While I am a believer of APT more than of CAPM, I will share some of my findings on CAPM. CAPM has many flaws: there are capital taxes, high transaction cost on illiquid securities with few floating shares, licensed leveraged funds do influence prices with outsized positions, different analyst has a different expectations of a fair price in the same time horizon, we see animal spirits overwhelming rationality after years of crowd risk aversion, stagnant performance following GFC, sophisticated quantitative funds do acquire proprietary data from wide ranging sources as inputs in their forecast model. By S&P Global, its follows the CAPM theory that the market portfolio (or tangent portfolio) consists of all assets in all markets, where each asset is weighted by its market capitalisation. This should include all types of asset including illiquid assets such as real estate, machinery, private companies and intangible assets like human capital and patents. However, such intangible and illiquid assets have a wide interval of expected price and lack of transactions transparency perpetuates the difficulty of its measurement. So most market participants turn to the public listed equities market with the largest overall market capitalisation, S&P 500.

Shown below, the market cap weighted SPX members is not equal weighted (100%/500=0.2%). While its earnings yield can be the same, market cap do increase as a result of increased issuance of shares rather than higher earnings. Apple has a 3.6% weight in SPX and attributes to **market cap weighted** SPX performance 18 times more than that of a **equal weighted** SPXEW. Weights of SPX portfolio constituents. So did CAPM contradicted its assumption of ‘broadly diversified’ when its tangent portfolio is market cap weighted, or 50 out of 500 companies totals 50% weight in SPX? Not necessarily… If we decompose the total volatility of SPX, smaller weighted companies may have a higher risk contribution to total volatility. That said, fears of a all time low VIX at 9.32 (aka a 32% chance that next day’s performance exceeds 9.32/sqrt(252)=0.587% calculated from 1m ATM options) led to whistle blowing from legendary macro trader Paul Tudor Jones that risk parity funds like AQR capital are supporting an ‘unsustainable equities valuations not seen since 2000’. Besides a cap on risk contribution from a single instrument after accounting for correlation between that instrument and others aka marginal VAR, security selection that concentrates exposure to those with positive forecasted alpha would help to prevent the worst scenario coming true. Given the postive tracking error or active return from SPX against benchmark SPXEW, we can deduce that this alpha is generated from being overweight in large market cap securities. This can be confirmed with Fama French 3 factor model, where one factor (**SMB**) measures the active return of **S**mall-**M**inus-**B**ig market cap securities, is negative.

CAPM also states that SPX (tangent portfolio) lies on both the** efficient frontier** that maximises **expected** returns relative to risk and on the **best possible capital allocation line with the highest sharpe ratio (CML)**, assuming a well diversified portfolio and the denominator, total volatility, is rid of idiosyncratic risk. From here onward, * Rm* refers to the

**expected**market return based on CAPM convex optimisation of its risk-return profile, which on the other hand, is based on individual assets

**expected**returns. This post don’t intend to highlight any particular benefits of forecasting and

**expected**returns here is defined as the historical 3m average returns.

**Beta**is a measure of the exposure to the

**systematic risk**that cannot be diversified away, which on the flip side of a coin, demands an

**equity risk premium**(ERP). If debt growth should align with asset growth, ERP should be tied to bonds yields such that bond selloffs should lead to higher ERP and higher expected returns or in other words, a decline in asset prices as debt, which ranks higher in seniority, is endangered.

Fortunately, convex optimisation was already implemented in **Python **in a package called **cvxopt**. Why convex? Asset prices are assumed to follow normal distribution. While countless arguments are against the former assumption, insisting that higher moments (skewness, kurtosis) matter given fat tails and wider returns dispersion, we test out simulated normal random variables as returns. Methodology from Quantopian was brilliant as it build portfolios aka linear combinations of individual assets, then create linear combinations of linear combinations of individual assets (aka of portfolios) , creating a linear algebra convergence on norm space. Weightings are supposed to be equally weighted, but are simulated as uniform random variables with jitter. Results are shown below. Firsthand, we observed the **benefits of diversification**. From top left to bottom right, simulated portfolio total volatility reduced from minimum volatility @ 0.400 with 10 assets to 0.10 with 100 assets, equally weighted with jitter, given expected portfolio return around 2%. There is no leverage with sum of weighs cap at 100% and long only.

*5, 10, 50, 100 assets in portfolio **(Simulated Portfolio: Individual asset returns ~ N, Equally Weighted)*

If we look at the indexed YTD returns of SPX members below, there is indeed a wide range of returns +/- 25% YTD. This shows that there are choice of assets available and for diversification and security selection with the goal of minimising portfolio variance.

Next, given the 80-20 skewed weightings in our tangent portfolio, market cap weighted SPX, pareto probability density was used to simulate members weight. Higher alpha will give higher weightings to a few randomly selected securities, a more concentrated exposure and less diversification. As seen below, we simulated increasing number of assets from 5, 10, 50 to 100 using alphas 1, 2 and 3 respectively.

*Portfolio with randomly Pareto weighted securities, alpha=1*

*Portfolio with randomly Pareto weighted securities, alpha=2*

*Portfolio with randomly Pareto weighted securities, alpha=3*

However, concentrated positions do move out the efficient frontier, indicating the benefits of security selection, being overweight/underweight in particular members of a tracking benchmark index. A general finding is that more concentrated positions (i.e. higher alpha) leads to a wider range of +/- portfolio returns and higher portfolio volatility. Alpha=1, 100 assets portfolio volatility is around 15%, whereas alpha=3, 100 assets portfolio volatility is around 23%. As mentioned earlier, since the efficient frontier is a norm space of linear combinations, it will not be affected by the choice of securities to be exposed as it represents the same space N=100. Variation in efficient frontier looks bounded by the most outward and most inward frontiers.

If we reshuffle the beta equation as a ratio, we get the **Security Market Line** (SML). It implies that the expected return of a risky asset is supported by a risk-free asset floor, plus the amount of exposure to systematic risk multiplied by ERP for holding the systematic risk. More exposure to systematic risk meant more compensation and higher expected returns. However, this theory has been debunked by the outperformance of active return funds betting against beta, suggesting that beta changes over time as do relationships across assets and volatility of assets i.e. good times for growth stocks are bad times for value stocks. This is also the main reason why I believe APT over CAPM, as risky assets carry a bundle of different risk exposure, not just ERP, which means it can be in overlapping good times to reap yields as well as in bad times to tolerate through drawdowns.

Using the latest SPX members’ returns as of 30 Jun 2017, ERP stands at 0.888%, and SPX index is expected to outperform risk-free rate by ERP, ignoring intercept as returns are adjusted to risk free rate. Yes, think about what happens when risky assets expected returns are less then risk free assets expected returns. Risk averse investors will look like gamblers when that happens!

If we look at sector indexes, risk premium for holding exposure to financials, energy and consumer staples are negative, while positive for healthcare and IT. **In other words, holding negative risk premia risky asset is expected to yield lower returns than risk free asset expected returns, a drawdown in bad times that won’t payoff in good times as higher beta leads to lower expected returns. However, negative risk premia doesn’t mean that currently investors prefer them over risk free assets, but that, as time pass and risk premia converges to 0, his expected return will be negative too. However, as long as risk premia don’t converge over time, staying negative ERP, its still good to carry.** Generally, the common interpretation is that securities above SML is considered undervalued, vice versa, regardless of the sign of slope given that for the same beta, you expect higher returns. Some digression here, negative relation between returns and beta for financials and consumer staples could be due to the fact that these industries earnings cycle are procyclical with the economy’s business cycle. Banks target a constant leverage ratios as part of its balance sheet active risk management. As the market rallies, banks holdings of equities (assets) rises and this give them capacity to increase their debt from deposits and short term borrowings to make further loans (assets). Say the bank is a not as risk averse as other banks and it takes more risk on its books. A bank’s beta above 1 means a more destabilising feedback loop where as the market rallies, it increases its risk more than proportionally. Risk premium for the bank has to rise and hence expected return has to fall.

Data is from google/finance. One day returns was used to determine the efficient frontier based on CAPM convex optimisation of mean-variance. **Expected** individual asset returns are 3m average of historical 1-day return. Time horizon is thus 1 day, only things is that expected individual asset returns are smoothed out for better visualisation. We see that forecasted returns for energy is below the frontier, as well as real estate. Forecasted returns for consumer staples and utilities are defensive stocks as their one-day variance are hugging the global min variance. While other sectors do have a handful of individual asset forecasted returns to be outside frontier, healthcare and industrial have more assets outside frontier, indicating that these 2 sectors are potential outperforming sectors.

As we saw the crowding in hedge fund industry, we also felt the oblivious closing of many funds before they survived for 5 years or maintained a 1b AUM. This push a trend of investors shifting from active fund management to passive index trackers, following the market portfolio. Is there really no more alpha to be harnessed from active fund management i.e. market have become perfectly efficient? Overall, evidence does suggest that betting against beta do deliver higher active return over benchmark i.e. +0.05% active return for diversifying away all systematic risk. This makes sense since

*alpha = E(asset return) – beta*ERP*

Below shows that choice of sector allocation and individual security selection within exposed sector contributes to a positive active return over its benchmark. Some sector favor more systematic risk, whereas most are otherwise.

A eod close-to-close backtest was done with the strategy being, long top K out of 500 securities ranked with highest alpha, to evaluate whether alpha does persist over time. Rebalancing is done on 1 day and 10 days basis i.e. select top K securities at time T, hold till T+10 days then rebalance to top K, process repeats. In other words, we only work with historical alpha with no hindsight biases. Result is, too frequent rebalancing won’t give enough time for highest alpha to deliver its alpha and holdings for 10 days give better performance, indicating high alpha does trend over time. Top 100 securities returns can be seen as benchmark SPX returns, as SPX weight distributions are like pareto.