32808336.deep-dive-into-modern-portfolio-theory.html

<p>Portfolio theory is usually done using a utility function - namely a mapping between wealth and happiness. A naive option is to use a linear function.</p><p>U(w) = w</p><p>100% loss of net worth is as unhappy as a 100% gain is happy. 80% loss is as unhappy as 80% gain is unhappy.</p><p>Other examples include:</p><ul><li><p>quadratic utility, U(w) = w - (s^2 / 2) w^2</p></li><li><p>power utility, U(w) = w^r / r</p></li><li><p>log utility, U(w) = log w</p></li><li><p>taylor series, U(w) = a_0 + a_1 w + a_2 w^2 + a_3 w^3 + …</p></li></ul><p>The intuition behind log wealth is that your net worth halving (50% loss) is as painful as your net worth doubling (100% gain). And that net worth going to zero is infinitely bad. Log wealth is also obtained by <a href="https://doi.org/10.1007/s11579-020-00270-1">assuming r tends to zero in power utility</a>.</p><p>Quadratic utility is an approximation where higher order terms are assumed unimportant in the taylor series. It assumes each investor has a value s (or sigma) which denotes the maximum variance in their portfolio that they can tolerate. Variance is the average of (squares of) deviations from what is expected. More possible deviation means a wider spectrum of outcomes. We square these deviations because larger deviations are especially bad as compared to smaller ones.</p><p>Sharpe ratio maximisation is another common technique. Sharpe ratio is the ratio of returns and variance. Sharpe ratio maximisation is also obtained by <a href="https://doi.org/10.1007/s11579-020-00270-1">assuming r tends to negative infinity in power utility</a>.</p><p></p><h3><strong>Markowitz efficient frontier</strong></h3><p>Markowitz proved that if each investor can be approximated to use quadratic utility with their unique s (variance), then you’ll obtain an “efficient frontier” for the market as a whole.</p><p>More specifically if you knew the probability distribution of every asset under the sun, then you could construct an optimal portfolio for each investor with their own unique s. In general the higher the s the more the returns.</p><p>This seems intuitive. Businesses are all about uncertainty. Those who can make bold gambles and take on more uncertainty are rewarded for this.</p><p>There will also always exist a lot of suboptimal portfolios - portfolios for which the variance s (uncertainty) is same but have lower reward.</p><p></p><h3><strong>The efficient frontier is dynamic</strong></h3><p>If the market is indeed so “efficient” - why aren’t all these high risk high reward consumed? The answer is obviously different investors have different s, some investors don’t want to tolerate high variance in their net worth. Being at risk of losing or gaining a lot of wealth is psychologically stressful. Many people prefer lower but more stable returns.</p><p>Investor's’ preferences can change with time. If more investors want to take on higher risk, they will flock to the opportunities with higher risk. But opportunites typically have finite capacity. If too many people buy a stock, the stock will appreciate in value, and now the reward for anyone else who buys the stock is lesser (although variance may still be the same). Therefore the efficient frontier can move in response to investor preferences. If it moves, the optimal strategy is to exit your portfolio and calculate the new optimal one for your risk tolerance. Assuming ofcourse, sufficient liquidity and sufficiently low fees.</p><p></p><h3><strong>Transaction costs and liquidity constraints</strong></h3><p>Pretty self-explanatory. These got ignored by the original portfolio theory although more modern analysis often include them.</p><p></p><h3><strong>Price processes</strong></h3><p>Markowitz assumed all investors have equal knowledge of a probability distribution for each asset’s price. This distribution is often taken to be a geometric brownian motion. This is a fancy way of saying that the (log of the) price does a random walk. Every moment it tosses a coin and moves a little up or a little down with seemingly no direction. Over time you will find the log of the price (on average) moving away from its original value at a slow rate.</p><p>If you are more familiar with statistics, a “continuous” random walk basically leads to the (log of the) price being normally distributed, where the variance of the distribution increases linearly as time progresses.</p><p></p><h3>Unknown unknowns</h3><p>Most theory assumes people know a probability distribution for asset prices with 100% certainty. In other words, the uncertainty in price and its nature is completely certain. This is not true in practice which is full of unknown unknowns. This is almost a philosophical problem. If I believe there’s a 90% chance this asset is normally distrbuted and 10% chance of “unknown” things happening, what do I do with the unknown part? Do I assume unknown distributions are always normal distributions? Or uniform? Or is log of price normally distributed? This uncertainty about uncertainty is hard to study, you can only bound it and avoid investing based on assumptions on it.</p><p></p><h3><strong>Technical barrier</strong></h3><p>Clearly some investors have better technical knowledge with which they can forecast this probability distribution than others. This means that the efficient frontier isn’t completely public. Imagine a grey region ahead of the public efficient frontier where privately known portfolios marginally outperform the publicly known frontier. Hedge funds spend a ton of money and recruit talent to help discover this. If enough people discover an investment it’s as good as public and just becomes a part of the efficient frontier. But it seems naive to assume that just because most people can’t seem to beat the publicly known frontier, that no one is beating it.</p><p>Professionals running companies often spend months to years deliberating on decisions. It seems naive to assume either a) no one knows better than these professionals or b) everyone knows better than these professionals, about the consequences of their decisions.</p><p></p><h3><strong>Efficient market does not mean fair price</strong></h3><p>The reason why having superior technical knowledge often doesn’t result in superior gains is noise. You may forecast the impact of one business decision taken by a company to be 20% higher than what other investors forecast. However the stock price is impacted not just by this decision, but by multiple other factors that impact the price. Even if knowledge of all those factors is public, they contribute sufficiently to variance that you aren’t gaining much more by holding this stock when adjusted for risk. Hedging all these factors is hard, and doing so usually leads to very low outperformane of the efficient frontier. Which means you often need to have superior technical knowledge over a vast number of companies, so that their combination gives a portfolio with both decent returns and risk that can outperform the frontier.</p><p></p><p>The “fair value” of a company is estimated by predicting how much revenue the company will make in its lifetime. Efficient frontier theory does not dictate that the stock will reach this fair value. Suppose the fair price of a stock is calculated to be $20, but it is currently worth $19. The risk-adjusted returns of buying this stock may be sufficiently low that buying it at $19 is not an optimal portfolio. Neither is a portfolio that in part buys this stock among other assets. It not being optimal doesn’t mean it’s not profitable, but that there exist other better opportunities for the same level of risk. Clearly we can see that stock prices have a certain price band around the fair price in which they can move around while being efficient. Assets with higher volatility have wider bands and bands typically range more on the upside. This is because shorting is typically more constrained as compared to longing - not to mention an interest rate being paid. There is also the phenomenon of reflexivity of longs, discussed later in this article.</p><p></p><h3><strong>Market behaviour informs expectations regarding risk</strong></h3><p>Let’s assume a crazy market where nobody knows what cashflow is or how to value companies. A company makes a bold new decision that you think is good. You are able to obtain shares for this company that’s worth $20 according to you, at $5. The problem with buying this is that you need to now hold these shares for years, possibly decades. Why? Because you have no faith in the market ever becoming rational - which means the only way you profit is if the company indeed makes enough revenue to deserve its fair price of $20. Clearly that has some risk and is not for everybody. Contrast this with the share price being worth $16 when you think it should be worth $20. Now you can have more faith that soon enough the market will get smarter and reprice the stock. That way you don’t need to hold the stock for years to profit - just long enough till the market reprices. A faster repricing means you make your gains faster and free up capital for the next opportunity. This is one place behavioural trends come into the picture. Volatility tightens when investors get more rational.</p><p></p><h3>Investments can be reflexive and self-fulfilling</h3><p>It is generally assumed that if the fair price of the company is $20, and you buy a lot at $15, the price eventually moves to $20 and that’s it. In practice, you are enabling the company to get more investment. If the company gets more investment it can generate more revenue. This in turn increases the fair price. Fair price can reflexively keep increasing as more investment enters and is approportiately utilised (possibly but not necessarily by issuing more shares). The effect of investments is especially pronounced in small companies. Even if a company’s leadership is incompotent and cannot fulfill their targets given a certain level of funding - if you over-fund them they will do something after all. Or atleast they might. This also explains why venture capitalists often tailgate each other’s investments, and why fundraising is a very important role for startups to play.</p><p></p><p>Reflexivity also works in the opposite direction. A signficant short interest on a company indicates that the company will struggle to raise new investment - since that short interest needs to be cleared first. Low expectation of being able to raise new investment can in turn depress the stock price, making the shorts more profitable.</p><p></p><p>Clearly this reflexivity can be more strongly utilised by larger funds. There is a school of thought which believes that exploiting this beyond a certain limit is unethical or harmful to society - especially in the case of shorting.</p><p></p><h3>Tail risk</h3><p>So far we have assumed a single parameter that determines risk tolerance - usually variance s from quadratic utility, or alternatively the exponent from power utility. However we should always pay special attention to the constant term in utility. Namely what happens to utility when wealth goes close to zero, or hits zero. Even a small probability of this happening is dangerous, as the utility from this happening is assumed negative infinity or atleast highly negative. If we assume geometric brownian price process, then tail risk is well-defined, but clearly this is only an approximation. Deviations from this model are most important when it comes to tail risk - the probability of an asset or portfolio going to zero or close.</p><p></p><p>Estimating the tail risk of an asset often requires fundamentally different analysis than estimating either returns or variance. Examples of tail risks would be earthquake or war. More common ones could include the CEO dying, theft/sabotage, a larger company entering your sector, or a power grid failure. It is impossible to hedge against all forms of tail risk. Therefore if we are considering a second parameter to define risk, and we should, it would be tail risk.</p><p></p><p>Taking on unhedgeable tail risk is very rewarding, often even more rewarding than just exposing yourself to variance (or volatility). Very few people can take on concentrated bets with high tail risk, so they end up diversifying their high tail risk bets to oblivion instead. Fundamentally new innovation typically has high tail risk - since reducing tail risk has either not been studied well in that sector or is impossible to do without further innovation. Clearly this kind of research and innovation is also what often rewards the world the most, if there exist those who are willing to undertake it or invest in it.</p><p></p><h3>Time horizons</h3><p>Firstly, different assets can have different levels of volatility in different time horizons. A total stock index has higher volatility when considering a daily or monthly timeframe as compared to a 30 year timeframe. This is because the monthly timeframe will likely be in either a bear or a bull market whereas a 30 year timeframe will contain enough of both and hence smooth out volatility.</p><p></p><p>Secondly, a utility function seems to have no notion of time. It assumes an immortal being who will keep maximising utility forever. In practice, people acquire wealth to attain goals. Sometimes these goals are well-defined, sometimes they are not. In other words people change both their risk tolerance and investment horizon with time. Someone who wants to be able to buy a house within 10 years with 90% certainty will invest differently from someone who wants to just be able to buy a new watch within the next 3 months with atleast 30% likelihood.</p><p></p><h3>Capital costs</h3><p>Efficient frontier tends to assume everybody can enter every opportunity. This is not true. An opportunity that will close up with a million dollar investment may not even be worth researching for a $100 billion hedge fund. A guy with $1000 on the other hand may do well to invest in this opportunity. This gives smaller investors an advantage in smaller markets, if access is allowed.</p><p></p><p>On the other hand, larger opportunities often require a larger investment to start off. Most trading we consider today is secondary markets of publicly traded companies - which are more open to people of all investment sizes. But for instance for a startup raising a seed round, they need a minimum amount of investment. It’s X or nothing, X/100 won’t help. There are also communication costs (including regulatory) associated with getting 100 retail investors to invest X/100.</p><p>Even if larger opportunities don’t require a larger investment, they can benefit from it. Assume a portfolio composed of indepth research into multiple companies - which overall outperforms the efficient frontier by 1%. Suppose it takes 1000 man hours to do this research. Someone investing $1000 will probably not bother to do this. A fund with $1 billion on the other hand can easily spend $400k hiring the educated manpower needed to do this research. This gives larger funds a major advantage in terms of research. Research is a fixed cost whereas returns are a fraction of investment.</p><p></p><h3>The everything bubble - a macro perspective</h3><p>Portfolio theory assumes opportunities are finite, but that there exists an infinite capacity for money to “sit out” of the opportunities. In other words, fiat money is infinitely liquid and infinitely stable.</p><p>In practice this is not true, fiat too is an asset of its own with finite capacity. Capital entering or exiting fiat can in fact affect the “price” of fiat. Here one will have to assume something like purchasing power to be the true wealth and fiat to be yet another asset. In this perspective, everyone needs to be 100% invested - fiat counts as an investment too. This means even if one’s wealth is growing slowly, or worse, expected to shrink, there is no “sit out” option. This makes sense in a recession or an apocalyptic event, where global wealth and purchasing power shrink. People will try to pick the least worst option here. Minimise risk-adjusted loss instead of maximise risk-adjusted profit.</p><p></p><p>A macroeconomic perspective will consider forex - movements between one fiat currency to another. It will consider monetary and fiscal policy, government being akin to a “company” that runs the fiat business. It will also consider the theoretical possibility of fiat collapse due to hyperinflation - a highly reflexive tail risk.</p>