- 4 Moment CAPM
The 4-Moment Capital Asset Pricing Model is based on two academic papers (Jurcenzko and Maillet, The Four Moment CAPM: Some Basic Results , working paper, 2002, and Hwang and Satchell, Modeling Emerging Market Risk Premia Using Higher Moments , working paper, 1999).
When financial assets are normally distributed, the historical asset return, the asset standard deviation and its covariance with the market are enough to estimate the asset expected return. This model is the 2-Moment CAPM developed by Sharpe (1964), Lintner (1965) and Mossin (1966). We claim that the risk in not only in volatility and linear correlation, but in skewness, kurtosis, systematic skewness, and systematic kurtosis. The model developed below and applied in AlternativeSoft’s platform is the Four-Moment CAPM, which accounts for the beta, the co-skewness, the co-kurtosis dependencies between the assets and the market portfolio.
The 2-Moment CAPM has the following form:
with systematic beta:
The risk premium of a non-normally distributed asset is equal to:
:: its market risk multiplied the market premium b1 plus
:: its systematic skewness risk multiplied with the systematic skewness market premium b2 plus
:: its systematic kurtosis risk multiplied with the systematic kurtosis market premium b3
If we assume that it is possible to construct a portfolio with zero beta, zero systematic kurtosis, and unitary systematic skewness, then the market premium of this portfolio will be b2. If we assume that it is possible to construct a portfolio with zero beta, zero systematic skewness, and unitary systematic kurtosis, then the market premium of this portfolio will be b3. Given these two assumptions, the market premium, b1, b2, b3, can be computed in a security market hyperplane as:
where E(R m ) is the world market expected annual return and R f is the expected annual risk free rate of return. SZ1,m is the systematic skewness between the portfolio Z1 and the market. SZ2,m is the systematic kurtosis between the portfolio Z2 and the market. By substituting b1, b2, b3, SZ1,m , and KZ2,m , in the Four-Moment CAPM equation and developing the terms, we finally obtain the asset i Four-Moment CAPM required rate of return:
We see that an asset required rate of return is composed of the risk free rate plus three premiums:
:: The first premium is the reward of having in the portfolio an asset which is contributing positively to the world market beta
:: The second premium is the reward of having in the portfolio an asset which is contributing negatively to the world market skewness
:: The fourth premium is the reward of having in the portfolio an asset which is contributing positively to the world market kurtosis.
To determine the expected returns for hedge funds, fund of funds or mutual funds using the Four-Moment CAPM, AlternativeSoft’s software platform is available.
- Bear Correlation
Bear correlation is the correlation between two marginal asset distributions during the first asset negative returns.
Instead of computing a classical correlation between an asset and an index, it is better to compute the correlation during marginal asset distributions or conditional asset distributions. This will capture both assets non-normalities and codependence asymmetries. If the conditional correlation is higher than the classical correlation, then the dependence between the asset and the index is unstable and asset’s extreme risks should be measured.
Correlation matrix between hedge funds and indices:
Bear correlation matrix: when the numbers below are higher than the numbers above, the assets are more correlated during equity market negative returns:
The Black-Litterman model was published in the Financial Analyst’s journal in 1992. It provides an estimate of future expected returns by combining equilibrium returns with an investor’s views. The resulting new vector of returns leads to potentially well diversified, intuitive, and stable portfolios. Implied returns are computed by using the following equation (reverse optimization):
with risk aversion coefficient (δ), asset covariance matrix (∑), and equal asset weights (w). Since the true expected returns (µ) are unknown, it is assumed that the equilibrium returns are reasonable estimate of the true expected returns:
ϵΠ ~ N (0,τ∑).
The matrix τ∑ corresponds to the confidence in how well the implied returns can be estimated.
K views in the model can be expressed as
P is a K x N matrix, each row of which provides a combination of assets for which we have a view.
q is a K-dimensional vector of the views about expected returns.
εq is a K-dimensional vector of errors, ϵ ~ N(0,Ω).
Ω is a K x K matrix expressing the confidence in the views.
Equations (2) and (3) can be stacked in the form
withI denoting the N x N identity matrix. Calculating the Generalised Least Squares estimator for µ results in the following Black-Litterman combined expected returns:
AlternativeSoft’s software platform uses the Black-Litterman combined returns as the input to compute optimal portfolios on the efficient frontier which will have: (i) low volatility (Mean-Variance Optimization), or (ii) small average extreme loss (Mean-Conditional VaR optimization), or (iii) low volatility, high skewness, and low kurtosis (mean-Modified VaR optimization). As an example of diversification achieved by using AlternativeSoft’s Black-Litterman based optimization, two figures are provided below. Figure 1 exhibits the portfolio weights on the efficient frontier obtained from classical mean-variance optimization. Portfolio weights computed by using BL model based mean-variance optimization are shown in Figure 2. Figure 2 displays that BL model based optimization produces a portfolio with more hedge funds in the portfolio leading to highly diversified optimized portfolio.
Figure 1: Distribution of weights based on classical MVO
Figure 2: Distribution of weights based on BL MVO
Read the full paper here.
- Cholseky Matrix
The matrix M has some interesting properties. It is used when simulating a portfolio of assets. The Cholesky decomposition separates a matrix E in two identical matrices M:
The matrix E is a matrix including the assets’ correlation.The matrix M is called Cholesky matrix. It is a lower triangular matrix. Multiplying the Cholesky matrix M with simulated random vectors will give linearly correlated random returns (see P.Wilmott, Derivatives , 1998, p.682).
These two measures, developed in 2010, are mixing beta (i.e. relation between an asset and a benchmark) and skewness (i.e. how two assets are related between themselves during extreme negative returns of the first asset).
Beta Coskewness measures how much an asset, within a portfolio, has historically reacted to extreme benchmark negative returns. An asset with a positive beta coskewness provides portfolio protection when the benchmark has extreme negative returns. Positive beta coskewness is good.
Beta Cokurtosis measures how much an asset, within a portfolio, has historically reacted to extreme benchmark returns. An asset with negative beta cokurtosis provides portfolio protection when the benchmark has extreme positive or negative returns. Negative beta cokurtosis is good.
In the example below, the benchmark is the S&P500 and the graph displays the beta coskewness for each fund in the portfolio. This means during S&P500 extreme negative returns, the funds BlueTrend and MAN AHL are protecting the heavy S&P500 losses.
The Beta Coskewness and Beta Cokurtosis measures are available in AlternativeSoft platform.
Non-normal distribution models are used more and more in order to price financial assets. We provide some basic questions and their respective answers on non-normal distributions:
(1) What is the distribution skewness when you see more returns on the left of the mean?
(2) What is the probability of having a return lower than -2.33 standard deviations?
(3) What is the kurtosis of a normal distribution ?
(4) Which is the less dangerous for a risk averse investor?
a) Positive skewness with kurtosis>3
b) Negative skewness with kurtosis>3
c) Negative skewness with kurtosis<3
(5) Assume a normally distributed fund with an historical annualized return of 10% and an annualized volatility of 5%. How many years should you wait in order to have a monthly return of -5%?
a) 37 years
b) 137 years
c) 3137 years
(6) Assume you invested in the 3 best S&P500 monthly returns and you have shorted the 3 worst S&P500 monthly returns, since 1990. What is this 6 dates cumulative return?
(4) Positive skewness with kurtosis>3
(5) 2822 years (i.e. [1 / normdist(-5%, (1+10%)^(1/12)-1,5%/12^0.5,1)] / 12 )
(6) 98%. This means you made 98% in 6 months.
Diversification, a finance concept developed by Markowitz (born August 1927, PhD thesis in 1955, Nobel in 1990), is the basis for portfolio construction, assuming normally distributed assets. In order to measure if a portfolio is diversified enough, a technique is to ‘push’ the assets correlation to 1.0 and compare the portfolio with the historical assets correlation and the portfolio with the assets correlation equal to 1.0. The following formula allows to measure how much the portfolio is diversified (i.e. 100% implies highly diversified, 0% means no diversification):
where Ω is the portfolio covariance matrix and w is the portfolio weights vector.
Computing the rolling diversification ratio allows to visualize if a portfolio has been diversified enough and is worth the fees paid by the investor. For example, in the graph below (i.e. 100% means a highly diversified portfolio, 0% means no diversification), one could argue this portfolio could and should be better diversified:
The diversification ratio is available in AlternativeSoft platform.
- Exposure Analysis ‘Understanding fund exposure is the first step to understanding risk. However, contrary to popular belief, exposure does not equal risk. Risk is only introduced through exposure to risky or volatile assets. As indicated in the graphic, exposures can vary significantly over time, so single point references often do not show the complete picture. Investors will want to know the portfolio’s exposure over the period in which performance is being presented. While averages are useful, presenting the information in time series is preferred as it allows investors to view intra-month exposures and to check for style drift (Catalano et al, January 2013)’.
Exposure analysis allows the user to answer the following questions to form a vital component of investment due diligence for both Asset Selection and Portfolio Construction.
Who are we exposed to..? (E.g. Which funds/managers)
What are we exposed to..? (E.g. Instruments, Sectors, Strategies, Categories)
Where are we over/under-exposed to..? (E.g. Geographies, Sectors)
When were we exposed to..? (E.g. How have exposures evolved over time?)
Why were we exposed to…? (E.g. Does my investment thought process coincide with my actual exposures?)
Please see the following brief presentation to understand more:
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- Fund of Funds
Investing in funds requires qualitative and quantitative analysis. The investor should construct a portfolio combining different strategies and the latest quantitative techniques.
AlternativeSoft provides a software platform to construct optimal portfolios with hedge funds, UCITS III, ETF, mutual funds and fund of hedge funds. The software platform is dedicated to portfolio managers, advisors, banks and other financial institutions investing in funds.
In an Investment Process, AlternativeSoft advises investors to follow the following five quantitative steps.
- Modified VaR
In risk measurement, the Value-at-Risk concept is more and more used. Nevertheless, many working papers use the Value-at-Risk concept when return distributions are not normal or when the multivariate return distributions are not elliptical.
Remember that Value-at-Risk (daily or monthly), with normally distributed returns is:
μ = expected asset return (daily or monthly)
zp = distance between u and the VaR in number of standard deviation. In other terms, number of standard deviation at (1-zc) or -1.96 with p = 95% probability
σ = standard deviation (daily or monthly)
The Modified VaR (MVaR) uses not only the volatility as risk, but as well skewness and kurtosis. The Modified VaR is:
where S is the skewness, K is the excess kurtosis, and zc is the distance between the portfolio returns and its mean in terms of standard deviation number. If the risk is measured only with the volatility, the risk is often underestimated. The assets returns are negatively skewed and have fat tails. Consequently, the volatility alone is not able to account for that.
The Modified VaR model has been developed by AlternativeSoft and then published in the Journal of Alternative Investments, Modified Value-at-Risk Optimization with Hedge Funds, Fall 2002. The Modified VaR is applied in AlternativeSoft’s platform. The skewness and the kurtosis effect is high if the VaR is computed at 99%. For example, for a portfolio invested in 50% stocks and 50% government bonds, computing the risk with standard deviation only underestimates the risk by more than 35%.
AlternativeSoft’s platform considerably reduces this problem by computing optimal portfolios using the mean, the standard deviation, the skewness, the kurtosis and by minimizing the probability of having extreme portfolio losses.
- Optimization Techniques
AlternativeSoft’s ‘Optimization’ Button (Button 07) is dedicated to delivering a range of easy to use and powerful optimization techniques which can be utilised by users. This is an ideal place to create optimal portfolios which conform to many constraints such as weights, liquidity and contribution to volatility. Many users also like using this to create benchmark portfolios which they can then compare against their real-world portfolios and therefore this button is excellent for bringing together modern portfolio theory & professional portfolio construction. By hovering your mouse over an optimization technique below, you will see a brief description about the optimization technique.
Our optimization algorithm is recognised as being both fast robust.
The beta of the CAPM measures the linear market risk. The first to assign that the beta was not linear were G.Pettengill, S.Sundaram and I.Mathur, “The Conditional Relation Between Beta and Returns”, Journal of Financial and Quantitative Analysis , 1995. They showed that the beta of a stock is different depending if the market is up or down. They did two regressions: one with positive market returns and one with negative market returns. Another approach has been developed for hedge funds by Favre and Galeano, “Hedge Funds Analysis Using Loess Fit Regression”, Journal of Alternative Investment , Spring 2002. They showed, with a powerful statistical technique, called Local Regression,, that several hedge fund indices have non-linear option payoffs.
By using polynomial regression between the stock and the market returns, it is possible to see if an asset has a higher exposure on the downside.
As many hedge funds managers use derivatives or are active managers, their returns are not linearly related to the SP500, for example. This is why the polynomial regression is appropriate. The figure below exhibits that the asset (vertical axis) is not exposed (even non-linearly) to S&P500 negative returns (left on the horizontal axis).
- Sharpe Vs Modified Sharpe Ratio
The Sharpe ratio is used to compare portfolios, stocks or indices. This ratio is:
The Sharpe ratio is useful when assets are normally distributed or when the investor has a quadratic utility function. This means that the portfolio is completely described by its mean and volatility. As soon as the portfolio is invested in technology stocks, distressed companies, hedge funds or high yield bonds, this ratio is not anymore valid. In that case, the risk comes not only from volatility but from higher moments like skewness and kurtosis.
The Modified Sharpe ratio is the ratio of the excess return divided by the Modified Value-at-Risk. The modified Sharpe ratio is:
where the Modified VaR is equal to:
A simulation gives a portfolio view for the future with some given probabilities. The investor must be aware of the probability to be below a given return in the next x years. The simulation below shows that the portfolio has 50% probability to be higher than 20.9% in 2 years.
Skewness measures the distribution asymmetry. A risk-averse investor does not like negative skewness. A distribution with positive skewness has more returns far away to the right of its mean return, as shown below.
Kurtosis measures the fat-tail degree of a distribution. A risk-averse investor prefers a distribution with low kurtosis (i.e. returns not far away from the mean). A distribution with positive kurtosis exhibits a peak in the middle and fat tails versus a normal distribution. A distribution to be normal should have an excess kurtosis equal to 0. In case of a positive skewness, it is possible to have a high excess kurtosis and to have no future extreme negative returns. The extreme returns will only be positive. This is only possible when the skewness is positive. As soon as the skewness is negative, the impact of a high excess kurtosis affects the extreme negative returns. The most interesting thing is when the return distribution has a skewness lower than -1 and an excess kurtosis higher than 1. In this case, the probability to have sudden high negative returns increases. For a distribution with a skewness of -1 and an excess kurtosis of 5 (for example, technology stocks, media stocks, telecom stocks or hedge funds in arbitrage strategies), a claasical approach will conclude that the investor will not lose more than -3.5% in the next 1 day with 99% probability. An approach, accounting for skewness and kurtosis, shows a -7.4% loss in the next 1 day with 99% probability. This is exactly what one observes on the equity market. The difference is large: an underestimation of 111% of the downside risks (i.e. 7.4%/3.5%-1=111%) when using volatility only to measure the risk. To conclude, for optimization and simulation, the technique must account for volatility, skewness, and kurtosis as soon as they are significant and as soon as the investor is risk averse.
- Stress Test
According to dictionary, Stress Testing is a form of testing used to determine the stability of a given system or entity. In our case, finance, Stress Test allows to measure how much return an asset or a portfolio would deliver in case of dramatic change in one or several economic factors like S&P500, EUROSTOXX50, volatility VIX, 3m T-bill, 10Y T-note, Inflation, Aaa Yield or Baa Yield. For examples:
- What happens if equity markets crash by more than -30% in April 2011?
- What happens if 10Y interest rates go up by at least +1% in March 2011?
- What happens if oil prices rise by 100% in 2011, inflation increases by +3% and gold collapses by -50%, altogether?
Some software provide the asset stress return with some probabilities, use simulation or use copula to describe the relation between assets. Other software are so complex that it takes two hours to stress test one asset or they do not allow the stress test of several economic factors or they do not stress the correlation between the economic factors (i.e. coefficients correlation between economic factors increase when the market crashes). A good stress test model should be transparent to the user, easy to explain to a client, allow to stress test an asset which has never had an historical negative return and correctly forecast future asset stress returns.
The following model computes an asset or portfolio stressed return with a scenario on multiple economic factors and on their respective correlation coefficients:
Computing the stress test for a hedge fund (10 years track record, max historical monthly loss of -3.82%) using the above model would give the following results:
1) The green cells represent the economic factors’ assumptions
2) The hedge fund return (i.e.-26.07%) over one month is shown in the last column, assuming the above assumptions are realized:
3) The hedge fund return (i.e.-58.47%) over one month is shown in the last column, assuming the above assumptions are realized and the economic factors’ historical correlation coefficients are increased:
The Stress Test model is available in AlternativeSoft platform.
- 4 Moment CAPM
- Bear Correlation
- Cholseky Matrix
- Exposure Analysis
- Fund of Funds
- Modified VaR
- Optimization Techniques
- Sharpe Vs Modified Sharpe Ratio
- Stress Test
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