All right. Let's discuss assignment four. This was the analysis of the 50 balanced funds. The key motivation behind this assignment was to test your understanding of the capital asset pricing model, the 3-Factor Model, differences between the two, what differences in Alphas between the capital asset pricing model and 3-Factor Model, what they represent. To do this, I gave you return data over the last 20 years for 50 balanced funds with varying degrees of performance, varying degrees of investment styles. Don't be alarmed, but there are two slight errors in the discussion of Assignment Four and Lesson 3-7. They appear on this slide as well as a couple other points in this discussion. The average raw return of the 50 balanced funds is 7.2 percent per year, not 7.5 percent, and the average CAPM Alpha is minus 0.3 percent per year, not zero percent. I hope these slight errors weren't keeping you up at night. Note that these two errors have no effect on answers to the questions of Assignment Four. I reported in the Excel spreadsheet, excess returns. Returns in excess of the one-year Treasury bill rate on an annualized basis. It's useful in this discussion going back, by scrolling the spreadsheet now, I probably would have just given you the raw returns. So, for the average 50 balanced funds in the spreadsheet, the average raw return was 7.5 percent per year. The excess return of 4.8 plus the 2.7 here risk-free return. So, the CAPM Beta 0.52 on average and the CAPM Alpha of zero percent on average. So, the average excess return of the 50 funds was 4.8 percent. To get this to a total raw return, add in the 2.7 percent of the risk-free rate to give us this average fund raw return of 7.5 percent per year across the funds. The CAPM Beta is about 0.50, 0.52 to be precise, and the CAPM Alpha, the average of that across the 50 funds literally zero, okay? Comparing this to the overall environment here, what were other assets yielding stock market,12 percent on average over this 20-year period, 1995 to 2014. Treasury bills 2.7 percent on an annual basis. Small stocks outperformed big stocks over 1995 to 2014 on the order about two percent per year while value beat growth on the order of about three percent per year over this period. So, here, again, we're looking at the raw returns per year. So, this is basically taking the excess return from the spreadsheet, adding back in the risk-free rate, which averaged 2.7 percent over this period. Now, these numbers down here on the X-axis are not fund ID numbers, instead this is just the ranking of the funds. One is giving you what the lowest return was, the lowest return fund, 50 is just giving you the highest return. So, what do we see for these balanced funds that have some combination of US stocks and Treasury bills as investments? Remember, the average raw return was 7.5 percent, that's roughly what the median return is here. The high return is just under 10 percent of the funds, and the worst-performing fund yielded a raw return of four percent per year. Now, what if we put these funds over 1995 to 2014 using the annual data? What if we put them through a capital asset pricing model calculated in Alpha and Beta to benchmark adjust because different funds have different equity tilts, maybe investing in different types of equities, so what if we adjust for the systematic risk of the different investments across the balanced funds? Then we see, remember the average Alpha, and this is Alpha from the CAPM in percent per year, and remember, this is the average over a 20-year period, right? These were based on returns for 1995 to 2014. So, it's not a one or two-year snapshot, this are these Alphas were estimated using annual data over a 20-year period. So, the average Alpha was zero. You can see here for the median fund, it looks like the Alpha is slightly less than zero, but around the middle of the distribution, here the Alphas are pretty close to zero. The best-performing fund in terms of this risk-adjusted return is a little over just under 2.5 percent per year. The worst-performing fund is underperforming its benchmark by three percentage points per year on average over a 20-year period, so this historically has really been a dud of an investment here. Okay. So, let's do some analysis of these various balanced funds, and what these various parameters from the models represent. The CAPM Beta, this is giving us the average tilt of the portfolio toward equities. So, for example, a portfolio that's 50 percent US stocks, 50 percent Treasury bills would have a Beta of 0.5. Okay, if it was 80 percent stocks, 20 percent Treasury bills, that would have a Beta of 0.8. Okay, what I'm assuming when I say US stocks, invest in the US stock market index. Now, if we look at the CAPM Beta across the 50 balance funds, it varies from 0.32 to 0.68. Okay, so that means for one fund, the systematic risk of that fund is equivalent to a fund that's a third in the US stock market, two-thirds in the risk-free rate. Where there's another fund, that's exactly the opposite. On average, it has the same systematic risk as a fund that's and third in Treasury bills, two-thirds in the US stock market. Okay, the CAPM Alpha, so if we look at, let's adjust. We want to compare fund managers across the different balanced funds, some funds are going to have higher returns because they're taking on higher risk. Okay, they're investing more in Tiffany's less in McDonald's or they're investing more in stocks in general less in T-bills. So, on average, we'd expect those fund managers to generate higher returns because they're taking on higher risk. The CAPM Alpha is adjusting for that, and by risk, in the CAPM model, it's just talking about systematic risk, market risk, sensitivity of the performance of your asset or security to the overall market. Once we account for the appropriate CAPM benchmark, the systematic risk of the investment, the equity tilt or the fund, we're getting outperformance as high as 2.3 percent per year, again, on average over this 20-year period, and underperformance as low as a negative 3.2 percent underperforming the benchmark by 3.2 percent per year for one of the funds. But again, the average across all the funds happened to be about zero. So, when we go to the 3-Factor Model, the Betas, for example, on the small minus big factor, the high minus low, which I would have called value minus growth factor, they give us clues about the underlying investment style of the portfolio. Okay, when we look at the Alpha on the 3-Factor Model, that adjusts to take into account this investment style, okay? So, a way to think about when you do the 3-Factor Model, fund managers don't get credit for investing in small value stocks. In the 3-Factor Model, that's taken into account. Instead of appearing in Alpha, that appears in the Beta on the size and value factors. Likewise, investment managers that are focusing on large cap growth stocks don't get penalized for that in the 3-Factor Model. We know those are assets or securities that have underperformed historically, that's taken into account in the 3-Factor Model. It shows up in the Betas, it doesn't show up in the Alpha. So, simply investing in a small value index fund won't yield any Alpha on the 3-Factor Model although it has historically in the capital asset pricing model. Okay. One simple measure that we started out with the Sharpe ratio, this just gives us the simple bang for the buck, the average excess return of the stock divided by its standard deviation or total volatility on that metric. Now, that isn't differentiating. All risk is kind of treated the same because the standard deviation of the returns reflects both kind, the idiosyncratic risk and systematic risk. So, the Sharpe ratio, excess return over the total risk, if you will, or the total standard deviation, volatility of the investment. Fund one, fund seven, fund 47 all have particularly high Sharpe ratios, in terms of bang for the buck, excess return over the volatility. How about taking into account the market risk of the portfolio? Different portfolios have different loadings on equity versus treasury bills, some of them are investing more in stocks, some are investing less. Those who invest more in stocks given the average stock return is 12 percent over this 20 year period they're probably going to do better than firms that invest less in stocks over this 20 year period. The capital asset pricing model will take that into account so we can more fairly compare asset managers of portfolios that have different investment strategies. So, let's compare for example to illustrate this two funds. fund 47, the raw return again the excess return plus the average T-bill rate over the 20 years gives us a raw return, 9.6 percent for fund 47. For Fund one, the raw return is 8.3 percent. Again it's excess return plus the average annual T-bill rate gives us 8.3 percent. So, a difference fund 47 yielding a higher return than fund 41, a difference of 1.3 percent each points per year on average over the 20 year period. So, a pretty substantial outperformance here, but what that doesn't take into account, what this 1.3 percent here doesn't take into account is it fund 47 is putting the foot on the gas pedal, putting the foot on the accelerator a little more than fund one is. So, for fund 47, the beta is 0.5, for fund one, it's 0.38. So, fund 47 is generating higher returns, but it is taking on more risk and that it has a higher tilt toward equities on average. So, once you take that into account in terms of the capital asset pricing model in calculating an alpha, the outperformance relative to the benchmark that takes into account market risk, takes into account the equity tilt both of these historically have done very well. Both fund 47 and fund 41, but now the differences in their performance risk adjusted is only 0.2 percent as opposed to this original 1.3 percent. Fund 47 has an alpha 2.3 percent, fund one has an alpha 2.1 percent. So, another way to think about this, of this difference in returns, a 1.3 percent, 1.1 percent of that, 1.3 minus 0.2. 1.1 percent of that, 1.3 is accounted for just differences in underlying risk of the investment strategies between the two funds. So, we just talked about to grade funds. At least great in terms of the historical performance. Fund number one, fund number 47. On the flipside, fund 23 has performed terribly. On average it invest 50 percent in the stock market reflected by a beta 0.54. Another way to say this is the risk taken on by fund 23 is the same as, systematic risk that take on by fund 23 is the same as a fund that's 54 percent invested in the US stock market, 46 percent invested in Treasury bills. That's what the beta 0.54 represents. So, the equity tilt toward this fund is about average across the funds in the sample are actually a little higher than that average beta 0.52. So, low higher than average equity allocation relative to the other funds. Despite that, its performance is the second worst. It's raw return is 4.5 percent per year, the excess return of 1.8 plus the 2.7 risk-free rate, 4.5 percent raw return. Second lowest in the group of 50 even though it has a slightly higher than average tilt toward equities. So, it's really doing poorly. The average return across all the funds was seven and half percent. Once you take into account its tilt toward equities, it has this benchmark adjusted return of minus 3.2 percent per year. In other words, given its tilt toward equities on average it should have been yielding 3.2 percent higher return, 3.2 percent each point higher return on average per year for 20 years than it actually did. Instead of generating 4.5 percent given its risk in investing in equities it should have been generating 7.7 percent. So, question one was, why are there differences in alphas across the CAPM model and the three-factor model and I identified six funds here with kind of dramatic changes here. Let's go through these funds one by one. Fund seven, you see the alpha increases by 0.8 percent per year, percentage points per year when you go from the CAPM model to the three-factor model. What is leading to this increase in outperformance of the benchmark? Well, what the three-factor model is taking into account is fund seven has a tilt toward growth stocks, indicated by this negative beta on the value minus growth or high minus low factor. So, this negative coefficient represents when growth stocks are doing well. HML is negative, negative times a negative coefficient is positive. So in other words, this negative coefficient here means when growth stocks are doing well that's a period where this fund is doing well so it has a growth tilt. Given growth stocks historically have outperformed value stocks, this 2.1 percent outperformance in the CAPM is even more impressive. It's even more impressive to generate this high return given you're kind of swimming upstream investing in growth stocks which have underperformed during this period. So therefore, when you go to the three-factor model, the alpha goes up a bit because it's taking into account that this is a manager who is investing in growth stocks given this negative coefficient on the value minus growth factor. Growth stocks have underperformed relative to value so therefore we adjust the outperformance up or another way to say it is, we lower the benchmark given the tilt toward growth stocks were fund seven, so the three-factor alpha is greater than the one factor CAPM alpha. Fund nine, the opposite happens, when you go from the CAPM model the alpha, the outperformance of 1.2 percent per year drops to 0.7 percent the three-factor model. Well, why is this? Well, this is because one of the reasons for the good performance in the CAPM for fund nine was the sensitivity to small stocks. When small stocks are doing well SMB is positive, positive coefficient on small minus big means this fund is doing well. So, one of the reasons that this fund did well is it invested in this small strategy which historically has beat large-cap stocks. So, we need to make our benchmark a little higher for fund nine and the three-factor model therefore the alpha falls. So, the performance of the manager of fund nine is less impressive once we take into account the investment style. For fund 26, we have an improvement in performance. They're negative in both but the CAPM alpha minus 1.8 percent, we actually bump that up to minus 0.7 percent, the three-factor alpha. Why? Because fund manager 26 was really swimming upstream. The investment tilt was away from small stocks toward large stocks given by this negative coefficient or indicated by this negative coefficient also toward growth stocks given by the negative coefficient here on the HML or value minus growth factor. So, fund manager 26 seemed to be investing in both large-cap stocks and growth stocks. Both of them have outperformed so we should lower our benchmark for this manager when we go to the three-factor model therefore the alpha is reduced as a result. So, in the case of fund manager 26, the investment performance is less bad once we adjust for the investment style that is swimming upstream investing in both large-cap stocks and growth stocks. For Fund 27, the kind of investment performance is viewed as less impressive once you go to the three-factor model, the alpha becomes negative because this was a manager that was investing in stocks that historically outperform. A positive tilt towards small stocks, a positive tilt toward value stocks. So, given you are generating an alpha of zero and the capital asset pricing model, that's not very impressive because you are investing in stocks that typically have positive CAPM alphas. The small stocks and value stocks we need to raise your benchmark up when you go to the three-factor model once we do that your alpha becomes negative here, performance is less impressive given the investment style. Fund 31 we can see here by the negative coefficient on the small minus big factor, the negative coefficient on the value minus growth factor means fund 31 is positively correlated with the performance of large stocks as well as growth stocks, and we know that both large and growth over this period underperformed. So, this CAPM alpha of 0.1 actually that's not that bad. So, we're going to lower your benchmark given you are investing in these types of stocks that underperform. So, the alpha goes from 0.1 to one. Performance is more impressive when you go to the three-factor model and we take into account your investment style. Then finally, fund 50, this is a fund manager who's doing very poorly in the CAPM model in terms of underperformance by two percent per year on average over this 20 year period, big tilt towards small stocks and value stocks. So, should be doing pretty well on average in a CAPM model is generating a negative alpha even though you're swimming downstream here. You're investing in assets that were this period tended to do well, small stocks and value stocks. So, it should be easier going downstream but you're underperforming, you're doing poorly in the CAPM world which means given your investment tilt, you're going to be doing even worse in a three-factor model because we're going to raise your benchmark up. We're going to raise your hurdle rate up given you're investing in small stocks and value stocks which historically have done well you're lackluster performance is viewed even worse given your investment style. So, question two, which of the 50 funds would you most like to invest in going forward? Then three, which of the funds would you least like to invest in going forward and why? Well, before I answer this I thought it would be useful just a few funds that caught my eye, the key thing is based on past performance some notable funds here, so on the positive side and black funds 1, 7, and 47. If you look at particularly for fund one, it has a small allocation to equities a CAPM beta is less than 0.5, despite that has a very high Sharpe ratio, a pretty high Raw return of eight percent per year, CAPM alpha of kind of two percent beating this benchmark, so it has kind of low risk sensitivity to the market, but is performing very well, so it has this high CAPM alpha, when you go to the 3-Factor model the alpha stays exactly the same, so that would be consistent with fund one manager, or the manager of Fund One is a manager that just knows when to get in and out of the market, doesn't have a particularly investment tilt in terms of small big value growth just as good at market timing, when to get in the S and P 500, when to get out of it for Fund one. For Fund 47 for example, they're taking on more risk as a result generating a higher return, this is kind of the example we went through earlier in this lesson comparing one versus 47, if you like kind of investing in large cap stocks that seems to be what Fund 47 is given this negative coefficient on the Small Minus Big beta. Focusing on on some of these kind of you know duds, based on the past performance 23, 42, 50, 23, Fund 23 we talked about earlier has this above average tilt toward equities. But despite that gives this you know kind of second worst a Raw return leading to this gigantic negative alpha in the CAPM model for Fund 50, it has this big tilt, it's investing in small stocks on average. It's consistent with investing in small stocks given this positive coefficient on the Small Minus Big factor. Positive coefficient on the value minus growth factor, so consistent investing in small value stocks. So, given the investment in small and value, it should be doing pretty well, but it's not you know it's CAPM alpha is minus two percent per year on average lackluster return. Once we account for kind of swimming downstream investing in assets that over this 20-year period, they did pretty well its 3-Factor alpha accounting for this favorable investment style even worse down to minus three percent. But the key issue, how well does the past predict the future is past prologue? That really goes into, do you want to simply rely on your future investment choice? Being tied to past returns. So, key question. Which of these funds would you invest in going forward? Which would you avoid given the historical data? The answer, any of them, none of them. Okay. So, say what's going on we do all this work? The answer is any of them are fine to invest in going forward or none of them, any of them are fine to avoid none of them, what's going on here? Okay. Well it turns out each of these funds the pattern in returns that was put into the spreadsheet, each fund, each year, I drew a random number between zero and one from a uniform distribution. So, that determined for that fund for that year what fraction was invested in the U.S. stock market what fraction was invested in Treasury bills. Okay, number drawn was 0.7 for that fund for that year, 70 percent in the broad U.S. stock market, 30 percent in treasury bills and then another random number drawn for the next year for that fund for the next year for the next year for all 20 years than random numbers drawn for the 20 years for the next fun. So, so on and so forth, so I guess 50 funds Times 20 years a thousand random numbers so then with all these allocations between stocks and the risk-free rate estimate cap-M regression estimate 3-factor model. Calculate Sharpe ratio just like you would with any return series, okay. So given these are random numbers uniform distribution drawn between 0 and 1 the expected value of that distribution 0.5, so they expected investment is 50 percent in treasury bills 50 percent in stocks. Pretty close with this average market beta of 0.52 percent for the sample remember we're drawing from random so it doesn't have to be exactly 0.5 but it's pretty close to that. Now a 50-50 US stock market Treasury bill portfolio would give an average return of 7.3 percent per year over the period. Pretty close to our average of 7.5 percent. So, what's the point thinking about the analysis of past fund performance. All of these numbers were random. They did not reflect any active decision-making on market level research or firm level research. So, I told stories, like hey here's a value tilt for this one, a growth tilt for this one, this one's good at timing the market just looking at the past interpreting the regression results. But it turned out the data underlying the regression results was just driven by a collection of random numbers. Okay? But, even though this portfolio allocations, were driven by random numbers the performance that we observe looks pretty similar to if we look at the performance of actual mutual fund managers. So, the point is if you're trying to make investment decisions based on past performance, particularly you're trying to make investment decisions for actively managed mutual funds, and you're basing that on past performance you need to be very careful because I gave an example here where funds, some funds have good past performance, some have bad, but it's clearly not predictive of future performance because it is all driven by random number generator. Okay? So the firm that did very well like Fund One could do very poorly in the future, if it just happens to draw bad numbers. Fund 50 which I believe did very poorly in the past with the random numbers, it happened to get could do very well in the future and in fact our expectation is all these funds have the same expected return in the future because they don't know which random draws they'll get whether they get a high equity allocation in the year stocks go up or whether they get a high equity allocation in a year like 2008 when stocks fall a lot, it's just driven by chance. Okay. So, the ultimate point is it's very hard to differentiate luck from skill simply looking at past performance. Okay? Let me give you another example of luck or skill. Are there any American football fans? Even if you're not, you're probably aware of the Super Bowl which is the championship game in which the winner from the American Football Conference, AFC, plays a winner from the National Football Conference, NFC. They meet up for the ultimate title in the Super Bowl. So, in one examples of some AFC and NFC teams 2015 Super Bowl winner, New England Patriots is an AFC team, Pittsburg Steelers, Denver Broncos also AFC teams. For the NFC, I have to go to my homestay, Green Bay Packers here, the famous NFC team, Dallas Cowboys, Chicago Bears and also be NFC teams. Okay. So, there's a lot of bets during the Super Bowl on the game. But there's also crazy bets during the Super Bowl that have nothing to do with the game. So, people can bet for other, whether the NFC or the AFC is going to win the coin toss at the start of the game. So the Patriots are playing the Packers. They did play once a Super Bowl, Packers won that game. Do the Packers win the coin toss? Do the Patriots win the coin toss? Does NFC team or does AFC team won? Or you can just bet is it heads or tails? Or does the team that wins a coin toss won the Super Bowl? All these are bets. So, typical wager would be like, if the gambler is correct, they get $100. If they're wrong, they have to pay $105. So, the casino on average is winning $2.50 on this bet here. The wagers losing, the gamblers losing $2.50 on average. But it's maybe just fun to put this together, Right? It's just driven by chance. So, the historically profitable strategy that has been very lucrative in the past is actually generated from betting on this coin flip. Okay. In particular, the historically profitable strategy is bet the NFC team in the Super Bowl will win the coin toss. So, if it's a Green Bay Packers, or Dallas Cowboys, or Chicago Bears, whoever the NFC team is, bet that they will win the coin toss in the Super Bowl. Okay. The NFC is on a role of the first 48 Super Bowls. The NFC has one 32 times or two thirds of the coin flips including a streak of 14 in a row, 1998 to 2011. So, for example, the likelihood of winning the coin toss 14 years in a row like the NFC did for '98 to 2011 is one in over 16,000. Another way to think about this, if you placed an original $100 bet in 1998 Super Bowl, that the NFC would win, and you kept on doing double or nothing with that bet at the end of 2011, that would have gone to a final balance of $1.6 million. So, this is a pretty lucrative investment strategy. But other coin flip bets, remember, all this is driven by chance. Other coin flip bets aren't as lucrative. So, of the first 48 Super Bowls, heads 50% of the time, tails 50% of the time. Literally, 24, 24 for heads and tails of the first 48 Super Bowls, the team that won the coin flip, won the game 50% of the time, the team that won the coin flip lost the game 50% of the time. So, for all this is driven by chance, it just happens as one bet based on the NFC has had this nice run, but obviously that's not driven by scale, it's just driven by luck. So, if we have a group of 100 people, some of those will flip a coin and get five heads in a row. Not too many but a few, just driven by the luck and the randomness of the process here. So, bottom line, luck versus skill. Pretty hard to differentiate a lucky money manager from a skill money manager. In particular, you just don't want to look at past returns and say that means money manager, let me put my money with them, because we saw that positive alphas, very positive alphas on the order of over two percent per year over a 20-year period can be generated by random numbers by chance. Okay. Assignment Four shows us with these 50 balanced funds, that adopted random investment strategies in terms of their allocation of equities versus treasury bills, some of those fund managers outperform for a long, long time. So, this doesn't mean you shouldn't invest with active fund strategies. I'm not saying that at all. What I am saying is, if you want to invest with an active fund, and the manager has done well in the past and you're investing based on that, hey, good past performance, I bet it's going to continue. Do some more research. Understand the source of the mutual fund of manager's outperformance, what's the investment idea, what's being exploited, and does that make sense to you a priori? Because remember, in a fully efficient market, some active fund managers will do well, some will do poorly, just like in the assignment with the 50 balanced funds. If you want to invest in a fund that will do well going forward, you want to be sure you understand the investment strategy, what behavioral biases it exploiting, why does it make sense a priori, and why did it generate profits in the past and likely continue to generate profits in the future. Okay. Another factor when we're thinking about active management and its successes, we remember all the Warren Buffetts of the worlds, I think, wow, investing is like easy, look how well Warren Buffet has done. We forget all the managers who have done poorly and gone out of business. It's like a selection bias like value versus growth stocks in terms of their returns. At some point, Microsoft, Google, when they started out, they were definite growth stocks. Microsoft and Google, Apple have done extraordinarily well. So, it seems like the returns of those companies were on average would be higher than the cement company than the value stock. But while we remember Microsoft, Google and Apple, we forget dozens of Pets.com of the world that failed. So, it's a selection bias, or selection bias in our memory is maybe playing into this as well. So, key lessons, learn from our discussion here of Assignment Four is understanding the differences between the capital asset pricing model and the Three Factor Model, what the differences and alpha represent, how the coefficients on the various factors in the Three Factor Model give us clues about at least what the potential investment style is of the funds, or at least in this case, all the equity allocations were driven by random, the Three Factor Model is just telling us, hey, how is the performance of our fund related to the performance of small stocks or value stocks? This is what's revealed by the Three Factor Model. Then, ultimately, the takeaway was be careful, just simply extrapolating good past performance of a fund manager into the future. It can be difficult to differentiate luck from skill. If you're going to invest with an active management strategy, be comfortable that you know what the strategy is, and ahead of time, why you think it's likely to be profitable going forward.
