So in this lecture, I will discuss how one might want to think about cryptocurrencies possible place in an optimal portfolio. So in the introduction, I laid out two traditional finance views. One was more theoretical, one empirical. The theoretical view is very simple, and corresponds to some of the quotes that I mentioned at the beginning of this set of lectures. Because cryptocurrency has no intrinsic value and pays no dividends, it should not be part of the optimal portfolio period. Here, I'm going to develop the empirical view. So in Lecture 2, we saw that cryptocurrency has high average returns. However, we saw that the standard deviation on those returns is also very high. We found that returns are positively skewed. Now another fact about cryptocurrency is that they have relatively low Betas. So now we're going to use these statistical results and combine them with the portfolio theory to think about crypto as an asset class. So one way to adjust for the risk of the portfolio is to use something called the Sharpe ratio, which is something that I introduced in the last lecture. It's named for Bill Sharpe, the Nobel Prize winning economist and inventor of the CAPM. Now, the Sharpe ratio is the slope of the capital allocation line. So what does that mean? Well, is the expected return minus the risk-free rate divided by the standard deviation of return. What it does is it tells you the extra return you receive for the asset per unit of standard deviation. So it's telling you how good is the risk return trade off of this asset. So now we're going to use the statistics that I showed you last time to compute the Sharpe ratio on the three major cryptocurrencies. Now, because these are daily returns, and because the interest rate is basically zero anyway, we will use a risk-free rate of zero. Thus, we'll compute the expected return divided by the standard deviation. Now recall that these returns tell you a percent gain on a strategy that exchanges dollars for cryptocurrency, and then converts this cryptocurrency back to dollar. It does assume no trading costs. Now, it's going to use average returns. So on most days, you will lose money perhaps even if the Sharpe ratio is positive, and then a big worry is the survivor bias. We are looking at the three best cryptocurrencies. Just by definition, they're going to represent the ones that have ex-post, the highest returns. Okay. So with all of those caveats, let's look at some Sharpe ratios. So on Bitcoin, the average daily return is 0.2 percent, whereas the standard deviation is 4.3 percent, giving us a daily Sharpe ratio on Bitcoin of 0.06. On Ether, the average return is about 0.6 percent and a standard deviation of 7.2 percent, giving us a Sharpe ratio of 0.08. On Ripple, the average daily return is 0.5 percent with a standard deviation of 8.8 percent, giving us a Sharpe ratio of 0.06, about the same as Bitcoin. Now, as a comparison, the Sharpe ratio on the S&P 500 is 0.056. So Bitcoin and Ripple have about the same Sharpe ratio as the S&P 500, and Ether has a slightly higher Sharpe ratio. So if we are going to base our asset allocation on the Sharpe ratio comparison, I would say this argues against investing in cryptocurrency. So whereas it's true that these cryptocurrency returns are high, if we look at the risk, the return per unit standard deviation is not very high. So once we take issues of survivor bias and trading costs into account, the Sharpe ratio analysis argues against investing in cryptocurrency. Now, remember as we discussed in the last class, we might expect based on the capital asset pricing model for the market portfolio to have the highest Sharpe ratio anyway. In fact, we do see that compared to cryptocurrency, the Sharpe ratio on the S&P 500 does not look so bad, and basically reflects the fact that the S&P 500 offers a very good risk return trade off. So if we're going to base the analysis on Sharpe ratio, perhaps we should not invest in cryptocurrency. I would argue though that we don't want to base the analysis on the Sharpe ratio comparison. Instead, we want to use another measure. We should use the Alpha. What is the Alpha on an asset? Alpha measures something called the abnormal return. To understand abnormal return, we need to think about the capital asset pricing model, which was introduced in the last lecture. The capital asset pricing model is the most popular theory in financial markets. What does it says, is that for any asset with return R, the expected return on the asset equals the risk-free rate plus the Beta multiplied by the market risk premium, namely the expected return on the market minus the risk-free rate. For the risk-free, rate you could take the treasury bill return. The market is the return on the aggregate stock market. We can consider the S&P 500. The Beta is the covariance of the market with R divided by the variance of the market. In short, the Beta measures how much an asset moves when the market moves. Now, the way we can measure the Beta is by taking a scatter plot of the returns on say Ripple's cryptocurrency or any cryptocurrency and on the market. So here's a scatter plot where we have points that correspond to daily returns on the S&P 500 and on Ripple. Ripple returns on the y-axis, S&P 500 returns on the x-axis. We can see the enormous one-day Ripple return that we discussed in Lecture 2. Based on this, we can see that there really is not that much of a relation between market returns and Ripple returns. Mainly, it's one big cloud. So now let's talk about the Alpha. So the Alpha is the part of the return that the Capital Asset Pricing Model does not explain. That's why we call it the abnormal returns. So we take the return that would be expected return and we subtract the part of the return that we would expect from the Capital Asset Pricing Model, that's the normal return. Now, according to the Capital Asset Pricing Model, namely if the theory is true, all Alphas should be zero. So in any one sample, we might measure a positive Alpha but that should simply be statistical noise or survivor bias. So recall the traditional view that I mentioned before. According to the Capital Asset Pricing Model, cryptocurrency has no place in the optimal portfolio. But let's take a purely statistical view. These cryptocurrencies actually do have positive Alphas. They do exhibit abnormal returns on Bitcoin at 0.24 percent, on Ether at 0.57 percent, and on Ripple as 0.50 percent. So these Alphas, these daily Alphas are actually pretty high compared to other strategies that investors pursue. So which measure should we use? Alpha or Sharpe ratio? It makes a difference. Crypto looks better as an asset class if we think in terms of Alpha and I would argue Alpha is better. Now, why is Alpha better? Recall Portfolio Theory. What portfolio theory says is that investors should seek out the highest Sharpe ratio. But the highest Sharpe ratio of what? The highest Sharpe ratio of the total portfolio. The Sharpe ratio is an intuitive measure of the trade-off between risk and return. It's tempting to use it for any assets. But it is not the most useful measure when applied asset by asset, because it does not take into account covariance whereas Alpha does. Crypto, as we saw with the Ripple plot, has a low covariance with the market, and that makes the returns more impressive because it can be used to diversify the market portfolio. So in other words, if we're talking about a specific asset, we don't want to look at the Sharpe ratio which is the measure for the overall portfolio, we want to look at the Alpha. Now, these are closely related, because if you have a high Sharpe ratio portfolio and if you include an asset that has a positive Alpha in that portfolio, what that does is it pushes the investment opportunity set outward and increases the slope as shown in this diagram. So in other words, the two measures are linked. What you want is the highest Sharpe ratio portfolio. How do you get there? Well, if you have you start with the highest Sharpe ratio portfolio you can find and then if you include an asset that has a positive Alpha, what that asset does is it will make your Sharpe ratio even higher. So by this discussion, what should happen is that if you include cryptocurrency along with the S&P 500, you can actually get a higher Sharpe ratio portfolio by combining these cleverly together. Now, caveats. It is certainly not the purpose of this lecture to advocate for investing in cryptocurrency, these are just principles for how to think about the problem. Survivor bias, transaction costs, and risk measurement especially due to the short sample all might argue against investing and taking these expected returns seriously. So let's go back to this issue of covariance. We see that the return on the top three cryptocurrencies has been high. Now, this may in part reflect a resolution of the risk of these currencies, so in that sense they might be unlikely to be repeated, who knows? But they're still risky. So there may still be more high expected returns to come. But it's hard to measure expected returns in a small sample. Perhaps, we've mismeasured the expected returns. However, the covariance with the market and the associated Beta is less likely to be mismeasured. So the Beta on cryptocurrencies is about 0.5, it's not very high. This Beta is important; if the Beta is low and 0.5 is a pretty low Beta, then cryptocurrency has value as a hedge. So how can we think about this Beta? Well, if we want to really get to what's driving Beta, we have to think about what's driving the price of cryptocurrency. So let's for a moment go with a substance, a simpler and more basic benchmark theory called the Gordon Growth Model. What the Gordon growth model says is that, if an asset pays dividends that grow every year at rate g, and the expected return on the asset is r, then the price of the asset if we take next years dividend D, is very simple. It's D divided by r minus g. This comes from an infinite series formula. Now, strictly speaking, according to this formula, crypto's price should be zero all of the time, because there's no dividends to cryptocurrency. That's the theoretical view I mentioned before. However, if Bitcoin might be useful someday as a medium of exchange, then maybe we can think about the dividend a little bit differently. The dividend reflects the convenience of exchange. Now, this is not such a crazy concept. For example, it is typical to have a bank account even though the rate paid by a bank is often below that of the treasury bill rate and the treasury bill rate itself is often below the rate paid on say AAA corporate bonds which are very safe. So it's pretty well established empirically, that there's something called a convenience yield for cash-like assets. So one way to think about this dividend is it's a convenience yield should Bitcoin ever become a cash-like asset. So if this is the case, what does the pricing formula tells us about the covariance? Well, there are two offsetting effects. On the one hand, the greater economic activity, the greater the demand for cash of all kinds and thus the greater demand for currency. Also, both cryptocurrency and the aggregate market depend positively on technological innovation. Both of these would argue for positive covariance and high Betas. The better the economy does, the more demand there is likely to be for cryptocurrency, and the more likely crypto will be used as a medium of exchange. This should drive up the price. On the other hand, when is it likely that cryptocurrency would become a medium of exchange? Related, when might people really consider cryptocurrency as a store of value? I would argue that the times when those things might occur are likely to be times that are not very good. They might be times of disruption, times of high risk, times when governments are not reliable, namely bad economic times when the market is doing poorly. This is going to be what pushes the covariance and the Beta lower thus it is likely that a Beta of 0.5 reflects these two forces. What it means is that cryptocurrency possibly has a role on a portfolio as a hedging security. Thus to conclude, cryptocurrency returns implies Sharpe ratios close to those of the market. They have positive Alphas, abnormal returns, which suggests that they are good investment. No, this does not take into account survivor bias. Now, the traditional theory says, do not hold cryptocurrency no matter what, it pays no dividends, it has no role in the market portfolio. But traditional theory says, don't hold money of any kind. So crypto's role as a hedge against bad economic times makes it intriguing as an investment. But what's needed is a quantitative theory of cryptocurrency to take this into account.