All right. Hopefully, you are able to make progress on assignment two. I put together a little discussion of my analysis, hopefully it matches up closely with yours and then Simon too really had some interesting real-world potential implications in terms of should you have an asset like gold in your portfolio? Doesn't make sense to have international diversification, doesn't add a lot if you're already investing in the US stock market. I think these are lessons that help motivate asset pricing models like the CAPM, capital asset pricing model which we'll develop in module two. So, let's look at my discussion here for assignment two. The first series of questions was considering a portfolio possibilities where you have large cap stocks, small cap stocks, so stocks that have a large market value equity think Apple, small-cap stocks like very small firms, value growth, cement company versus the Internet start-up, and now we add gold into the mix and we're using annual return. Well, here we use a remember for this one actually using monthly return data 75 to 2014 so over those 480 months of past data, we have historical estimates that we use as our assumptions for the future in terms of average returns, standard deviations, correlation of the assets that were present 1975 to 2014, we're running with those being our assumptions for the future and we form portfolios accordingly. So, in column one here, same risk is large stocks. So, what portfolio do we put together where we have the same risk of large stocks? So, a standard deviation of 4.3. Remember this is on a monthly basis that's why the average return one percent per month about 12 percent per year. What portfolio comes out given we want to maintain this standard deviation of 4.3 percent and we went to compare the return of this portfolio to the return currently offered by large stocks of 1.01 percent per month or at least its historical monthly return. So, that's shown in the first column, and you can see what does this portfolio look like, it has the same risk portfolio standard issue 4.3 percent but offers a return based on our assumptions on average of 1.2 percent per month. So, almost 20 basis points per month, on an annual basis about two and half percent difference per year non-trivial. What's going into this portfolio? Well, 48 percent value stocks, 19 percent gold. Okay. Actually large stocks are only 22 percent of this portfolio and interestingly growth stocks, they come in with a weight of zero here. Just as I mentioned I jumped ahead of myself I'm so excited to go through this example here, but if we want a portfolio that has the same risk is large stocks but offers almost a 0.2 percent or 20 basis points per month higher return is a portfolio that's 22 percent large, 11 percent small, 48 percent in value stocks, and 19 percent in gold stocks. This portfolio clearly dominates the large only portfolio on the order of about a fifth of a percent 19 basis points per month. Interestingly, gold is almost one-fifth of this portfolio, remember we talked about putting the team together even though gold has historically has had the low average return, it's zero correlation with the other assets makes it appealing part to add to the portfolio view. View gold is like the basketball player that gets in and gets the rebound and dies on the floor to get the loose ball or balls for you. Gross stocks has zero weight in this portfolio. Column two, we want to maintain the same risk as large stocks, the standard deviation of 4.3 percent is what we want to stick with, but now we're free to pick any portfolio weights including negative portfolio weight. So, we're allowed to short but again we're not worried about the cost of shorting in terms of margin requirements or such, we are abstracting away from that so the results are in column two here. What do you notice in column two? Well, first let's focus on the return. Being allowed to short relative to not being allowed to short only increases the average expected return by 0.05 percent per month, five basis points per month on average, maybe about a little over half a percent 0.65 percent on an annual basis. So, some increase, but is it really that large and look at the gigantic change in portfolio weights. In this portfolio here, we're shorting gold stocks 86 percent and a gigantic allocation to large cap stocks is a result. So kind of a pretty wild portfolio,111 percent in large stocks, 30 percent in small, 31 percent in value, 14 percent in gold, these weights add up to 186, our portfolio can't give more than 100 percent, so we're getting the additional 86 from the shorting of growth. Okay? There's a portfolio indicated there. So, why do we have this big switch of large beating of large allocation going up a lot, growth having this gigantic short when we're allowed to short? Well, this reflects from the optimization programs perspective it looks at the assumption and large and growth are very highly correlated with each other. Large stocks and growth stocks are moving together very closely, their correlation 0.93. So, the optimization routine sees that and sees that large beats growth on average by about six basis points per month. So, it's saying let's do this big allocation toward large shorting growth stocks to capitalize on this difference in average returns. Okay? But what if when we are making the assumptions, the future is somewhat different growth stocks do better, remember where shorting growth stocks big-time, this portfolio could be a gigantic disaster for us. So, this is where when you're looking at the optimization routine the numbers it spits out are correct given the assumptions but taking a step back you say,"Do I really want this portfolio that has such a gigantic short position with gross stocks? I'm not getting that much difference in performance relative to the portfolio without the shorting going on. What if my assumption for the growth stock returns going forward? What if it should be higher than how growth has performed in the past? Do I really want to go this route in column two, maybe the portfolio allocation in column one is good enough?" Then finally here, what's a portfolio that gives us the biggest bang for the buck, the biggest Sharpe ratio, where I'm also putting in what has been the average rate on treasury bills over the period 1975 to 2014, so we can calculate the excess return which is the numerator of the Sharpe ratio that's shown in column three. So this optimal or this portfolio that yields the biggest bang for the buck, the maximum Sharpe ratio, what are the components? Seventy-five percent value, 15 percent small, 10 percent gold, so gold still makes a team. Remember, the basketball team always needs the player to grab the rebounds, to dive on the floor that get the loose ball, not everyone on the team can be a top scorer, you need some gritty players as well. Remember there's no shorting allowed here, so large stocks don't make it into this portfolio, growth stocks do not either. Okay? So, why do we have this allocation for this maximum Sharpe ratio portfolio, high allocation to value stocks? This just represents their high average monthly return in the past which we're assuming happens on average going forward and also their own high bang for the buck, they have the highest Sharpe ratio. So that's why value stocks or three-quarters of this maximum Sharpe ratio portfolio. Why does gold make the cut? Why is gold on the team? Do you know? DRO stocks aren't on the team. Large stocks aren't on the team. Gold has a smaller average risk turn historically than both gross stocks and large cap stocks by a wide margin, why is it make the team? Well, even though its return is low, it is giving a return higher than the risk-free rate over the last 40 years and we're making the assumption that that continues on average, it has a higher return in the risk-free rate, even though it's uncorrelated with the performance of these other stocks, large value small growth. So, when the other stocks are falling in value, gold is holding its own. When the other stocks are going up, gold is also having a lower return. So, it's some downside. When the other stocks are doing well, gold is not doing so great, but when the other stocks are falling, there's no correlation here with how gold is doing. So, that lack of correlation causes gold to be a valuable part of the portfolio. Another way to think about this, over the last 40 years, both gold and Treasury bills have basically a zero correlation with these stocks. Gold has a little higher return than Treasury bills during this 40-year period. So, that's why gold makes it into this maximum-Sharpe-Ratio portfolio comprised of these possible five risky assets that I gave you. Gold is a basketball player grabbing the rebounds, diving on the floor, always a valuable part of the basketball team, also a valuable part of the portfolio here. The following revised table has an added row at the bottom, listing the portfolio standard deviation for the large, small, value, growth, gold portfolios. This new table contains a portfolio standard deviation, you need to answer question three of Assignment two, with the relevant portfolio standard deviation highlighted by a red box. The monthly portfolio standard deviation is 5.22 percent for the portfolio in question three. So, on the next few screens here, I just gave you some Excel screenshots for how did I produce the estimates that came up in the first column, in the second column, and in the third column for the first part of assignment two where we're dealing with what portfolio allocations should you have when you have large, small, value, growth, and gold as possibilities. For the second part of assignment two, we talked about international diversification. So, US stock assumptions, US stock market average return, 4.34 percent. Remember, this is monthly returns again. July of 1990 through the end of 2014, average return,0.89 percent per month for the US. So in column one, we want to have the same risk as the US stock market, but we're open to international diversification. What portfolio gives us the highest return given the constraint of we want to have the same risk as US stocks? Column two is just saying give us a portfolio with the maximum-Sharpe-Ratio. What's the key thing we see, US stocks are still a gigantic part of both of these portfolios. The average expected return in each of these two cases is basically the same as when we're 100 percent in US stocks. The Sharpe ratio also basically the same as being 100 percent in US stocks. So, if you're already investor in the US market, adding exposure to Asia-Pacific, non-Japan, Japan and Europe doesn't really add much to your portfolio. Now, if you're from Denmark and you're considering this exercise, there probably is going to be more benefits to adding international diversification. But the US market is already so large that adding exposure to these other regions really doesn't do that much for you. Maximum-Sharpe-Ratio portfolio, 87 percent in US stocks. Now, Japanese stocks are excluded from the maximum-Sharpe-Ratio just their return over the last 25 years is so low that the optimizer no way is putting that in the maximum-Sharpe-Ratio portfolio, if those poor returns are to continue. So, that leads to question six. We just talked about questions four and five. For question six, what if those poor returns that Japan has had in their stock market in the last 25 years, don't continue. What if Japan going forward has the same expected or average return, I think it's 0.96 percent per month, as Asia-Pacific excluding Japan. Then things change a lot. Japan goes from zero percent of the maximum-Sharpe-Ratio portfolio to 30 percent. Asia-Pacific falls out less than one percent. So, what makes Japan an attractive stock, in this case, relative to Asia-Pacific? It has the same expected return, but it has lower correlations with the other regions of the world. We're keeping our correlation assumption and standard deviation assumption the same. So, that makes Japan a more attractive asset than Asia-Pacific excluding Japan. Both have the same return with this new assumption, but Japan has less correlation with the other assets. So, it adds more diversification benefit to our portfolio. The examples with gold and Japan here with this new assumption, really get at the heart of the capital asset pricing model. What assets should be valued more highly by investors, and by being valued more highly, investors probably then require a lower return to hold them. So, useful examples to motivate module two and the capital asset pricing model. The following revised table has an added row at the bottom listing the portfolio standard deviations for the US, Japan, Asia-Pacific, Europe portfolios. This new table contains a portfolio standard deviations, you need to answer questions five and six of assignment two. With the relevant portfolio standard deviations highlighted by a red box. The monthly portfolio standard deviation is 4.37 percent for the portfolio in question five and is 4.11 percent for the portfolio in question six. So, now is a good time to wrap up and highlight what we learned in Lesson 1.8, assignment two. Discuss through the various scenarios, adding gold, adding international stocks and it really highlights the importance of correlation between the assets in putting together the optimal portfolio.
