lecture 6 Page 6

7 Histograms of Rates of Return for The FLATTER the DISTRIBUTION the greater the standard deviation which means GREATER RISK! 1, US small stocks are very flat, very risky., US large stocks not as flat as small, less risky than small. 3, world large stocks less risky than US large. 4, 5, and 6 less risky than the previous. Risk Premiums Rate of return on T-bills is essentially risk-free 9as long as you hold it to maturity). Investing in stocks is risky, but there are compensations The difference between the return on T-bills and stocks is the risk premium for investing in stocks An old saying on Wall Street is You can either sleep well or eat well. Say I want risk-free for up to 0 years, what product is available? Strip bonds (zerocoupon), they are available in up to 0 year terms. Investing in anything other than government securities is risky. But there are returns for this risk. The difference between the return on a T-Bill and a risky stock is the Risk Premium, the return in excess of the risk free rate, the difference between the FNCE30: Investments Lecture 6 Page 7

8 expected payout and the certain payout, the additional amount needed to compensate for risk. Investor Attitude Towards Risk RISK AVERSION assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities. you are risk adverse if you prefer the risk free return over the gamble. RISK PREMIUM the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities RISK NEUTRAL Indifferent to risk such as US government. A Quick Illustration Choice 1: throw a fair die, if 1, 3, 5 we get $6 million otherwise we get 0. Choice : given $1 million. GAME 1: We take choice because we are risk adverse and this is a substantial amount of money. If choice is taken away we are still willing to play, still happy but not as happy. We stand to loss nothing but do not have the option of the sure $1 million. How can we reduce risk? One way is to come together as a group and each roll, splitting our winning evenly among ourselves. As a group of 6 we have the expectation that in 6 rolls at least 1 person will win, and we will split the winning for $1 million each. GAME : Modify the game. We will each still roll the die but before we do a coin will be flipped. If the coin is heads we go forward with the game. If the coin is tails we lose, no flipping, get nothing. Is there still value in playing as a group? Yes, because if heads and we play the game we have eliminated risk by playing as a group. The risk associated with flipping the coin is called SYSTEMATIC RISK because it effects everything in the game. The risk associated with the die roll is called FNCE30: Investments Lecture 6 Page 8

9 UNSYSTEMATIC RISK. In financial markets it is called Company or Asset Specific Risk. The die roll is an example of asset specific risk, aka unsystematic risk, aka diversifiable risk, This is the part of the risk we can diversify away. The question is: How to ELIMINATE? The coin toss is an example of systematic un-diversifiable risk. The only way to avoid this risk is not to play the game (of course there is no return!). The question is: How to PRICE? Something that effects the whole marketplace (like the FED jacking up interest rates). Many forms of risk fall in between the two. Something which effects an industry or an entire country. Not quite as clear cut as market specific and asset specific. Systematic versus Unsystematic Risks What changes stock prices? How can we categorize these things? Total risk of individual security = portfolio (systematic) risk + unsystematic (diversifiable) risk News: is it market moving news or company specific news? News, new information, effects stock prices but not entirely, only about 30% of movement from news. Lead in to Diversification Consider an investment in a single stock What would be the sources of risk to this position Two broad sources of uncertainty Risk that comes from conditions in the general economy Risk that is firm-specific influences This firm-specific affect the stock without noticeably affecting other firms in the economy Some risks effect the general economy, FED actions, inflation reports, etc. There are other risks which are firm specific. The risk in a particular investment has both components, asset specific and market specific. When speaking of stocks we can also include company or firm specific risk. Market specific effect everyone to some degree. Firm specific effect only one company with no noticeable effect on other similar companies. Boils down to the idea that if you are not diversified you accept much greater risk but remember a few classes back, this risk due to failure to diversify is not compensated by FNCE30: Investments Lecture 6 Page 9

10 the market. From a diversified investors point of view most of the risk is assets specific and is therefore irrelevant. WHEN SPEAKING OF RISK IT'S ESSENCIAL TO INDICATE DIVERSIFED OR NOT! Start with a single, then ADD Another Stock For example, place half your funds in the first stock and half in a different stock. What should happen to portfolio risk? it should go down. To the extent that the firm-specific influences on the two stocks differ, diversification should reduce portfolio risk. There is asset specific risk in both stocks but they are not correlated. If something happens to the first stock which is specific to that stock it is not going to be correlated with an asset specific shock to the second stock. Therefore there will be some reduction in overall risk. The event of one stock going down in price due to an asset specific influence will usually not imply that the second stock will go down. So by adding the second stock we have reduced risk as long as long as the two are not perfectly positively correlated. Remember there is industry specific risk which can have a different effect. We will look at ways to try and reduce this form of risk using index models. Typically 0 stocks are all that is needed to create a diversified portfolio. If you are willing to invest across countries you can get an even better diversification. The question becomes, do I invest in countries and try to ignore industry or do I try to invest in a range of industries across a range of geographic regions? This is an interesting problem in diversification. So we know that portfolio risk decreases, the rate at which it decreases is dependant upon industry and that sort of thing. Two Asset Portfolio AVERAGE RETURN of a Two Asset Portfolio μ P = w A μ A + w B μ B VARIANCE of a Two Asset Portfolio: p = wa A + wb B + wa w B AB The CORRELATION is given by Covariance / (Std dev of first x Std dev of second), or AB = AB / ( * B ) The mean can be based on historical values 9average return) of a forecast. So if you have the average return of both stocks your portfolio is going to be the weighted average of the two returns. The weights are calculated according to the amount of that asset in the portfolio (by $?). Variance is the square of standard deviation and is also a way of measuring risk. This is a matter of connivance in calculations. Given that the FNCE30: Investments Lecture 6 Page 10

11 Covariance is not bounded in any way Correlation can be read in an absolute sense. Correlation is always between +/- 1. Covariance is difficult conceptualize, not so with correlation. Gives a feeling for how the two random variables move in tandem. Correlation Between Two Securities Positively correlated: 0 < AB < 1 PERFECTLY POSITIVELY CORRELATED: AB = 1 Negatively correlated: -1 < AB < 0 NOT REALISTIC PERFECTLY NEGATIVELY CORRELATED: AB = -1 Uncorrelated: AB = 0 As long as two securities are not perfectly positively correlated (r AB = 1), diversification will work. Perhaps you could find a 5 day period where two stocks have been perfectly positively or negatively correlated but anything longer than that is impractical. You will rarely find negative correlation. The majority of all stocks have a positive correlation (between 0 and positive 1). As a result we can see that diversification does lower risk in a portfolio. Portfolio Expected Return as a function of Standard Deviation Two risky assets, does not matter what they are. Here we are mapping expected return versus standard deviation. Given any return we prefer the lowest risk. If I like risk I want the highest return for a given risk. The straight black line of row=1 never happens. If the correlation is +1 you get the mix of returns but you also get the mix of risk. Once the correlation is less than +1 the line begins to bow outwards. This bowing allows us to reach a certain risk level return level while lowering our risk! This is due to the lower correlation (less than 1). The lower the correlation the more we eliminate standard deviation. At the extreme, if one asset were the FNCE30: Investments Lecture 6 Page 11

12 perfect hedge of the other, the correlation is -1. If we could find assets with a correlation of -1 we could construct a portfolio which has standard deviation of 0 (no risk). But it does not exist. Yet we still get the benefit of diversification. Perfect Positive Correlation Notice from the two asset case above And P = wa A + wb B + wa w B AB AB A B ( ) A B Therefore P = wa A + wb B + wa w B B AB So that if AB = we can factor to get: P = (wa A + w B B ) take square root of both sides Therefore P = w A A + w B B when AB = This is saying that in the correlation=1 case the standard deviations are just the weighted average, we are on the straight line and can move up and down it by adjusting W's. Therefore, the standard deviation of the portfolio with perfect positive correlation is just the weighted average of the component standard deviations, you save nothing, you share the returns and the standard deviation. In all other cases, as long as the correlation coefficient is less than 1, making the portfolio standard deviation less than the weighted average of the component standard deviations. You get the shared return but you get less than the shared standard deviation. This is why diversification works. Again this is saying that adding assets to a portfolio whose standard deviation is less that 1 will reduce risk. Always works no matter how many asstes you already have in the portfolio. FNCE30: Investments Lecture 6 Page 1

13 Less Than +1 Correlation A portfolio's expected return is the weighted average of its component expected returns, And its standard deviation is less than the weighted average of the component standard deviations So, portfolios of less than perfectly correlated assets always offer better riskreturn opportunities than the individual component securities on their own The lower the correlation between the assets, the greater the gain in efficiency Example with Negative Weights Suppose I purchase stock in a margin account using 30% borrowed funds, and 70% of my own funds which total my funds = $77,000 The margin account charges annual interest at a fixed rate of 6% pa The stock has an expected return of 1% and a standard deviation of returns predicted at 35% What is the expected return and standard deviation of returns for the portfolio? We have two assets, one negative, one positive. 77, 000 stock purchase 100% $110, My Broker Account: $77,000 + x = $110,000 Barrowed = Debt = $33,000. Return on the stock predicted to be r S = 1%. Standard deviation of stock: S = 35% Now consider that the broker, B, has a return on his investment of 6% which is risk free, r B = 6%. and B = 0%. Now construct the portfolio Portfolio assets = $77,000. Amount of portfolio invested in stock is $110,000 where $33,000 belongs to broker 110, , 000 W S =1.49 W B= 0.49 WS W B=1 so the brokers 77, , 000 money is represented by a negative weight. Now we calculate the portfolio return r PORT =WS rs WBr B= 1% + 6% 14.6% now find the variance and use it to find the standard deviation. Note that B = 0% will cause some terms to disappear. FNCE30: Investments Lecture 6 Page 13

14 35% 50% We see that P WS S WB B WSWB SD P WS S the expected return has increased from 35% up to 50%. Example with Shorted Stock I have an account in which I have assets invested in a risk free asset earning 5% pa in the amount of $50,000 I wish to sell short stock with a current market value of $0,000 The proceeds can be invested in the same risk free asset noted above The stock has an expected return of 1% and a standard deviation of returns predicted at 35% What is the expected return and standard deviation of returns for the portfolio? Risk Free, this tells us it is a T-Bill investment, standard deviation is 0. The amount of my investment is $50,000 in an account which is invested in a 1 year zero coupon bond. I wish to sell short stock with a current market value of $0,000. I will sell stock which I do not own meaning I will receive $0,000 from another investor, I will owe the stock to somebody else. I will invest the $0,000 in the same risk free asset. This means the $0,000 proceeds from the short sale will also be earning 5% per year. What is in the portfolio? The portfolio has a value of $50,000, I sell stock short for $0,000 so I owe $0,000 and also receive $0,000 but the value of my portfolio, provided the stock price does not move, is still $50,000. Now consider that our weight in the stock is NEGATIVE, we have sold it short. At today's market value I owe somebody $0,000 worth of stock, I show this as negative weight: W S =0.4 W D= 1.4 WS W D= Now the return on portfolio is 0 70 R =0 F R =0 F D=0 S =0 r D PORT=WS rs WR r F R = 1% + 5%.% F We know the last two terms in the standard deviation calc are 0 so we do not bother to write them. We have a negative weight so we use the variance calc to calculate the 0 0 standard deviation: P ( ) (35%) 35% 14% We use the magnitude calculation to remove the negative sign. FNCE30: Investments Lecture 6 Page 14

15 Statistics of K Asset Portfolio Expected Return of a Portfolio containing K asstes assets: = w 1 μ 1 + w µ + + w K μ K Variance of a Portfolio p = w1 1 + w + + wk K + w1 w 1, + w 1 w 3 1,3 + + w 1 w K 1,K + w w 3,3 + w w 4,4 + + w w S,K + + w K-1 w K K-1,K (can also be written in covariance matrix form with the variances along the diagonal) Variance or Covariance? Suppose we have N securities each with the same variance and the same covariance with the others. Suppose we form an equally weighted portfolio of these securities. Therefore W 1 = W = = W N (weight of each is the same). The variance of the portfolio will be given by = w + w + + w + w w + w w + + w w p 1 1 S S S w w w 1S + w w w w S S + + w S-1 S S-1S (N total number of terms) There are N equal variance terms, call them var, and N N covariance terms, call them covar. Therefore. 1 var var 1 covar 1 1 P N N N covar N N N N So as N gets bigger (meaning we add stocks), the individual variance of a stock becomes less and less important (smaller) leaving only the average covariance. The covariance term dominates. Notice, if we assume var is the same for each stock, as is the correlation coefficient, then covar = * var, and var 1 P 1 var N N Result: when I diversify the standard deviation of an individual stock is not important. It's that stocks covariance with the other stocks in the portfolio, this is the portfolio risk. FNCE30: Investments Lecture 6 Page 15

16 Variance FNCE 30, Investments H Guy Williams, 008 If using average variances the covariance is equal to the correlation times the variance if we are using average variances, substitute covar = * var then if the stock has correlation of zero (then the portfolio now has a zero standard deviation (only left with 1 times the correlation times the variance). if corel=0 can remove risk. Variance or Covariance The next chart was constructed in the following way. Select the Dow 30 monthly returns for last 15 years Form portfolios containing the first security, then the first two, and so on. On average we have standard deviation of the individual security is 8.4%. On average we have the xy, (sqr root of covariance) of the individual security is 4.9%. Variance of 1 stock from the DOW30 Asymptotic trend line is the systematic risk var 1 P 1 var N N Number of Securities Randomly picked on of the set of DOW30 stock and ran a simulation, then ran again with another randomly picked stock, repeat The first stock represents a portfolio of 1 asset and has high variance. as we randomly add stocks to the portfolio the variance of the portfolio declines sharply. By the time you get to 15 stocks the transaction costs will eat up the reduction in risk. The value you ultimately trend to is the average covariance, that value is the lowest a particular set of stocks can go. Keep increasing the number of stocks in each portfolio (which is a point on the graph) until you reach the last data point which is a portfolio with 1000 stocks. As we do this we find that the covariance becomes important, not the variance. This is the reason why diversification works. The points trend to an average value of FNCE30: Investments Lecture 6 Page 16

17 4.9%. But we can see we get most of the benefit at 5 stocks! At 0 stocks there is no further gain to adding to the portfolio, no more benefit of diversification. This plot is the graphical form of the equation. Systematic versus Unsystematic Risks What changes stock prices? News, information, new data. How can we categorize these things? Portfolio risk which is systematic, market risk which is not diversifiable, unsystematic risk which is the trend line on previous page. Total risk of individual security = portfolio (systematic) risk + unsystematic (diversifiable) risk Portfolio Risk as a Function of the Number of Stocks in the Portfolio This difference is the company specific risk, 15 to 0 stocks kills off company risk. This is the MARKET RISK area. Cannot eliminate. This is the average covariance limit. AKA: Asset Specific Risk, specific to the firm. Adding stocks to a portfolio in a random manner. We start with a high standard deviation but as we add stocks std dev declines until it reaches the MARKET RISK limit. = Total Risk = Company Specific Risk + Market Risk (disappears as we add more stocks) this is The portfolio risk is equal to the square root of the average covariance. FNCE30: Investments Lecture 6 Page 17

18 Failure to Diversify If an investor chooses to hold a one-stock portfolio (exposed to more risk than a diversified investor), would the investor be compensated for the risk they bear? NO! Stand-alone risk is not important to a well-diversified investor Rational, risk-averse investors are concerned with σp, which is based upon market risk There can be only one price (the market return) for a given security NO COMPENSATION SHOULD BE EARNED FOR HOLDING UNNECESSARY, DIVERSIFIABLE RISK. This investor could earn the same expected return at a lower level of risk. This investor takes on more risk. No. The market prices risk from the lowest level and the lowest level is from the perspective of a diversified investment. The market expects investors to hold a well diversified portfolio, that is how it measures risk. The market is interested in the covariance's. The well diversified investor is in a stronger position therefore he is the one who dictates prices in the market. This is not the same as an investor who chooses to be leveraged, that is a different circumstance. What type of investor is forced to be undiversified? This can happen to a CEO, the investors will force the person to take a large percentage of compensation in stock because they want his interest aligned with those of the company and investors. They are compensated for this by being sold the stock at a low price. Can be below the market price. FNCE30: Investments Lecture 6 Page 18

19 The Minimum-Variance Frontier of Risky Assets Expected return vs standard deviation, the dots are individual assets. If we form different portfolios of these assets, because the correlation is not +1 they form little curve shaped portfolios, then you form portfolios of portfolios and migrate outward. Eventually you can't do it anymore. So you reach the limit of how much diversification you can achieve (you can run out of assets). If another stock goes public, because its correlation is less than +1 the whole line will shift by a small amount (as it will for each asset added). The boundary is all of the possible investments you can make when at the limit, it is the minimum variance frontier, the horizon. Since we are considering expected return to risk the only place we would want to make an investment is along the EFFICIENT FRONTIER which is the positive sloping portion of the frontier. Any other point would be sub-optimal. Want the lowest possible risk for the desired return. GLOBAL MINIMUM VARIANCE PORTFOLIO is the portfolio with the lowest possible variance. Why is this important? Later when we form tangential portfolios (because we believe in a CAPM world) the GMVP will be the optimal risky portfolio. UNSTABLE If you do this in practice using historical data the optimal point (tip) is very stable. But the efficient frontier is very unstable, it will move from year to year. Very STABLE FNCE30: Investments Lecture 6 Page 19

20 Expected Return FNCE 30, Investments H Guy Williams, 008 Foreign Investment Example I have taken monthly returns for the SP500 (used as index of US stock markets) and a Swiss market index for the last 1 years ( asset portfolio) I formed portfolios with varying weights in each asset. The below table shows the results of this analysis. 50% invested in the S&P500 and 50% in the Swiss index. Weight E{r} Std Dev Weight E{r} Std Dev -50% 1.19% 6.35% 50% 0.95% 4.14% -40% 1.17% 6.05% 60% 0.9% 4.07% -30% 1.14% 5.76% 70% 0.90% 4.04% -0% 1.1% 5.49% 80% 0.87% 4.04% -10% 1.09% 5.3% 90% 0.85% 4.09% 0% 1.07% 4.98% 100% 0.83% 4.17% 10% 1.05% 4.76% 110% 0.80% 4.9% 0% 1.0% 4.56% 10% 0.78% 4.43% 30% 1.00% 4.39% 130% 0.75% 4.61% 40% 0.97% 4.5% 140% 0.73% 4.8% 1.% 1.1% 1.0% 0.9% 0.8% 0.7% 0.6% 100% invested in the S&P500 and 0% in the Swiss index. All other results are a mix. 100% invested in the S&P500 and 0% in the Swiss index. Notice that the S&P500 gives less return than the Swiss index but the Swiss index carries a lot more risk (less stocks, more volatile). So the higher return compensates for the riskier Swiss market. Below we have the actual plot of this efficient frontier. Notice S&P500 gives us a return of.83% per month with a standard deviation of 4.17%. So why do investors only invest in the US market? It makes no sense, even if you just look at the Swiss market mixed with the S&P500, you are already on the lower limb. We haven't even considered other foreign markets. The point is that just adding one foreign market allows us to increase our return from.83% to close to.95% with roughly the same risk! 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0% 6.5% 7.0% Standard Deviation FNCE30: Investments Lecture 6 Page 0

21 Over 1% additional return for the same risk (monthly)! We can imagine that our position would improve drastically if we added more countries. Now we will add a risk free asset into the mix. A risk free asset will have 0 standard deviation and some sort of positive return. We will look at how we combine risk free with risky. Zero coupon government bond is risk free. (see below) Riskless Borrowing and Lending Suppose we form portfolios of a risky asset (say a stock) and a riskless asset (zero-coupon government bond). Suppose we invest with weights W and 1-W (where w is the weight in the stock). The EXPECTED RETURN of the portfolio is: (this is true for any combination of risky and risk free asset) [weight * expected return] stock + [weight * expected return] risk free Note: risk free expected return, μ f = actual risk free return, R f : no risk! μ P = W * μ S + (1 - W) * μ f = R f + W * [μ S R f ] This term is the Risk Premium The RISK PERMIUM is the return on the risky asset over and above the return on the risk free asset. The STANDARD DEVIATION of the portfolio is: 1 P w w w 1 w S f Sf The standard deviation of the risk free asset 0 (as is covariance). Therefore we are left with: Combining the two we have: P P = W * S R P f [ S R f ] S This is a linear equation, the mean of the portfolio is a function of the standard deviation of the portfolio. The standard deviation and the mean of this portfolio are described by a straight line. The line which is tangential to the efficient frontier provides investors with the highest return. FNCE30: Investments Lecture 6 Page 1

22 Riskless Borrowing and Lending This is the equation of a straight line. An investor can combine the riskfree asset with any risky asset in the opportunity set. However, the line that is tangent to the efficient set of risky assets provide investors with the highest return at any given standard deviation. The Investment Opportunity Set with a Risky Asset and a Riskfree Asset in the Expected Return-Standard Deviation Plane P R P f [ S R f ] S This region is over invested on margin. This line describes all of the different portfolios that can be constructed from this risky and risk free pair of assets. Point P is 100% invested in the stock. Point F is 0% invested in the stock (taking only the risk free return?). Line is called CAL: Capital Allocation Line The SHARPE RATIO or SHARPE INDEX or SHARPE MEASURE or reward-tovariability ratio is a measure of the excess return (or Risk Premium) per unit of risk in an investment asset or a trading strategy. Since its revision made by the original author in 1994, it is defined as : where R is the asset return, R f is the return on a benchmark asset, such as the risk free rate of return, E[R R f ] is the expected value of the excess of the asset return over the benchmark return, and σ is the standard deviation of the excess return (Sharpe 1994). Note, if R f is a constant risk free return throughout the period, Sharpe's 1994 revision acknowledged that the risk free rate changes with time. Prior to this revision the definition was assuming a constant R f. FNCE30: Investments Lecture 6 Page

23 EXAMPLE: Two Risky and One Riskless E Suppose we wish to invest in the S&P 500 ETF and the Nasdaq QQQs We also have available a risk-free asset paying an annual return of 5% The objective is to maximize the slope of the CAL (capital allocation line) for any possible portfolio p We will put together a portfolio of two assets and a risk free asset. This is a two step problem: 1. find the optimal risky portfolio. combine that with the risk free asset. We will form a portfolio in the two assets: S&P500 ETF and QQQQ ETR (both mimic their respective indices). We also have available the risk free 5% per year. So 3 assets total. Objective is to form an optimal portfolio for any given risk level. What is the optimal portfolio? The optimal portfolio is found by maximizing the slope of the CAL (capital asset line) also known as the Sharpe Ratio function: The Sharpe Ratio divides the risk premium by the standard deviation. It give us risk premium per unit of risk. This can be done for a portfolio, a single asset, or many things. The Sharpe ratio gives us the slope of the CAL, the steeper the p slope (bigger) the more return I get per unit of standard deviation. Given a choice an investor always wants the CAL with the steepest slope (largest Sharpe Ratio). So we maximize this eq. r p r f We begin by solving the problem for the risky portfolio. Look at the two risky assets (ignore risk free for now). If I combine the two risky assets the expected return of the risky portfolio is the weighted average of the risky returns. The risky stdev is similar. E{r P }= w A E{r A } + w B E{r B } P = wa A + wb B + wa w B AB Where W A + W B = 1 (For the risky portfolio considered on its own) For this example instead of solving algebraically we will plug in values from 60 months of returns for the S&P500 and QQQQ (below). FNCE30: Investments Lecture 6 Page 3

24 Data S&P500 QQQQ Mean 0.846% 1.07% Var StDev.60% 4.50% Cov Correl Jan % -.0% Dec % % Nov % -0.10% (partial list of points) (stdev of NASDAQ x S&P) As a practical matter this value of correlation is the closest to 1 we should expect to see. Now for the risky portfolio we have (substitute in the values): E{r P }= w A E{r A } + w B E{r B } = w A w B P = wa A + wb B + wa w B AB P = wA wB + x wA w B Now substitute in for W = 1 W B A E{r P }= W A (1 W A ) = W A p = WA WA Thus we need to maximize Sharpe Ratio: W WA WA A Take the derivative wrt W A and set equal to zero FNCE30: Investments Lecture 6 Page 4

25 The general solution to Maximize the Sharpe equation is: W A A f B B E r r E r rf AB B B B E r r E r r E r E r r A f f A A f AB Therefore, W A = 60.4%, and so W B = 39.6%, this is the best mix of risky assets to give us the maximum slope. W A = 60.4% W B = 39.6% Tangent Point Steepest line we could get We invest along this line to maximize Risk return Free 5% per year = 0.416% per month Portfolio of NASDAQ and S&P500 at various Ws This is the portfolio of the S&P500 and NASDAQ at the weights we have solved for. Our solved for tangential maximum slope line dominates every other possible investment combination of these three assets (including risk free). The risk free intercept is at 5%/1= =0.416%. (why is risk free expressed as a monthly amount?) GENERALIZE TO MANY SECURITIES Generalize the portfolio construction problem to the case of many risky securities and a risk-free asset Identify the risk-return combinations available from the set of risky assets. This means establish the covariance/variance matrix plus all of the returns of the stocks. Identify the optimal portfolio of risky assets by finding the portfolio weights that result in the steepest CAL meaning maximize the Sharpe Ratio for all of the possible portfolios. Choose an appropriate complete portfolio by mixing the risk-free asset with the optimal risky portfolio, this means along the tangential line. FNCE30: Investments Lecture 6 Page 5

26 Capital Allocation Lines (CAL) with Various Portfolios from the Efficient Set In this case CAL(P) is the risky portfolio with the highest Sharpe Ratio. Often people like CAL(G) because it is more stable, but it is not optimal! Separation Property Portfolio choice problem maybe separated into two independent tasks Determination of the optimal risky portfolio The best risky portfolio is the same for all investors, regardless of risk aversion Allocation of portfolio to T-bills versus the risky portfolio Depends on personal preference of each investor There is more than 1 separation property in finance, this is just one of them. This Separation Property is saying that when you invest you can break the investment problem into two pieces: 1. Find the optimal risky portfolio. This will be the best risky portfolio and it is the same for all investors. This is because what we are finding is P and CAL(P). No investor would not want P if able to accurately predict the future. Because any combination of risk free asset with P is going to dominate everything else. Every investor should identify the same P, should agree if they have consistent believes.. Once you've established the P you can decide on the level of risk you want, less or more, by mixing this (part 1) portfolio with the risk free asset. If you want less risk include more government bonds and less risky asset. If you want to be fully invested in only stocks you may prefer to be at P. If you want to take on even more risk you can invest on margin through an account with a broker. Optimal risky portfolio is the same for every investor. Deciding where to settle your investments on the line is up to the risk tolerance of each investor. FNCE30: Investments Lecture 6 Page 6

27 Problems with this Approach Problem with data collection (there is so much) And mathematical analysis of a dataset that large is daunting To build the efficient frontier means you must construct the variance/covariance matrix of all assets which you could invest in. If there are 5000 assets in the US market (there are actually more) you will end up with a matrix which is 5000x5000. This is a 5 million entry covariance matrix calculation. Then you have to invert the matrix. So it sounds easy in theory but is tough to do in practice. But you can also generate random portfolios and simulate. Requires a couple of million attempts but you may then have an approximation of where the efficient frontier resides. So in practice there are too many data points. This will transition us to CAPM and then some more exotic ways of pricing risk such as arbitrage pricing theories. Advantages of the Single Index Model Reduces the number of inputs for diversification Easier for security analysts to specialize (We will look at index models. Not as accurate but maybe good enough.) This stuff dates back to the 60's. Calculating portfolios in a reward / standard deviation framework, understanding how to find the tangential portfolio, and then calculating a portfolio of risk free assets with the risky tangential portfolio. Pure mathematics. Issues: There are problems with it. Looking at historical data, may not be a good picture. There is also a problem with data collection. To really calculate the tangential portfolio you will have to solve the variance/covariance matrix. Dimensions will be huge, getting better though as computers become faster. But keep in mind when doing this you should be considering the global financial market, so there are even more securities, larger matrix to solve. Huge data collection exercise. At least 5000 assets traded in the US alone. It is tough to do but could be done in theory. To counter this we will look at some other ways to measure risk and return. Will look at index models (which are kind of a short cut) and then CAPM. Later we will consider if arbitrage models might be more useful in actually allocating investments to a portfolio. FNCE30: Investments Lecture 6 Page 7

28 Reduces the number of inputs needed for diversification. Instead of needing 5000 calculations we would only need This Index form is also easier for security analysis to specialize, good for industry analysis. Problem is that with a single index model you may miss some risk. Saving on calculations but may miss some diversification ability. Single Factor Model We can decompose the rate of return on any security, i, into the sum of its expected plus unanticipated components (unknown) r i = E(R i ) + e i where the unexpected return, e i has a mean of zero and a standard deviation of i which measures the uncertainty about the security return. What you expect to get plus the uncertainty. The unexpected is the error term has a mean of zero. On average its value will be 0. We give it a standard deviation of i, it measures the risk or uncertainty of the unexpected term. But how do we calculate the expected term? We will see but the point is that we have created a framework to study how we can break down risk into systematic versus unsystematic within an INDEX MODEL. now Suppose there is a common factor that drives variation in security returns, some macroeconomic variable, could be GDP, inflation, Interest Rates, some macro thing we call this macroeconomic risk m m affects all firms though not by the same degree. Some firms are more effected, some less. Some of the firms will have bigger betas and some will have smaller betas. (we are using this in a very broad sense, not in CAPM sense. Beta can be used in any regression setting.) Then we can decompose the sources of uncertainty into uncertainty about the economy as a whole which we capture by the factor m which is the SYSTEMATIC RISK, cannot avoid. And uncertainty about the firm in particular is captured by the term e i, the UNSYSTEMATIC RISK, this we can avoid. we can amend the above equation to get FNCE30: Investments Lecture 6 Page 8

29 Single Factor Model Equation r i = E(R i ) + ß i F + e i ß i = index of a securities particular return to the factor [UNCERTAIN] e i is the firm specific uncertainty.(risk) F= global, macroeconomic risk; in this case F is unanticipated movement; F is commonly related to security returns. [PREDICTABLE] Assumption: a broad market index like the S&P500 is the common factor, it acts as a proxy to the macro factor F. The return on any particular security (using this single factor model) is going to be what we expect to get plus uncertainty about the global factor (such as GDP surprises, interest rate surprises, etc.) plus asset specific factors (firm specific, Ford with exploding gas tanks). Risk Single Factor Model In the equation: r i = E(R i ) + ß i F + e i. E(R i ) is constant, no uncertainty. and F and e i are independent (uncorrelated, covariance=0) this makes measuring the variance of these two terms much easier: Therefore all of the uncertainty is in these terms, the risk e risk of 1 st + risk of nd i i F i This equation is telling us that the systematic risk of security i is determined by its beta, β, coefficient. The higher the beta for a particular company the greater that companies systematic risk. "Cyclical" firms have greater sensitivity to the market/macro economic factors and therefore higher systematic risk, β. If we add enough firms to the portfolio the risk associated with e i (its σ in the above eq) will shrink to zero. We will be left with only β which is non-diversifiable. Another advantage of this formulation is that if we want to measure the covariance between two assets. (next page ) FNCE30: Investments Lecture 6 Page 9 The portion of the realized return outside of E(R i ) comes from the uncertain components.

30 Covariance Single Factor Model The covariance between any pair of securities also is determined by their betas, and only by their betas. Cov ri, rj Cov ifei, jfej i j F Why is this? Well the two terms (red and blue above) are independent and have no correlation, no covariance. So we are left with only the covariance of the first two terms. The result is the product of the betas and the variance of the factors. This makes calculations much easier. Note that using this "Single Factor Model" we are able to avoid the 5000x5000 matrix calculation of the full bore efficient frontier of all firms in the marketplace approach. This is a nice short cut. So how do we implement one of these factor models?... Regression Equation: The Single-Index Model Approach to making the single-factor model operational Assume the rate of return on a broad index of securities such as the S&P 500 is a valid proxy for the common macroeconomic factor. (using S&P500 is pretty standard but it is not absolutely necessary, we could use other indices. The point is that the macro factor is definitely going to affect a market index. We use the market index to track the market factor, we know it will do so pretty well.) We will run a regression comparing the premium on our asset to that of the risk premium of the market index. Denote the market index as M (macro influence) Excess return = actual - risk free = R M = r M r f aka: Risk Premium Standard deviation of M will run a regression comparing the premium on my asset to the risk premium on the market index. The model will return β which is the slope. Note the error term over the aggregate of the regression will equal zero. Something will happen with alpha too. Regress the excess return of a security R i = r i r f Against the excess return of the index R M The regression equation/model is: R i (t) = i + i R M (t)+e i (t) FNCE30: Investments Lecture 6 Page 30

31 EXAMPLE JP Morgan Year- Month Monthly T-Bill Rate annualizied R f Annual/1 S&P500 Total Return that Month Single Factor Model Return Excess Return aka risk premium Regression of excess return (?) R f (%) Annual r f r M r i R M R i % -6.1% 9.54% -6.34% 9.3% % -0.86% -4.31% -1.10% -4.54% % -4.40% -.94% -4.71% -3.4% % 1.48% 3.40% 1.17% 3.09% % 3.58%.9% 3.7%.6% % 1.9% 1.16% 0.94% 0.81% % -3.0% -8.44% -3.59% -8.83% We take monthly data, 60 months, 5 years worth of data. calculated the risk free rate on an annual basis using the T-Bill rate from the Federal Reserve, it is the monthly T-Bill rate annualized. Federal reserve gives you annual rates, it takes the monthly yield and multiplies it by 1. One month T-Bill is a pretty good proxy for the risk free rate for a one month period. r M : return on market, this is the S&P500 total return for that particular month. r i : return on the asset under consideration. R M : risk premium=actual value risk free rate. r M r f = = -6.34% R i : regression of excess return, = r i - r f = i + i E{R M } (on next page) α = β = We run the regression in Excel. In this case the explained variance (sqr of stdev) works out pretty well. The significance value indicates that this index model is explaining a great deal, says the relation is not just by chance. The important thing is INTERCEPT = 0.7% (pretty close to zero) and the beta (listed as RM in the table) of β = So in the regression model for this single index model for JP Morgan has a beta of This is important because the beta for the market as a whole is 1 and therefore the model is telling us JP Morgan is slightly riskier than the market and for this additional risk we are paid an additional 8% risk premium. FNCE30: Investments Lecture 6 Page 31

32 The result is significant, we can see this from the F value, which is not entirely unexpected because it is a one factor model. JP Morgan: α = Intercept =.754%, and β = RM = It is very simple to implement this single factor model, could do it for every stock in the marketplace as long as you have access to the data bases. Now we want to consider from the expected return viewpoint The Expected Return-Beta Relationship Because E{e i } = 0 we get E{R i } = i + i E{R M } Second term is the part of a security's risk premium is due to the risk premium of the index The market risk premium is multiplied by the relative sensitivity or beta, of the individual security This is the systematic risk premium What we are doing here is taking the expected value of the regression equation (using the alpha and beta we solve for). Because this equation uses the alpha and beta which came from a regression analysis it, the equation using these terms, will have an expected value of zero. E(R M ) = the expected value of the market risk premium. E(R M ) measures the systematic risk of the security. Note: expected value of regression equation is zero because the best fit of the line to the data points is accomplished with a least squares fit. This means that the average value of the difference between the line and the points is, by the very calculation, zero! The Expected Return-Beta Relationship Alpha is a nonmarket premium For example, α may be large if you think a security is underpriced and therefore offers an attractive expected return. When security prices are in equilibrium, such attractive opportunities ought to be competed away, in which case α will be driven to zero (this is the arbitrage being priced out). Alpha would be zero if the market were perfectly efficient (strong form), the alphas would be arbitraged out of the market. Alpha represents an extra unjustified return, this would drive the price of the stock up until the premium was priced out. The exception would be if your model is in some way mis-specified. A positive value for alpha tells us we are getting an extra return over and above the risk associated with the market. Stock analysis are always looking for this positive alpha stock. Ideally we want all positive alpha stocks, or could basis portfolio to hedge. But does it really happen? This is the question of efficiency. Would require a stock which all other security analysis had missed! This is very unlikely. The firm Janus use to say they could find these stocks, hasn't worked. They are still a typical return firm. They may be out there but hard to find. FNCE30: Investments Lecture 6 Page 3

33 Return and asset value: the lower the asset value the higher the return I am going to earn. In an ideal world there should be few positive alpha stocks. If there were many positive alpha stocks in the market mutual funds would always outperform the market (which they do not). Diversification Part The CAPM, Chapter 9. APT, Chapter 10. APT is difficult and tricky. What is it telling us? We are interested more in the intuition it gives us. Maybe we can break down the market risk into components? We need to understand how APT gives us insight into how things work. Assumptions CAPM Individual investors are price takers, they have to take the price given them. The flip of this is no individual investor can alter the market based on their trading alone. (There are research results which show that even large block trading cannot effect the price very much). Single-period investment horizon; we can only model one period into the future. Idea is if we only invest for one period the risk free rate is fixed and inflation will not kill us. Multi-period investing has reinvestment risk and inflation risk. Investments are limited to traded financial assets, when we select the optimal portfolio it is an optimal portfolio of traded assets. Some people try to extend CAPM to non-traded assets. No taxes and transaction costs, not realistic but the model can be extended to take these costs into account. Information is costless and available to all investors Investors are rational mean-variance optimizers There are homogeneous expectations, everyone believes the same thing about the variance / covariance matrix and the expected returns of assets. Resulting Equilibrium Conditions All investors will hold the same portfolio for risky assets market portfolio Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value FNCE30: Investments Lecture 6 Page 33

34 Same Risky Portfolio All investors will desire to hold identical risky portfolios If all investors use identical Markowitz analysis (assumption 5) Applied to the same universe of securities (Assumption 3) For the same time horizon (Assumption ) Use the same input list (Assumption 6) They all arrive at the same composition of the optimal risky portfolio Saying all investors use the same mathematical analysis to estimate the efficient frontier. This implies that if the weight of GE stock, for example, in each common risky portfolio is 1%, then GE also will comprise 1% of the market portfolio The same principle applies to the proportion of any stock in each investor's risky portfolio The optimal risky portfolio of all investors is simply a share of the market portfolio Now suppose that the optimal portfolio of our investors does not include the stock of some company, such as Delta Airlines When all investors avoid Delta stock, the demand is zero, and Delta's price takes a free fall As Delta stock gets progressively cheaper, it becomes ever more attractive and other stocks look relatively less attractive Ultimately, Delta reaches a price where it is attractive enough to include in the optimal stock portfolio Such a price adjustment process guarantees that all stocks will be included in the optimal portfolio It shows that all assets have to be included in the market portfolio Resulting Equilibrium Conditions RISK PREMIUM on the market depends on the average risk aversion of all market participants RISK PREMIUM on an individual security is a function of its covariance with the market. This is saying it is dependent on beta, β. Not dependent on standard deviation, not dependent on asset specific, not dependent on total risk. How the individual asset contributes to the overall risk of the marketplace, its covariance. Return and Risk For Individual Securities The RISK PREMIUM on individual securities is a function of the individual security s contribution to the risk of the market portfolio An individual security s RISK PREMIUM is a function of the covariance of returns with the assets that make up the market portfolio FNCE30: Investments Lecture 6 Page 34

35 The CAPM β Calculation The contribution of a security to the risk of a well-diversified portfolio is proportional to the covariance of the security's return with the market s return (this is the regression method above, also included the R i eq) This contribution is called the beta The CAPM states that the EXPECTED RETURN on a security is positively related to the security s beta R i : return on investment R F : risk free β i is the CAPM β R M : market return Well-Managed Firms R M -R F = Market Premium Cov( Ri, RM ) i ( R ) Ri R β ( RM R ) F Can WELL-MANAGED firms will provide high rates of return? True if one measures the firm's return on assets i The CAPM, however, predicts returns on investments in the securities of the firm Suppose our firm has superb management, does everything right. Yes, they can make excess returns if you measure their returns compared to the assets they control. What CAPM says is that the returns on the firms stock is driven by the equation above. So superb management generating excess return from the companies assets, the stock price will be driven so high that it (the price) is driven back to equilibium and the expected return on the stock is still going to be driven by CAPM. An efficient market argument, high return on good news but after that jump back to a CAPM based return from then on. Suppose that everyone knows a firm is well run Its stock price will therefore be bid up Consequently returns to stockholders who buy at those high prices will not be excessive (just earn a CAPM based return). Excess returns from the stock have been instantaneously rung out of the market. SECURITY PRICES, in other words, already reflect public information about a firm s prospects. F M FNCE30: Investments Lecture 6 Page 35

36 The Security Market Line EXPECTED RETURN versus BETA RELATIONSHIP is a reward-risk equation The beta of a security is the appropriate measure of its risk because beta is proportional to the risk that the security contributes to the optimal risky portfolio Risk-averse investors measure the risk of the optimal risky portfolio by its variance (in this case variance has removed all asset specific risk). Variance only measures the market risk of that portfolio. The reward, or the RISK PREMIUM on individual assets depends on the contribution of the individual asset to the risk of the overall market portfolio. Can measure in terms of the covariance or with beta directly. Capital Market Line, CML Security Market Line, SML Security Market Line measures beta against return. How is this different from the Capital Market Line? The CML has return vs standard deviation, risk! The Security Market Line The beta of a stock measures its contribution to the variance of the market portfolio. The CAPM states that the security's risk premium is directly proportional to both the beta and the risk premium of the market portfolio. The EXPECTED RETURN versus BETA relationship can be portrayed graphically as the security market line (SML) shown above. The beta of the overall market portfolio is 1. The expected return of a beta equals 1 portfolio is the expected return of the market as a whole (such as the S&P500). CAPM tells us for a given beta exactly what return we should expect to earn for any particular asset. FNCE30: Investments Lecture 6 Page 36

37 Alpha The difference between the fair and actually expected rates of return on a stock is called the stock's alpha Security analysis is about uncovering securities with nonzero alphas, stocks with positive alphas, additional returns. The starting point of portfolio management can be a passive market-index portfolio The portfolio manager will then increase the weights of securities with positive alphas and decrease the weights of securities with negative alphas The SML and a Positive-Alpha Stock Looking for stocks which reside above the security market line. The additional return (beyond that predicted by CAPM) is the alpha of the stock. Stocks with positive alphas are undervalued stocks. Getting more than you should based on CAPM. CML vs. SML The CML graphs the risk premiums of efficient portfolios as a function of portfolio standard deviation i.e., portfolios composed of the market and the risk-free asset Standard deviation is a valid measure of risk for efficiently diversified portfolios The SML graphs individual asset risk premiums as a function of asset risk The relevant measure of risk for individual assets held as parts of welldiversified portfolios is not the asset's standard The SML is valid for both efficient portfolios and individual assets Usually you would compare the Capital Market Line to a efficient diversified portfolio. FNCE30: Investments Lecture 6 Page 37

38 Problems with the CAPM Beta is not much use for explaining rates of return on firms' shares There appears to be another measure which explains these returns quite well such as BOOK VALUE to its MARKET VALUE. o On average, companies that have high book-to-market ratios tend to earn excess returns over long periods, even after adjusting for the risks that are associated with beta. Portfolios of high book-market ratio seem to earn greater returns, seem to have positive alphas. This is unusual, in efficient markets this should not be the case. The alpha value, if it exist for a particular firm, must be more than the transaction costs. On the other hand beta does not seem to explain a whole lot. People suppose the traded assets are not enough, need human capital, need non-traded assets. Also argue that we have not been around long enough to measure beta. Some say there is just too much noise in the market for any given stock, beta may be working but we cannot hear it. The CAPM and Reality Is the condition of zero alphas for all stocks as implied by the CAPM met? Not perfectly but one of the best available. Seems to be some abnormalities where positive alpha portfolios of stocks. Is the CAPM testable? Can we determine if it is testable? Proxies must be used for the market portfolio (eg S&P500). Most test of CAPM are really test of weather or not you are using the correct market portfolio. CAPM is still considered the best available description of security pricing and is widely accepted. It just does not do a great job of explaining risk in certain situations. Inter-Temporal Capital Asset Pricing Model Individuals optimize a lifetime consumption/investment plan Additional sources of risk when moving away from 1 period model Future risk-free rates, investment risk, reinvestment risk. Inflation risk, if faster than you had planned. Hedge this extra-market risk by forming other portfolios CAPM originally conceived as a single period model. People has extended it over the years, in this case it is extended to an inter-temporal model, looking at FNCE30: Investments Lecture 6 Page 38

39 more than one period, maybe over an entire lifetime. How can we measure risk and construct well diversified portfolios? (continued ) What Merton said is instead of constructing well diversified portfolios in a beta CAPM sense, may want to construct portfolios which move with the risk free rate. If risk-free rates go down and your portfolio suffers, you may have done better to construct part of your portfolio in assets which move in opposite directions to the risk free rate. Hedging against the risk-free rate falling over and above what we could do in the CAPM sense. (Nothing to do with beta here, we are outside the single period world). Inflation risk, you may not earn the return you were expecting if inflation increases. But you may be able to hedge inflation by incorporating stocks which outperform when inflation goes up. For example, what is a good hedge for oil prices? How can I set aside some of my investments so that I can hedge oil price risk? We want an asset which has a positive correlation to oil, it will increase when oil prices increase, will give us additional returns. Maybe go long in Exxon, buy gold, buy natural resource firms, buy natural resource ETFs. Allocate some of your portfolio to assets which go up in price when oil goes up in price. (petroleum based products such as plastics). CAPM wouldn't tell us to do this. CAPM knows nothing of these risks. In a CAPM world you can be as well diversified as you like but inflation can still hurt you. So measuring these other kinds of risk outside the CAPM world will allow us to allocate some of our portfolio to reduce the additional risks. To reduce this risk (inflation) I may have to reduce my overall return, but that reduction in overall return may be worth it (to a particular investor). The Story so Far Returns on securities are variable This variability is measured by STANDARD DEVIATION Returns on securities are INTERDEPENDENT The interdependence is measured by COVARIANCE and by CORRELATION This interdependence leads to a number of interesting results Diversification can eliminate some, but not all, risk Capital asset pricing model (CAPM), tells us how to measure risk (beta). Assets together in a portfolio reduces the overall standard deviation of the portfolio more so than just spreading the standard deviation across securities. As long as the correlation is less than +1 you can achieve some reduction of risk by adding the security to a portfolio. Now we are examining the question: Does CAPM capture all the risk? It may not. Maybe there are other risks we may want to hedge. Maybe there are other ways to construct portfolios with more efficient returns. This is APT FNCE30: Investments Lecture 6 Page 39

40 APT: Arbitrage Pricing Theory Announcements, Surprises, and Expected Returns The return on any security consists of two parts First the EXPECTED returns Second is the UNEXPECTED or risky returns A way to write the return on a stock in the coming month is: R R U where R is the expected part of U is the unexpected part of Any announcement can be broken down into two parts, the anticipated or expected part and the surprise or innovation: Same idea, for a given security I should be able to break down the returns into two pieces: the returns I expect to earn, E(R) calculated using the CAPM framework, and the surprises! Things we were not expecting. News changes stock prices, that is the uncertainty. Announcement = Expected part + Surprise The expected part of any announcement is part of the information the market uses to form the expectation, R of the return on the stock The surprise is the news that influences the unanticipated return on the stock, U Single Factor Model Equation R i = E(r i ) + Beta i (F) + e i the return the return R i = Return for security i Beta i = Factor sensitivity or factor loading or factor beta F = Surprise in macro-economic factor (F could be positive, negative or zero) e i = Firm specific events Return on security = expected + macro + firm specific FNCE30: Investments Lecture 6 Page 40

41 Arbitrage Pricing Theory (APT) APT relies on three key propositions (ASSUMPTIONS): Security returns can be described by a factor model, must believe that there are some drivers in the marketplace which affect all stocks to some degree and it can be measured. There are sufficient securities to diversify away idiosyncratic risk, enough securities in the market to allow us to get as close to zero risk as we like. If you look behind APT this is how it works. Well-functioning security markets do not allow for the persistence of arbitrage opportunities. Not as many assumptions. APT says you give me an error rate and I can get as close to that error rate as you want no matter how small it is as long as I have sufficient securities. idiosyncratic risk means e i 0. Arbitrage Pricing Theory Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit (no risk return for no investment). Since no investment is required, an investor can create large positions to secure large levels of profit. In efficient markets, profitable arbitrage opportunities will quickly disappear Law of One Price If two assets are equivalent in all economically relevant respects, then they should have the same market price The LAW OF ONE PRICE is enforced by arbitrageurs If they observe a violation of the law, they will engage in arbitrage activity-simultaneously buying the asset where it is cheap and selling where it is expensive In the process, they will bid up the price where it is low and force it down where it is high until the arbitrage opportunity is eliminated Same σ should give same E(R). FNCE30: Investments Lecture 6 Page 41

42 Portfolios and Diversification We know that the portfolio return is the weighted average of the returns on the individual assets in the portfolio: R R β F ε i i i i Fully Expanded Portfolio of individual stocks. The return on this portfolio is the weight times the security for each security. R R β F ε Each component R i is built from a factor model: i i i i Regroup for the weights times the expected return for each security plus the weight times the factors (beta) and finially the epsilons are the idiosyncratic risk. R X R X R X R P 1 1 N N ( X β X β X β ) F 1 1 X ε X ε X ε 1 1 N N APT says if I keep increasing the number of assets in this portfolio (orange) the first line does not disappear, you just end up with the average return of the stocks in the portfolio, the second line will trend toward the beta in the market, the third line, due to the independent nature of the idiosyncratic risk epsilon, by adding more assets to the portfolio we can make this term as close to zero as we want, bring the entire third line to zero. This is diversification, same as the other models. What are we left with FNCE30: Investments Lecture 6 Page 4 N N

43 Well-Diversified Portfolio The result of the above equation is: Expected Return of the Portfolio + the Beta of the Portfolio times the "Factor". The idiosyncratic risk epsilon still exist but we have made it arbitrarily small. DISEQUILIBRIUM EXAMPLE R This is where ARBITRAGE PRICING comes in. This is what we need to understand for the homework. Here we have a number of different models and a risk free assets which has a beta of zero (this is not the beta in the CAPM sense). This is a regression beta. Now suppose we have asset A and asset C. There is disequilibrium here because asset A has a beta of 1, it is earning a return of 10%. If I combine that with the risk free asset I can form a portfolio anywhere on the line. Asset C cannot be priced at that level because this portfolio of the risk free asset and asset A which has the same theta (beta?) as asset C gives me a higher return than asset C does. In the marketplace this cannot exist because the investor would sell it short and invest in A and the risk free asset, would have a zero investment, but based on the risk there would be a positive expected return. it should be arbitraged out, because we are buying one asset and selling another eventually the prices should converge. How do we construct this exchange? Mix A and the risk free rate so that I end up with the same beta, then show that the expected return of the constructed portfolio is either higher or lower than this other asset. Therefore if I sell short and buy long I should be able to make an arbitrage profit. (our HW problem will have 3 assets with betas and expected returns where one of them is out of equilibrium. We should be able to construct a portfolio out of the other two which gives the same beta but with a higher return. Higher return is the key. Example: We have asset 1,, and 3 with the shown returns and betas. If I construct a portfolio where I invest W weight in asset 1 and 1-W in asset the beta of the portfolio is given by Wβ 1 + (1-W)β = β Portfolio = β 3. We have intentionally selected our weights so that the beta of the portfolio equals the beta of the third asset. Then we know that we have WR 1 + (1-W)R = R 3. If this is not the case I can construct an arbitrage opportunity so that I can make a profit with zero investment. To demonstrate consider Wβ 1 + (1-W)β = β 3 and WR 1 + (1-W)R > R 3. What this is telling me is that I have a portfolio that has the same risk as this particular asset (in this world risk is measured by beta). Therefore, by the law of one price, beta is the only way we can measure in this world, the return on this P R P β P F FNCE30: Investments Lecture 6 Page 43

44 portfolio should be the same as the return on this asset. If it is greater I will implement a strategy to sell the asset R3 and reinvest the proceeds into WR 1 + (1-W)R because I am selling something where I am paying a return of R 3 and generating a return of WR 1 + (1-W)R which is bigger. I end up zero investment but a positive expected return. [lecture 7 recording 13:3]. HW: construct a portfolio of two assets with enough weight so that you get the beta of the third and see if any of the pricing things double up and then we can construct portfolios to make arbitrage profits. One-Factor Security Market Line Consider the market index portfolio, M, as a well-diversified portfolio The beta of the index portfolio is 1, the market portfolio. It is the most efficient way of measuring this risk. Beta of 1 means we are dealing with the market portfolio So if the line below holds true, if this tells me that is assets fall above or below this line I have arbitrage pricing opportunities. We are not talking about CAPM here, this is purely a mathematical argument. We should be able to construct this line and we should be able to use that line to price risk for any particular asset. If it falls above or below we can arbitrage it away. We are in a market setting with a market portfolio with beta of 1 and an expected return of that of the market. The difference between the risk free return and the market return is the MARKET RISK PREMIUM. The risk is associated with a factor. APT with Market Index Portfolio The equation of the return line is given by the formula below it which looks a lot like CAPM. The difference is that this formula in an APT world is constructed without all of the baggage associated with CAPM, no homogeneous beliefs nor any of the other stuff. All that is required is the ability to arbitrage and make profits and sufficient assets in the market to diversify. So we get a CAPM type result without the necessity for all the assumptions. E(r p ) = r f + [E(r M ) r f ]p FNCE30: Investments Lecture 6 Page 44

45 Multifactor Models Use more than one factor in addition to market return Examples include gross domestic product, expected inflation, interest rates etc. Estimate a beta or factor loading for each factor using multiple regression. When Ross put together APT he wanted to prove CAPM without the assumptions. Some people argue that there is more than one type of market risk that needs to be diversified. There is more than one type of risk then can be captured by a single factor. Mutli-levels of risk. Examples would be instead of just using a market factor I can pull in other factors such as inflation, GDP, interest rates, expected inflation. Thses are all things which I may want to hedge against unexpected movements. The market protfolio may do an adequate job but I may be able to put together other portfolios which will hedge against these other risks. May do a better job more efficiently. We get more explainatory power than just a CAPM model. How do we approach this? We can run multi=factor regressions and estimate betas for each of these factors. For example Multifactor Model Equation R i = E(r i ) + Beta GDP (GDP) + Beta IR (IR) + e i R i = Return for security i Beta GDP = Factor sensitivity for GDP Beta IR = Factor sensitivity for Interest Rate e i = Firm specific events We will use GDP and interest rates. These interest rates are often the difference between a two year bond and a ten year bond or a junk bond and a treasury security. It depends on what you want to use and how you think it can be modeled. There is no rule as to what you should use. So now we run a regression on model with more than one factor. No problem. Trouble is, it is really just a description. We can use an arbitrage argument which says if these factors are important then this is how risk should be priced. CAPM tells you what the factors should be, says the factor is the market portfolio. With APT you can pick whatever factor you want. Pick whatever you like and construct a model that says risk should be arbitraged away if you believe this model is true. What's more, in these models you still need an expected return and that expected return comes from CAPM. What people do is use CAPM to predict the expected piece and then use multi-factors to capture the unexpected pieces of the stocks returns. Gets to feeling like a data mining problem but sometimes data mining exercises work. No guidance concerning which factors to use. FNCE30: Investments Lecture 6 Page 45

46 Multifactor Model The above is no more than a description of the factors that affect security returns There is no "theory" in the equation Where does E(r i ) comes from? We need a theoretical model of equilibrium security returns Shortcoming of the multifactor APT No guidance concerning the determination of the relevant risk factors or their risk premiums Example of the multifactor approach is the work of Chen, Roll, and Ross IP = % change in industrial production EI = % change in expected inflation VI = % change in unanticipated inflation CG = excess return of long-term corporate bonds over long-term gov bonds GB = excess return of long-term government bonds over T-bills More factors you have the better the predictions you can make, at least based on the past. So this factor model has industrial production, how much it changes per period. Another is change in expected inflation. These days we use TIPS to predict inflation. Unanticipated inflation is also included. Long term corporate bonds over government bonds gives a type of credit risk premium, may capture something important about the marketplace. Then excess of long term gov bonds over T-bills will measure how steep the yield curve is. So we have a lot of factors measuring certain aspects of the economy. We are throwing in so many different components of the macro economy, it really would not be shocking that if we run regressions on this we will get a lot of predictive power. The argument of the researchers who did this is that market risk is important but people really care about these 5 things, this is the risk that they really want to hedge. The blunt object of the market is liable to miss something, this thing gets down at the essence of where the risk really arises from. Also consider that running a CAPM regression is simple, just download the info from yahoo finance. Collecting industrial production, inflation, changes in inflation, There is a lot of info to try and wrangle. Would be tough for a financial manager of a firm (not impossible). These types of models can give good accuracy results. But the problem arises when you try to collect the data for next year and make the necessary predictions, etc. FNCE30: Investments Lecture 6 Page 46

47 Alternative to Macroeconomic Variables Use firm characteristics that seem on empirical grounds to proxy for exposure to systematic risk The factors are chosen as variables that on past evidence seem to predict average returns well and therefore may be capturing risk premiums Another way to go. Instead of trying to grapple with the economic data, could look at proxies the same way we did for the single factor model when we looked at the market return as a proxy for the macro factor. There are various models. Look at past results and past ways of constructing portfolios of assets to see if they do a better job than the market portfolio itself. One way is the book-to-market size and large company versus small company below Fama and French Three-Factor Model SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the return on a portfolio of large stocks HML = High Minus Low, i.e., the return of a portfolio of stocks with a high book-to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio RM = Return on the broad market This model constructs portfolios based on small compared to big, high-book-tomarket compared to low-book-to-market, as well as the return on the broad market. They are trying to get the same result as above but without having to find all the macro data. Using finer proxies. This data is easy to get, can download it from websites. FNCE30: Investments Lecture 6 Page 47