Understanding Variance, Covariance, Correlation, and Efficient Frontier in Risk Management, Summaries of Design history

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Lecture Notes MTH6113: Mathematical Tools for Asset Management Dr Kathrin Glau; Dr Linus Wunderlich March 12, 2022 Contents 0 Preliminaries 3 1 Efficient Market Hypothesis (EMH) 3 1.1 The Weak Form of EMH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 The Semi-strong Form of EMH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The Strong Form of EMH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Criticism and Use of the EMH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Stochastic models of long-term behaviour of security prices 7 3 Risk and return 13 3.1 Shortfall probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Value at Risk and α-quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Stress test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Mean-variance portfolio theory 21 4.1 Introduction to portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 Mean & variance of the portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3 Attainable sets of portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.4 Minimal Variance Portfolio (MVP) . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.5 Short selling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.6 Efficient frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.7 Adding a risk-free security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5 Factor models of asset returns 28 5.1 Single factor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6 Pricing 32 6.0 Mean-variance portfolio theory for several assets . . . . . . . . . . . . . . . . . . . 32 6.1 The Captial Asset Pricing Method (CAPM) . . . . . . . . . . . . . . . . . . . . . . 34 6.1.1 CAPM formula: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.1.2 The security market line (SML) . . . . . . . . . . . . . . . . . . . . . . . . . 35 6.1.3 Efficient portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.1.4 How to use CAPM? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.1.5 Discussion of the validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.2 The arbitrage pricing theory (APT) . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7 Utility Theory 39 7.1 Reminder: convex and concave functions . . . . . . . . . . . . . . . . . . . . . . . . 41 7.2 Expected utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 7.3 Pricing lotteries based on utility theory . . . . . . . . . . . . . . . . . . . . . . . . 44 1 8 Behavioural finance 45 2 1.3 The Strong Form of EMH Nobody can consequently outperform the market with their investment. The line of thoughts is very similar to the other two cases. 1.4 Criticism and Use of the EMH In general, is difficult to test the hypotheses, as the primary information is not available. In regards to the semi-strong form of efficient market hypothesis, one can study the influence of information releases on prices of financial instruments. There is, stronger criticism against the validity of the strong hypothesis: if we believe that insider trading is not profitable, that has strong consequences. Many cases of insider trading are documented, so one cannot argue that they do not exist. In order to have sufficient insider trading so that prices reflect all insider information, a significant number of insiders need to trade, such that the price can reflect their private information. -10 -5 0 5 10 Log-return day t, % -10 -5 0 5 10 L o g -r e tu rn d a y t -1 , % Figure 1: Scatter plot of subsequent plots for GE’s returns within over 55 years. The empirical correlation 1, 46% is not statistically significant. There are also arguments based on data that support the EMH: the scatter plot 1 shows that returns of subsequent days are uncorrelated for a specific time series of prices. This means that the autocorrelation of the stock returns corr(Rt+1, Rt) ≈ 0. This empirical observation has been repeatedly made for other asset price time series as well thus underpinning that the future price cannot be predicted based on the past and today’s price. At least this is evidence against a very basic form of predicting the price based on the price history thus supporting the weak EMH. Some studies have investigated the possibility of outperforming the market by comparing the long-term performance of mutual funds with the one of the market, the latter here is represented by the 5 Wilshire 5000 Total Market Index, see Figure 1.4. In some years mutual funds outperformed the market, but no fund does so consistently, underpinning the strong EMH. The different forms of EMH follow intuitive rationales. This is highly beneficial for getting a first understanding of trading strategies and modelling purposes. We discussed rationales of trading strategies: Those based on the believe that one can consistently outperform the market using historical prices, those based on the believe that one can consistently outperform the market using publicly/ private information. The position of facing a strongly efficient market is that one of a market participant who does not believe to be able to use information to ”beat the market”. The benefit of the EMH for modelling purposes is the following: the financial market is utterly complex and simplifications need to be made before one is able to formulate a mathematical model. The different forms of the EMH give a reasonable rational to formulate such simplifications. In this sense, we are with this chapter at a stage where we set the ground for mathematical tools, for being able to formulate and justify mathematical models for the behaviour of financial quantities. Figure 2: Example of a binomial model over three time periods. The observation of uncorrelated subsequent returns andthe reasoning underpinning the EMH supports the random walk theory of stock prices. Here, we model stock prices randomly, und in a way that the daily increments are independent of the history of prices. An simple example of a model respecting these features is the binomial model, compare Figure ??. 1.5 Summary The implication on potential investments is of large interest for us: 1. Assuming none of the hypothesis holds, you can find investments, which are based on • patterns found in historical stock prices, or • any information concerning the company/the market and can consistently expect profits that are larger than the market average. 2. Assuming only the weak form is valid, you cannot find investments, which consistently yield superior profit and are based on • patterns found in historical stock prices, however, it can be based on • any further information concerning the company/the market. 3. Assuming the semi-strong form (hence also the weak form) is valid, you cannot find invest- ments, which consistently yield superior profit and are based on • any public information, 6 investment based on investor may believe in investor does not believe in historical stock prices no EMH weak form public information weak form semi-strong form private information semi-strong form strong form investor needs to in- crease risk, to increase expected payoff strong form - Table 1: Overview of the EMH however, it can be based on • any private information concerning the company/the market. 4. Assuming the strong form (hence also the semi-strong and weak forms) is valid, you cannot find • any investment that consistently yield superior profit. The only way to increase the expected return is to • increasing the risk. An overview on what can/cannot be used to design superior investment strategies is Table 1. Empirical evidence of the weak formulation of efficient markets can be found, but testing the hypotheses is difficult. The validity of the hypotheses is therefore also criticised. The EMH is useful to get an orientation towards the investment strategies and to simplify the complexity of real markets to set the ground for mathematical modelling. 2 Stochastic models of long-term behaviour of security prices Week 2 A consequence of the efficient market hypothesis is the random walk theorem, stating that the returns on subsequent days are independent of each other. An important model is the lognormal model. The Log-normal Model With (St)t∈N the daily stock price, we consider (Xt)t∈N the daily log-returns Xt = log(St+1/St). Log-returns for several days are obtained by summing up the daily log-returns: log(St)− log(Ss) = t−1∑ i=s Xi, i.e. St = Ss exp( ∑t−1 i=s Xi) for s < t. The key-assumption for the lognormal model is that • the daily log-returns Xt are iid (i.e. independent and identically distributed), and that • this distribution is a normal distribution N (µ, σ2). Parameter Estimation in the Log-normal Model Given N iid random variables Xi with an assumed distribution, e.g. N (µ, σ2), we need to estimate the model parameters, here µ and σ. Here, Xi for each i represents the daily log-return of a stock, and the model parameters are the mean µ and the volatility σ of the log-returns, and σ2 is its variance. Parameter estimation of a time series of data is a large and deep area of statistics. The estimation will only approximately represent the real time series, and in view of the limited number of observations, the error needs to be well 7 The consequences of underestimating large losses can be severe! ”[. . . ] Large fluctuations in the stock market are far more common than Brownian motion predicts. The reason is unrealistic as- sumptions – ignoring potential black swans. [. . . ]“ see https://www.theguardian.com/science/ 2012/feb/12/black-scholes-equation-credit-crunch If the Black-Scholes model is systemically used to estimate the risk and underestimates large losses, this leads to a systemic underestimation of large losses. This can have severe consequences as financial institutions thus may face a lack of risk capital in times of crises. This in turn can lead to a further destabilisation of the system and can advance a crisis. This does not mean that the Black-Scholes model is not a good model. It is a very good model in the sense that it displays some features of the stock market in a very simple way. However, it cannot serve all purposes. It has its clear shortcomings and when it is used systemically in the wrong way this can lead to damages on a global scale. It is therefore highly important that you understand the benefits and shortcomings of the Black-Scholes model. Moreover, it is important to realise that whatever model you use, it has its specific scope and you need to understand its benefits and shortcomings very well. This is of a urgent economic meaning, globally. (In)dependence and (no) autocorrelation and volatility clusters In Figure 4 we see clus- ters of high changes in the subsequent returns. This is known as volatility clusters. There presence indicate a dependence of subsequence returns, contradicting one of the basic assumptions of the log-normal model. Next, let us graphically study the autocorrelation of the returns, i.e. the correlation between subsequent returns. To do so, we build pairs (Rt, Rt−1) of all subsequent returns observed. We plot the value of Rt on the x-axis and the value of Rt−1 on the y-axis, thus obtaining the scatter plot Figure 6. The points are centred around zero, radially symmetric. This indicates that there is no linear dependence between Rt and Rt−1. Computing the empirical autocorrelation yields −0.016 confirming that this is very low, thus no indication of a linear dependence. Notice that this is a rudimentary approach, only to get a rough idea. To make this mathematically conclusive, one would need to employ statistical techniques, which goes beyond the scope of this lecture. This observation has been made consistently, Figure 6: Scatter plot of subsequent returns of the HSBC stock prices. To summarize our findings, returns of stock prices (and also log-returns as they are very similar) exhibit • No autocorrelation: corr(Rt, Rt+1) ≈ 0 (this is in line with the weak form of EMH) • Volatility clustering: corr(R2 t , R 2 t+1) > 0. We can observe periods of large volatility and of small volatility; • Heavy tails / spikes: High losses and gains much more likely than for normally distributed random variables. The presence of volatility clusters indicates a dependence of subsequence returns. However, we also observed no autocorrelation. Correlation and dependence of random variables is closely linked. If two random variables are independent they are uncorrelated. The contrary, however is 10 not always true. For instance consider X standard normally distributed. Clearly X and X2 are dependent. What is their correlation? For returns, we look for such random variables, which have no autocorrelation, but which are dependent. Stylized Facts To model the stock price evolution in an appropriate way means to balance model complexity against realistic features. Researchers have established a list of stylized features, that is features that stock prices typically exhibit. This step is helpful in modelling as it establishes the features that a model should reproduce. In practice, the actual goal of the model determines which features are most important and which ones may be ignored. Building a good model is a highly nontrivial task, each model will clearly not be perfect: Each model is flawed. But which model is good enough for the actual tasks at hand? This type of work, modelling, is done in financial institutions when internal models are build and validated, it is also a vivid research area. Here, we have listed three of the most important stylized features of the daily returns that we have observed. Deeper discussion and more stylised facts: R. Cont, Empirical properties of asset returns: stylized facts and statistical issues, Quantitative Finance, Volume 1, 2001 https: //www.lpsm.paris/pageperso/ramacont/papers/empirical.pdf Autoregressive Model A better fit of the data is available with more complex models, e.g. the autoregressive AR(1) process. There the volatility (i.e. standard deviation of the log-returns) is a stationary autoregressive stochastic process: Xt = µt+ σtZt, Zt ∼ N (0, 1) iid , σt = α+ βσt−1 + vεt, εt ∼ N (0, 1) iid , |β| < 1, with Zt and εt being independent of each other and of σt−1, Xt−1. The autoregressive model introduces a positive correlation of the volatility and hence the magnitude of returns. This way volatility clusters are introduced. Challenges are the fitting the parameters and a more complex evaluation compared to the lognormal model. Comparison AR(1) Model to Market Data In order to obtain a first impression on the behaviour of the AR(1) model compared to market data, we simulate log-returns in the model, for an arbitrary choice of the parameters. Note that we did not fit the parameters, so the comparison is in a preliminary stage and we can only have a glimpse on the behaviour in respect to stylized facts. We display the time series of the related stock prices, the time series of the log-returns, in comparison to one empirically observed time series of market data in Figure 7. From the time Figure 7: Log-returns Top: AR(1) model; Bottom: Empirical data series of stock prices itself it is hard to extract stylized facts, similarities or differences. Turning to the time series of log-returns, however, we observe that the AR(1) model reproduces clusters, i.e. periods of a large number of high returns in absolute values and periods of lower numbers of high returns. We also observe some positive and negative spikes. Both features are more extreme in the 11 empirical time series, but note that we did not fit the parameters and we have only one example here, so we should not draw any conclusion from this single observation. Next, we display the histogram of log-returns from the empirically observed prices, in the AR(1) model, in a log-normal model in Figure 8. We observe that the shape of the empirical distribution of the log-returns is better reproduced, it is steeper around the mean and higher returns are more likely than in the log-normal model. Both the steeper form in the middle and the slower decay of the tails are visually more similar to the market data then the histogram of the log-normal returns. Figure 8: Histogram of Log-returns Estimation of Parameters in the Autoregressive Model The general approach to derive the model parameters α, β, v in σt = α+βσt−1+vεt is the following two-stage procedure. First, estimate the expectation value E(σt), the variance Var(σt) and autocorrelation corr(σt, σt−1). Second, derive the parameters β = corr(σt, σt−1), α = (1− β)E(σt), v2 = (1− β2) Var(σt). The challenging step is to estimate the empirical volatility. Remember how we estimate the em- pirical variance in the log-normal model, σ2 ≈ 1 N−1 ∑N i=1(Xi − X̄)2. The subtle point is that this is a good estimator if the sequence Xi is iid. However, the AR(1) model is build in such a way to create dependence of the log-returns. The main difficulty thus is that • Xi are not independent in the AR(1) model, and • σt is different for each Xt, but it is impossible to estimate the variance with a single data point only. Figure 9: Time series of locally estimated volatilities. 12 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03 -4 -2 0 2 4 6 8 10 10 -4 BMW VW Dailmer Deutsche Bank Commerzbank Lufthansa Adidas Siemens Figure 11: Several stocks in the σ − µ-plane and their efficient subset. risk measures it is crucial to understand the scope of the measure, what does it reflect and what does it not reflect? Remember, there is always something that the risk measure does not capture, so always be sure to understand very well what it does and what it does not reflect. Ultimately, the development and the understanding of measuring financial risks is highly rel- evant for financial institutions and sound risk measures are required to control the risk of invest- ments. On a systemic level, controlling the risk of the investments of all institutions is required to guarantee the stability of the financial system and of the economy as a whole. Next, we consider the semi-variance as a slight adaptation of the variance as a risk measure and then we turn to the most commonly used risk measures in practice, the shortfall probability and the value-at-risk. Finally, we briefly discuss the concept of stress testing, which is one of the pillars of financial risk assessment. Semi-variance The first drawback of the variance as risk measure listed above is that losses and gains equally contribute to the variance, while investors will welcome gains and suffer from losses. In order to adapt the semi-variance is defined by E(min{0, X − µ}2). It measures the downside-risk. As a major drawback, we observe that it is still highly dependent on the mean µ, also the other two criticism listed above are still valid. 3.1 Shortfall probability The variance is a very simple measure of investment risk. While it enables us to easily compare stocks, for a more detailed investigation more advanced risk measures need to be considered. Shortcomings include: • due to it’s dependency on the expected value, assets with a larger expected value may seem riskier although they are not; • unexpected large gains are valued the same as unexpected large losses; • the variance does not give any information about the size of the risk or their probability. A likely small loss can have the same variance as a less likely huge loss. 15 Figure 12: Illustration of the empirical shortfall probability. To solve the problems, the shortfall probability and the Value at Risk can be considered. They answer the questions • how likely are large losses (shortfall probability); • how large are likely losses (Value at Risk). Both are based on the realised loss L = −R (note, that we can use either the return R or the log-return X in the definition of the loss, depending on the situation; the results will differ only slightly). The shortfall probability can best be evaluated using the distribution function of the return R: FR(x) = P (R ≤ x): SF(b, R) = P (L ≥ b) = FR(−b), see Figure 13 for an illustration. Roughly, the shortfall probability measures how likely large losses are. More precisely, it measures how likely losses larger than a pre-specified threshold are. How to compute the shortfall probability? If we have a model at hand, we can do that with the help of the density, or the distribution function directly. If we have an observation of a time series of daily returns Xt for days t = 1, . . . , N instead, we need to evaluate the empirical shortfall probability instead. This is given by SFe(b) = |{t : 1 ≤ t ≤ N, s.t.−Xt > b}| N . Figure 12 illustrates how to obtain the empirical shortfall probability for 20 samples and the threshold b = 0.1. We count 4 samples below the threshold, which is 20%, therefore SFe(0.1) = 20%. 3.2 Value at Risk and α-quantiles Week 4 The shortfall probability quantifies how likely losses beyond a given threshold are. Asking differ- ently, we may want to know with which level of loss do we have to probably deal? For instance, we would like to be prepared to compensate all likely losses with cash, while we leave it open how we move on when a larger loss happens, because the scenario is unlikely. The notion of value-at-risk makes this mathematically precise. First, we have to specify what we mean with likely losses. This is done by specifying a confidence level, for instance 95%. The value-at-risk is the maximum amount to be lost with a specified likelyhood, i.e. at a pre- defined confidence level. For example, if the 95% VaR is 1 million, there is 95% confidence that the portfolio will not lose more than 1 million. The Value at Risk is defined as VaRα = inf{b : P (L > b) < 1− α}. 16 If the distribution function of the return FR is continuous and strictly increasing, we can use the inverse function to evaluate the value at risk: VaRα = −F−1 R (1− α). Note: we usually evaluate VaRα for α > 0.5, e.g. 95% or 99%, which yields 1 − α < 0.5. See Figure 14 for the illustration of the evaluation using the density function. -b=-1.5 P(-X>b) Figure 13: Evaluating the shortfall probability SF(X, 1.5) using the distribution function (here X log-return) Reminder: The distribution function FX(x) := P (X ≤ x) is left-continuous Definition 3. For α ∈ (0, 1) the number qα(X) = inf{x : α < FX(x)} is called upper α-quantile of X. qα(X) = inf{x : α ≤ FX(x)} is called the lower α-quantile of X. Any q ∈ [qα(X), qα(X)] is called α-quantile of X. • If FX is continuous and strictly increasing, qα(X) = qα(X) = F−1 X (α). • VaRα = −q1−α(X). Note: different notations are used in practice, e.g. VaR95% is sometimes denoted VaR5%. Examples: 1. uniform distribution → Tutorials 2. normal distribution → Homework 17 Figure 16: Illustration of the empirical value-at-risk as quantile of the empirical distribution. 3.3 Stress test The idea behind stress testing is to model important possible scenarios and the compute the related risk. A possible implementation is done along the following steps: 1. Build a factor model for the ingredients of the portfolio. 2. Specify a set of stress scenarios S ⊂ Ω. (for instance high/moderate/low interest rates and high/moderate/low inflation rates) 3. For all ω ∈ S compute the future portfolio gain G(ω). 4. For losses L = −X compute worst case loss %(L) = sup{L(ω)|ω ∈ S} when we restrict our attention to those element of the space of possible events that belong to S, our selected scenarios. Example: Consider one stock S and a risk-free asset with rate r. We assume the stock-price is given by the random variable S1 = { S0(1 + µ+ σ), p = 1/2, S0(1 + µ+ σ), p = 1/2. where the current mean and variance are µ = 0.05 and σ = 0.1. The risk-free rate is r = 4% and we have invested £1, 000 each in the stock and the risk-free security. The stress test defines certain scenarios and returns the worst-case lost. Then one needs to check, whether the result is acceptable (passing the stress test) or not (failing it). In our case these scenarios could be Ω = {“µ = −0.5, σ = 0.05, r = 0.03”, “µ = 0, σ = 0.2, r = 0.01”, . . .} 20 In each case the maximal loss is computed, e.g. R(“µ = −0.5, σ = 0.05, r = 0.03”) = (1000(1− 0.5− 0.05) + 1000(1 + 0.03))/2000− 1 = 0.26 R(“µ = 0, σ = 0.2, r = 0.01”) = (1000(1 + 0.0− 0.2) + 1000(1 + 0.01))/2000− 1 = 0.095. In this case the largest loss would be L=-9.5%. 4 Mean-variance portfolio theory Week 5 • What is a portfolio? – A collection of investments hold (here stocks/risk-free securities) • Why is portfolio theory so interesting? – A lot more can happen than with single assets. Illustrative example: Ice-cream sellers & umbrella sellers. We model that the following summer will be either rainy or sunny, both equally likely. If it is rainy the umbrella sellers make a larger profit, while ice-cream sellers make a loss. In a sunny summer the situation is the reverse. rainy summer (p = 50%) sunny summer (p = 50%) Return ice-cream sellers (R1) -5% +10% Return umbrella corporation (R2) +10% -5% Both investments have an expectation of 2.5% and a standard deviation of 7.5%. If we buy equal parts of the ice-cream seller and the umbrella corp. we have a return of 1 2R 1 + 1 2R 2 = { 2.5%, p = 50%, 2.5%, p = 50%. , i.e. a safe return of 2.5% (standard deviation zero). Why does this happen → both investments are negatively correlated: corr(R1, R2) = Cov(R1, R2) σ1σ2 = E ( (R1 − µ1)(R2 − µ2) ) σ1σ2 = −1. Note that a correlation of −1 is a very extreme case unlikely to happen in practice. Let us have a look at an example with no correlation: R1 = { 10%, p = 1/2 −5%, p = 1/2 , R2 = { 10%, p = 1/2 −5%, p = 1/2 , which are independent of each other. Due to their independence, the joint distribution now has four cases: probability: 25% 25% 25% 25% R1 10% 10% –5% –5% R2 10% –5% 10% –5% 1 2 (R1 +R2) 10% 2.5% 2.5% –5% For the portfolio this yields E( 1 2 (R1 +R2)) = 2.5% and √ Var( 1 2 (R1 +R2)) ≈ 5.3% < 7.5%. In general we see that a portfolio can have a smaller expected value than the individual assets, while having an expectation value as large as both assets. This is called diversification. 21 4.1 Introduction to portfolios In the following, we try to optimise our portfolio. We will figure out how to distribute the money and what other choices remain. Let us consider • 2 Stocks S1, S2 with expected return µ1, µ2, variances σ2 1 , σ 2 2 and correlation ρ; • 2 dates t ∈ {0, 1}: – t = 0 is today, i.e. values S1(0), S2(0) are deterministic; – t = 1 is some point in the future, i.e. values S1(1), S2(1) are random variables. A portfolio consists of buying/owning x1 stocks of asset 1 and x2 of asset 2. The current value is known: P(x1,x2)(0) = x1S 1(0) + x2S 2(0), and the future value is a random variable: P(x1,x2)(1) = x1S 1(1) + x2S 2(1). Note that the amount of shares x1, x2 can be quite disproportional to their value, e.g. with x1 = 1 and S1(0) = £150 the value is higher than for x1 = 10 and S1(0) = £10. We therefore introduce weights w1, w2, which represent proportion of our wealth P(x1,x2)(0) invested in the two assets: w1 = x1S 1(0) P(x1,x2)(0) , w1 = x2S 2(0) P(x1,x2)(0) , with w1 + w2 = 1. The weights allow us to conveniently express the return RP(w1,w2) of our portfolio in terms of the individual returns R1 = S1(1)/S1(0)− 1 and R2 = S2(1)/S2(0)− 1: RP (w1,w2) = P(x1,x2)(1) P(x1,x2)(0) − 1 = x1S 1(1) + x2S 2(1) P(x1,x2)(0) − 1 = x1S 1(1) P(x1,x2)(0) + x2S 2(1) P(x1,x2)(0) − 1 = x1 S1(0) P(x1,x2)(0) S1(1) S1(0) + x2 S2(0) P(x1,x2)(0) S2(1) S2(0) − 1 = w1 S1(1) S1(0) + w2 S2(1) S2(0) − w1 − w2 = w1R 1 + w2R 2, i.e. RP = w1R 1 + w2R 2. If we know the proportions we wish to invest in each asset, i.e., we have the weights w1, w2 given, we can compute the amount of shares as x1 = w1P (0) S1(0) , x2 = w2P (0) S2(0) . We typically ignore the fact that we cannot buy fractions of a share. Example 1. We wish to invest £1 000 in equal parts in two assets with S1(0) = 10, S2(0) = 100. With the initial wealth P (0) = 1 000 and the weights w1 = w2 = 1/2, we can compute the amount of stocks: • Asset 1: x1 = w1P (0) S1(0) = 500 10 = 50, i.e. we buy 50 shares of company 1; • Asset 2: x2 = w2P (0) S2(0) = 500 100 = 5, i.e. we buy 5 shares of company 2. We can compute the return either based on new prices or based on the given return: 22 Theorem 3. Let σ1 > σ2 be the standard deviations for both assets and ρ ∈ (−1, 1) their correla- tion. Then the MVP is given for the weights w1 = max ( σ2 2 − ρσ1σ2 σ2 1 + σ2 2 − 2ρσ1σ2 , 0 ) . and w2 = 1− w1. Proof. See Tutorial 5. 4.5 Short selling Assuming no restrictions on short-selling, negative weights are positive w1 ∈ R, w2 = 1− w1 ∈ R. Negative weights include a leverage (borrow the less profitable asset & sell it to buy the more profitable one.) Theorem 4 (MVP, general case). With no restrictions on short-selling and ρ ∈ (−1, 1), the MVP is given for the weights w1 = σ2 2 − ρσ1σ2 σ2 1 + σ2 2 − 2ρσ1σ2 , and w2 = 1− w1. 4.6 Efficient frontier Definition 4. The efficient subset of the attainable set is called efficient frontier. Observe: The efficient frontier is the part of the attainable set, which connects the MVP with the asset of the highest expectations (and continuing beyond if short-selling is possible). When no short-selling is possible, it is a closed set (including the end points), otherwise it is half-bounded with the MVP as its end-point. 25 4.7 Adding a risk-free security As risk-free asset is like a bank account, where we earn a fixed interest on our money. In the case of short-selling, we assume that we can borrow money for the same rate and without any restrictions. Combinations of an asset with the risk-free asset form a straight line. We can view any portfolio of assets 1&2 as an individual asset and then apply the known theory. Example 3. We compare two cases with a wealth of £1 000: a) investing £700 in BP and £300 in Shell b) investing w0 = 1/2 into the bank account and investing the rest to BP and Shell with the same ratio’s as in part a). Investment (b) then means to put £500 into the bank account, and investing £350 in BP and £150 in Shell. In the mean-variance diagram, portfolio (b) is in the middle of portfolio (a) and the riks-free portfolio. See the slides for further examples. Efficient frontier with a risk-free asset The efficient frontier is a tangent to the attainable set of all portfolios that include assets 1&2. (Special cases do exist in the case of no short-selling and are outlined in the lecture slides). The tangent point, which connects the risk-free security with the attainable set of assets 1&2 is called Market Portfolio (MP). Theorem 5. If µ0 < µMVP the weights of the market portfolio are w1 = c c+ d , w2 = d c+ d , where c = σ2 2(µ1 − µ0)− ρσ1σ2(µ2 − µ0), d = σ2 1(µ2 − µ0)− ρσ1σ2(µ1 − µ0) 26 Proof sketch (This has not been part of the lecture; it is only included for your own interest) Maximise the slope of the line connecting any portfolio with the risk-free security to find the tangent. The slope is given in dependency of w1 as s(w1) = ∆µ(w1) ∆σ(w1) , and we need to solve the optimisation problem max w1∈R s(w1). We note graphically that there is a unique solution, so we can find it by looking for roots of the derivative: findw1 s.t. s′(w1) = 0, i.e. 0 = s′(w1) = µ′(w1)σ(w1)− (µ(w1)− µ0)σ′(w1) σ2(w1) , where µ(w1) = w1µ1 + (1− w1)µ2, σ(w1)2 = w2 1σ 2 1 + 2ρw1(1− w1)σ1σ2 + (1− w1)2σ2 2 . For the derivatives of µ and σ this yields µ′(w1) = µ1 − µ2, σ′(w1 = d dw1 (√ σ2(w1) ) = 1 2 √ σ2(w1) d dw1 ( σ2(w1) ) = 1 2 √ σ2(w1) ( σ2)′ (w1). This yields s′(w1) = µ′(w1)σ(w1)− (µ(w1)− µ0) 1 2σ(w1) ( σ2)′ (w1) σ2(w1) = 2µ′(w1)σ2(w1)− (µ(w1)− µ0)(σ2)′(w1) 2σ3(w1) . The fraction can only be zero if the nominator is zero, so we solve 2µ′(w1)σ2(w1)− (µ(w1)− µ0)(σ2)′(w1) = 0, where (σ2)′(w1) = 2w1σ 2 1 + 2ρσ1σ2 − 2ρ2w1σ1σ2 − 2(1− w1)σ2 2 . Inserting and solving this finishes the proof. 27 2. Fundamental factors are not observable. Examples are the industry sector, the country or the continent. Here the factor loading is known, but the factors need to be modelled. Example: consider the assets (a) HSBC, (b) Barclays, (c) Shell, (d) BP, (e) Bank of America, and the factors (a) Company in the financial industry; (b) Company in oil&gas; (c) Company located in Europe. Then we have three factors F = (F1, F2, F3), where the factor loading is given by B =  1 0 1 1 0 1 0 1 1 0 1 1 1 0 0  . The factors F need to be modelled. 3. Statistical factors are factors purely based on statistical evidence. Neither B nor F are observable. Examples include PCA (Principal Component Analysis) or statistical factor analysis. Week 8 As an example, let us consider PCA for a single factor. There the stock prices are combined linearly and we seek the stochastic processes, which can best explain the stocks (for processes with mean zero). This is an optimisation problem, where we minimise the mean-squared-error of the combined process to the individual stocks as follows. Based on the historical data Si(t), i = 1, . . . , d, t = 0, . . . N we find ci, i = 1, . . . , d and B = (bi1)i=1,...d ∈ Rd×1, such that N∑ t=0 d∑ i=1 ∥∥∥∥∥bi d∑ j=1 cj(Sj(t)− µj)︸ ︷︷ ︸ F1 −(Si(t)− µi) ∥∥∥∥∥ 2 is minimal. Statistic factors can be very powerful, as they need only few factors for a good approximation. However, the factors lack interpretability. 5.1 Single factor models A single factor model is the most simple form of factor models. It restricts the possible values of the correlation, but allows for a clearer interpretation. In this subsection, we consider single factor models of different complexities. With p = 1 we have Xi = ai + biF + εi, i = 1, . . . , d, where F is a stochastic joint factor. An example would be a broad market index, e.g. the FTSE All-Shares index. Mean, variance and covariance are: 30 • E(Xi) = ai + biE(F ); • Var(Xi) = b2i Var(F ) + Var(εi) as εi and F are uncorrelated; • Cov(Xi, Xj) = bibj Var(F ) for i 6= j (also due to εi and F being uncorrelated). A special one-factor model is Sharpe’s Single-Index Model (SIM), which brings an economic interpretation: Ri − µ0 = αi + βi(RM − µ0) + εi. • µ0 is the risk-free rate; • RM is the return of the market portfolio; • εi ∼ N (0, σ2 i ) independent of each other and of RM. • αi: the stock’s alpha: abnormal results • βi: the stock’s beta: responsiveness to the market return. The model was developed by William Sharpe in 1963 and is widely used in practice. • All returns are corrected by the risk-free rate • With the market portfolio as the single factor, the return of the stock is decomposed into three parts: – The abnormal return: any returns that outperforms the market consistently is influenced by the alpha αi; – The return may have a different responsiveness to market movements. With a small value of β the stock price reacts only slightly to changes of the market. With β larger than one, the stock price reacts stronger than the market; – Each stock has an idiosyncratic risk, independent of the market and other stocks. We will come across β again in our next chapter on CAPM. Equicorrelation model (extra reading) A second, even simpler, single factor model is the equicorrelation model: Xi = √ρF + √ 1− ρεi, i = 1, . . . , d, where F is the single factor with F, ε ∼ N (0, 1) iid, ρ ∈ (0, 1). As the name suggests, the random variables Xi have the same mutual correlation coefficient: E(Xi) = √ρE(F ) = 0; Var(Xi) = ρVar(F ) + (1− ρ) Var(εi) = 1; Cov(Xi, Xj) = Cov(√ρF + √ 1− ρεi, √ ρF + √ 1− ρεj) = ρVar(F ) = ρ, i 6= j, hence corr(Xi, Xj) = %. Where is this useful • if you have several equal risks, e.g. with homogeneous credit portfolios; • as a (drastically) simplified version of Sharpe’s SIM: Ri = √ρRM + √ 1− ρεi, where εi, RM ∼ N (0, 1) and µ0 = 0 are assumed for simplicity. 31 Due to it’s simplicity, the model allows us to consider huge portfolios by hand: Let us consider d stocks in an equicorrelation model and a portfolio of equal parts: RP = 1/d d∑ i=1 Ri and Ri = √ρRM + √ 1− ρεi. Then expectation and variance are given as E(RP) = 1 d d∑ i=1 E(Ri) = 0, Var(RP) = 1 d2 Var ( d∑ i=1 Ri ) = 1 d2 d∑ i,j=1 Cov(Ri, Rj) = 1 d2 d∑ i,j=1 (ρ+ (1− ρ)δij) = 1 d2 (d2ρ+ d(1− ρ)) = ρ+ 1− ρ d →d→∞ ρ. We immediately note two interesting facts: 1. With large portfolios the individual variance Var(Ri) becomes negligible in comparison to the covariances Cov(Ri, Rj), due to the curse of dimension. 2. Investing into many assets eliminates the risk only up to a certain limit (dictated by the market return). This shows two different kinds of risk: • The systemic risk which is due to the common factor, the market return RM. • The specific risk which is individual to each asset and independent of other assets. By diversification only the specific risk can be reduced. The systemic risk remains and will be estimated further in the next chapter. 6 Pricing Pricing models aim to explain the expected return of assets. 6.0 Mean-variance portfolio theory for several assets Note: This subsection contains interesting insights and prepares for CAPM, but is not relevant for the exam. Before introducing the Capital Asset Pricing Model (CAPM), let us extend the portfolio theory of Chapter 4 to several risky assets. Let d risky assets be given by: • vector of their expected returns µ = (µ1, . . . , µd)>= ( E(R1), . . . ,E(Rd) )>∈Rd • matrix of the pairwise covariances (symmetric and assumed to be invertible) C =  Cov(R1, R1) Cov(R1, R2), · · · Cov(R1, Rd) Cov(R2, R1) Cov(R2, R2), · · · Cov(R2, Rd) · · · · · · . . . ... Cov(Rd, R1) Cov(Rd, R2), · · · Cov(Rd, Rd)  ∈ Rd×d. 32 Thus, the slope of the tangent is ∂µw ∂w ∂σw ∂w ∣∣∣∣∣ w=0 = µi − µMP Cov(Ri,RMP)−σ2 MP σMP . As it is the same as the slope of the CML, we can set it equal to µMP−µ0 σMP and solve for µi: µi − µMP = µMP − µ0 σMP · Cov(Ri, RMP)− σ2 MP σMP = (βi − 1)(µMP − µ0) = βi(µMP − µ0)− (µMP − µ0), Thus µi − µMP = βi(µMP − µ0). Remark 1. The same formula holds for portfolios: µP = µ0 + βP(µMP − µ0), where βP = Cov(RP,RMP) σ2 MP . In the CAPM formula, we see that the expected return of any investment is only determined by the covariance with the market portfolio. Interpretation • β(µMP−µ0) is called risk-premium. It rewards investors, who expose themselves to a higher market risk. • the idiosynchratic term εP, such that RP − µ0 = βP(RMP − µ0) + εP, represents the specific/diversifiable risk. We can compute the variance: σ2 P = β2 Pσ 2 MP + Var(εP). As the specific risk εP can be diversified (by buying the market portfolio), it is not rewarded by a higher expected return. Computation of the variance: Var(RP) = Var(µ0 + βP(RMP − µ0) + εP) = β2 P Var(RMP) + Var(εP) + βP Cov(RMP, εP), where Cov(RMP, εP) = Cov ( RMP, RP − µ0 − βP(RMP − µ0) ) = Cov(RMP, RP)− βP Cov(RMP, RMP) = 0. 6.1.2 The security market line (SML) CAPM yields a linear relation between the expected return and the beta of a portfolio. Plotting several portfolios in a β-µ-diagram, they should form a line: the security market line. 35 6.1.3 Efficient portfolios A portfolio is efficient if and only if corr(RP, RMP) = 1, see coursework. This yields βP = Cov(RP, RMP) σ2 MP = σPσMP corr(RP, RMP) σ2 MP = σP σMP . Thus for efficient portfolios there is no diversifiable risk and µP − µ0 = βP(µMP − µ0) σP = βPσMP. 6.1.4 How to use CAPM? In the following, we discuss some ways how CAPM can be used for portfolio analysis. • Value portfolios using the Sharpe ratio: µP − µ0 σP . The larger the Sharpe ratio is, the more efficient is the portfolio. For efficient portfolios, we have µP − µ0 σP = βP(µMP − µ0) βPσMP = µMP − µ0 σMP , where the Sharpe ratio of the market portfolio is maximal: wMP = arg max w,w>1=1 w>µ− µ0 w>Cw (This holds by construction of the market portfolio, as (w>µ− µ0)/(w>Cw) is the slope of the line connecting a portfolio with the risk-free asset.) • Valuing the stock price: CAPM models the required return for the taken risk. Strong deviation from the model can be used as an investment strategy. – If µi−µ0 > βi(µMP−µ0), the stock is underpriced and you could buy it (as the expected return is larger than the required return) – If µi − µ0 < βi(µMP − µ0), the stock is overpriced and you could sell it or short-sell it (as the expected return is less than the required return considering the taken risk). Remark: These transactions will push the market price towards predicted value and the market gains efficiency. 36 • Performance measure of a stock using Jensen’s alpha: Jensen’s alpha is the difference of the realised return and the required return: α = RP − (µ0 + βP(RMP − RP)), where for this application, RP and RMP are the realised returns, i.e. empirical data. Jensen’s alpha measures the past performance in comparison to the required return. 6.1.5 Discussion of the validity As a mathematical model, CAPM has only few assumptions, e.g., • No trading costs (including brokerage fees, bid-ask-spread, taxes, etc); • No restrictions on short-selling and borrowing money for the same rate as lending money; • Available values for all covariances and expectations. Knowing the parameters is crucial to compute the market portfolio, but not required beyond that. A practical approach is thus to replace the “mathematically computed” market portfolio wMP by a market-index. (Reminder: If all investors buy efficient portfolios, everyone buys portions of wMP, hence it reflects the whole market and can be replaced by an index) This requires severely more assumptions, e.g., • all investors have the same time horizon; • all investors can borrow or lend money with no risk at the same rate; • all investors are non-satiated, risk-averse and trade purely based on σ and µ; • “perfect market” – information is freely and instantly available, no investor believes they can affect the price by their actions • all investors have the same estimates for the parameters • all investors measure in the same currency (e.g. pounds/dollars/ “value”). We see that the assumptions to have wMP reflect the whole market are quite strong and often criticised as being unrealistic. How can we test CAPM? Idea: Plot the security market line. Remember: The CAPM formula µi − µ0 = βi (µMP − µ0) predicts a linear relationship between the excess return µi−µ0 and βi = Cov(Ri, RMP)/σ2 MP. The line in the β − µ-plane is called security market line (SML). We can plot the line by estimating these parameters. Let (Sit)t=1,...N+1 be the historic time-series of asset prices of asset i and their log-returns (Xi t)t=1,...N , Xi t = log ( Sit+1 Sit ) . With the log-returns of a market index (XMP t )t=1,...N , we can estimate µi = E(Xi t) ≈ 1 N N∑ t=1 Xi t , µMP = E(XMP t ) ≈ 1 N N∑ t=1 XMP t . 37 Both measures are equivalent, if we assume all other investments to remain at the same value: u(payoff) = ũ(wealth + payoff)− ũ(wealth) ũ(wealth) . This means that the payoff of a particular payoff is a scaled version of the utility function of the wealth. We therefore only consider the utility of the payoff for a certain action. Definition 5. A utility function is a function u : R→ R which is monotonic increasing. Example 6. With this first example, we demonstrate which utility functions suit different risk attitudes. Let two lotteries be given with the payoff L1 = { £2, probability 50%, −£1, probability 50%, L2 =  £2, probability 25%, £0.5, probability 50%, −£1, probability 25%, Both lotteries have the same expected value £L1, while L1 bears more risk. • If you are risk-seeking, you prefer L1, • If you are risk-averse (risk-avoiding), you prefer L2, • If you are risk-neutral, you are indifferent. To decide for one of the lotteries, we value the expected utility: E ( u(L1) ) vs. E ( u(L2) ) . (Note that in general E ( u(Li) ) 6= u ( E(L1) ) for nonlinear functions u.) • E ( u(L1) ) = 1/2u(2) + 1/2u(−1) • E ( u(L2) ) = 1/4u(2) + 1/2u(1/2) + 1/4u(−1) 1. If you are risk-seeking: E ( u(L1) ) > E( ( u(L2) ) : ⇔ 1/2u(2) + 1/2u(−1) > 1/4u(2) + 1/2u(1/2) + 1/4u(−1) ⇔ 1/2 ( u(2) + u(−1) ) > u ( 2− 1 2 ) . This is a convexity condition for u: 40 2. If you are risk-neutral: 1/2 ( u(2) + u(−1) ) = u ( 2−1 2 ) , which means linearity of u. 3. If you are risk-averse: 1/2 ( u(2) + u(−1) ) < u ( 2−1 2 ) , which is a concavity condition for u. 7.1 Reminder: convex and concave functions Definition 6. A function u : R→ R is called • strictly convex, iff u ( t x+ (1− t) y ) < tu(x) + (1− t)u(y), for all x, y ∈ R, t ∈ (0, 1). A famous example is u(x) = exp(x). • strictly concave, iff u ( t x+ (1− t) y ) > tu(x) + (1− t)u(y), for all x, y ∈ R, t ∈ (0, 1). A famous example is u(x) = log(x) (for x > 0). If u is twice differentiable, we can use the sign of the second derivative as a test: • u′′(x) > 0 for all x ∈ R ⇒ u is convex, • u′′(x) < 0 for all x ∈ R ⇒ u is concave. 7.2 Expected utility Remember, that a utility function is required to be monotonic increasing. Definition 7. • A utility function, that is strictly concave is called risk-averse, (Check if u′′(x) < 0). • A utility function, that is strictly convex is called risk-seeking, (Check if u′′(x) > 0). 41 • A utility function, that is linear is called risk-neutral. We compare the expected utility for two cases: 1. A lottery L 2. A fixed payments of E(L). • If you are risk-averse E(u(L)) ≤ u(E(L), i.e. you need to be rewarded for taking risks. • If you are risk-seeking E(u(L)) ≥ u(E(L), i.e. you would pay to get the risky option. • If you are risk-neutral E(u(L)) = u(E(L), i.e. you do not care about the risk, only about the expected value. Proof. We proof the inequality for the case of L being a discrete random variable and u being risk-averse (without loss of generality). Let xi be the possible outcomes of L, each with a positive probability of pi, i = 1, . . . , N , and∑N i=1 pi = 1. Then E ( u(L) ) = N∑ i=1 pi u(xi), u ( E(L) ) = u ( N∑ i=1 pi xi ) . Thus, we need to show N∑ i=1 pi u(xi) ≤ u ( N∑ i=1 pi xi ) for u concave, pi > 0 and ∑N i=1 pi = 1. We proof this by induction over N .1 • Base case N = 1: p1u(x1) ≤ u(p1x1) with p1 = 1 is trivially true. • Induction step N − 1 7→ N : We assume that inequality holds for N − 1 terms and then conclude that it is still valid for N terms. This means, we assume N−1∑ i=1 qi u(xi) ≤ u (N−1∑ i=1 qi xi ) for any qi > 0 with ∑N−1 i=1 qi = 1. and show that N∑ i=1 pi u(xi) ≤ u ( N∑ i=1 pi xi ) for any pi > 0 with ∑N i=1 pi = 1. To show this, we first regroup the sum of N items as a sum of N −1 item with the remaining item: N∑ i=1 pi u(xi) = N−1∑ i=1 pi u(xi) + pN u(xi) = (1− pN ) N−1∑ i=1 pi 1− pN︸ ︷︷ ︸ =:qi u(xi) + pN u(xi) 1https://en.wikipedia.org/wiki/Mathematical_induction 42 which yields w0 = 75/108 ≈ 69%. Thus our expected utility is maximised with 69% of our money being invested risk-free and 31% in the market portfolio. As a comparison, we can see if we can maximise the expected payoff: E ( w0P0 + (1− w0)P1 ) = 1 2(2− 0.9w0) + 1 2(0.5 + 0.6w0) = 1.25− 0.15w0 →∞, for w0 → −∞. The expected payoff can be arbitrarily large, but the (risk-averse) expected utility is bounded. 8 Behavioural finance Behavioural finance is a modern extension of classical market models, taking into account the irrationality of the market participants. Two examples are the following: Example 9 (Allais paradox). Named after Maurice Allais, who published it in 1953. Consider two different situations, each of them being the choice between two lotteries. Situation 1: Lottery 1A: { £1 000 000, probability 100%. vs. Lottery 1B:  £1 000 000, probability 89%, £5 000 000, probability 10%, £0, probability 1%. Most people prefer Lottery 1A over Lottery 1B Situation 2: Lottery 2A: { £1 000 000, probability 11%, £0, probability 89%, vs. Lottery 2B: { £5 000 000, probability 10%, £0, probability 90%. Most people prefer Lottery 2B over Lottery 2A. Can we explain this using utility theory? Let’s assume we have a utility function u that explains this choice. The choice L1A vs. L1B yields u(£1m) > 0.89u(£1m) + 0.1u(£5m) + 0.01u(£0) (1) The choice L2B vs L2A yields 0.1u(£5m) + 0.9u(£0) > 0.11u(£1m) + 0.89u(£0), which yields 0.01u(0) > 0.11u(£1m)− 0.1u(£5m) Inserting in (1) yields u(£1m) > 0.89u(£1m) + 0.1u(£5m) + 0.11u(£1m)− 0.1u(£5m) = u(£1m), which is a contradiction. We see that the choice made by most people cannot be explained by classical utility theory. The reason is a different reception of small probabilities compared to larger ones (i.e. the difference between 0% and 1% is a lot more significant different than the difference between 89% or 90%) 45 Example 10 (Different perception of gains and losses). Again, we consider two different scenarios. Situation 1: You receive £1 000 and have two options: Lottery 1: { £500, probability 100%. vs. Lottery 2: { £500, probability 50%, £0, probability 50%. Most people prefer L1 over L2 (risk-averse). Situation 2: You receive £2 000 and have two options: Lottery 1: { −£500, probability 100%. vs. Lottery 2: { −£500, probability 50%, £0, probability 50%. Most people prefer L2 over L1 (risk-seeking). Although the final outcome is exactly the same in both cases: Lottery 1: { £1 500, probability 100%. vs. Lottery 2: { £1 000, probability 50%, £2 000, probability 50%, the choice of many people depend on a reference point (here £1 000 or £2 000). Also we see that gains and losses are valued with a different risk attitude: • risk-seeking for losses, and • risk-averse for profits. One way to incorporate these effects is the cumulative prospect theory (CPT) It has three main features: • A reference point in wealth, defining profits and losses (framing) • S-shaped utility functions, i.e. functions that are (locally) concave for profits and (locally) convex for losses, e.g. • A non-linear transformation of the probability measure which increases the weight of small probabilities. For more informations, see, e.g. Xun Yu Zhou - Mathematicalising Behavioural Finance, 2010. 46