Beleggingsleer Study Notes PDF

Beleggingsleer geschreven door studenttew De website voor het Kopen en Verkopen van je Samenvattingen Op Stuvia vind je de beste samenvattingen, notities en ander studiemateriaal. Voor alle toetsen,...

Beleggingsleer geschreven door studenttew De website voor het Kopen en Verkopen van je Samenvattingen Op Stuvia vind je de beste samenvattingen, notities en ander studiemateriaal. Voor alle toetsen, examens en cursussen. Bekijk het aanbod op Stuvia. www.stuvia.com Gedownload door: hannahducatteeuw | [email protected] Wil jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Beleggingsleer Chapter 1: The Investment Environment What is the difference between corporate finance and investment analysis? Corporate finance: mainly concerned with cash management, ensuring liquidity and raising funds, how and where are we going to get money? Investment analysis: deals with the problems of investing funds, what do we do with the money that we have? Investments and financial assets Why are financial markets interesting? There may be situations where you earn more than you need for consumption, in that case you may wish to shift part of your income to the future. Other times what you earn will be less than what you need for consumption, in that case you might want to bring part of your future income to the present. The first problem can be solved by real assets, whereas financial assets can help in both situations… Real assets: assets that help to produce goods and services (real estate, machines, factories, etc.) Financial assets: represent claims on real assets Major classes of Financial Assets 1. Debt / Fixed income securities Money market instruments Bonds Debt has a limited lifetime (with very few exceptions), you give away money for a couple of years and you get a fixed interest rate every year as a return. 2. Stocks / Equity Common stocks Preferred stocks Stocks/Equity has an unlimited lifetime and you don’t know in advance what you will receive, you get what is left from the profit after taxes if the company decides to turn out a dividend, or nothing at all. 3. Derivative securities 4. Foreign currencies The main subsets of asset classes in the investment universe are classical/traditional investments and alternative investments… Gedownload door: hannahducatteeuw | [email protected] Wil jij1€76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen TRADITIONAL INVESTMENTS These are long term investments, you invest and wait/hope for success. You buy it now and hope that in the future the price of it will increase, then you have a profit. Classical investments include three asset classes:  Stocks (equity)  Bonds (or fixed income)  Cash (money market instruments, usually treasury bills) ALTERNATIVE INVESTMENTS These cover a wide range of investments and use unconventional strategies and/or unconventional investment vehicles. They include:  Hedge funds  Commodities (e.g. gold)  Managed futures  Private equity  Currencies  Gems and stones  Derivatives  Fine art, wine *Managed futures: money managers who invest in derivatives, most of them are trend followers and react on rising or falling markets. They go long and short. In alternative investments there is a huge variation in terms of standardization, liquidity and valuation. Commodities, foreign currencies and derivatives are highly liquid and standardized, and their price can easily be observed. Others, such as private equity and particularly collectibles are illiquid, unique and difficult to price… Liquidity may be very difficult in alternative investments, currencies is one of the most liquid markets, but fine art is one of the most illiquid things… in classical investments liquidity is always high. Fine art is also very heterogeneous, while this isn‘t the case for currencies for example. Both heterogeneity and illiquidity increase if you go down in the list of alternative investments. What all alternative investments have in common is that they have a low correlation with traditional investments. The pictures shows all the correlations with high capitalisation stocks. The dark ones are classical investements and the grey ones are alternative investments. Alternative investments are valuable because they give you a huge diversification benefit. They allow us to diversify the risk because they have a low correlation with our existing investments (large capitalisation stocks). Gedownload door: hannahducatteeuw | [email protected] Wil jij2€76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Money market instruments The money market is the market for short-term borrowing and lending. It is dominated by banks’ lending to each other. Typical instruments: Treasury bills: short-term security issued by the U.S. government Commercial paper: securities issued by companies or banks Certificates or deposits Bankers acceptances … Everything below one year to maturity is a money market instrument. Treasury bills are often used as a proxy for the risk free rate of return. You know what you will get back at the end with treasury bills, you buy at a discount and get more money back in the end then what you paid for it. There is no risk because you know you‘ll have a profit. Bond market: different issuers Bond market = debt market = fixed income market: longer term securities than on the money market Bonds can be issued by governement, by federal agencies, by companies… Plain vanilla bond: you know when you’ll get your money back, how much you will get back, what interest rate will be used etc. Zero coupon bond: you pay something today and you get money at the end but no interest payments in the meantime. … The foreign exchange = FX Why is the foreign exchange so important? It affects inflation through the cost of imports, it affects national capital flows through risk and returns of different assets. If the stock market keeps falling or raising for a long time, companies might adjust the way the fund their activities. Therefore it is better for companies to buy different assets from different countries to diversify their risk…. But the foreign exchange has a more direct impact through import and export and the risk for foreign investments. The foreign exchange is a very important market. Turnover ratio How often is every stock sold on the market? Gedownload door: hannahducatteeuw | [email protected] Wil jij3€76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Chapter 2: Risk and return from the historical record Portfolio theory breaks investments down mainly to risk and return. Return: the amount of money gained or lost on an investment relative to the amount of money invested. Which factors do impact the interest rate? Supply of capital: by households Demand of capital: by businesses Government’s net supply and/or demand: actions by the ECB C OMPARING RETURNS : ZERO COUPON BONDS Assume that you have the choice between several zero-coupon bonds with different maturities and a par value of 100. How can you compare them? The return that you get is the holding period return over the period from t=0 to t=T. It is a bit like an interest rate that you get throughout the period. The difference is that a normal interest rate is fixed at the beginning. PT P T −Pt HPRt = –1= Pt Pt Pt = price today  initial investment PT = price at the end of the holding period  final payout HPRt = holding period return BOND TIME TO MATURITY PRICE Pt HPRt A 1 90 (100/90) – 1 = 0.11 B 5 80 (100/80) – 1 = 0.25 C 10 70 (100/70) – 1 = 0.43 The holding period returns increase for bond A to C. this does not necessarily mean that bond C is better than the others. We can’t compare these bonds because they holding periods differ. The holding period return is a good measure ONLY if the holding period is the same for all assets under consideration. Otherwise we need to normalize the returns by the holding periods. You can do that by calculating ‘one-year-equivalent-returns’. Gedownload door: hannahducatteeuw | [email protected] Wil jij4€76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen HPR (1+ EAR)T −1 A simple way is to use the annual percentage rate: APR = = T T T = time to maturity However, the APR does not consider re-investments, if you would reinvest the money you would get different results at the end. Assume you buy bond B for €80, after 5 years you get €100 back so you have earned €20. If you would reinvest in the same bond immediately, you would now be able to invest €100 which is 1,25 times what you previously invested. This will lead to a payout of €125 instead of €120… To account for that you can use the effective annual rate: EAR = (HPR + 1)1/T - 1 TIME TO PRICE Pt HPRt APR EAR MAT. 1 90 (100/90) – 1 = 0.11 0.11/1 = 0.11 (1+0.11)1 – 1 = 0.11 5 80 (100/80) – 1 = 0.25 0.25/5 = 0.05 (1+0.25)1/5 – 1 = 0.0456 10 70 (100/70) – 1 = 0.43 0.43/10 = 0.043 (1+0.43)1/10 – 1 = 0.0364  If the time to maturity > 1: EAR < APR The longer to maturity, the larger the difference between the two. P 1−P0 + D1 Holding period return can also be calculated by using dividends: HRP = P0 T HE RELATION BETWEEN INTEREST RATES AND INFLATION Fisher-equation: rnominal = rreal + πexpected  π = expected inflation rate If you lend money to someone you ask a compensation because you have to wait until you get your money back before you can use it yourself. You also ask a compensation because you take a risk by giving someone your money. And you also ask a compensation for the possible loss of purchasing power. If you give away €1000 for 10 years, and you compare what you can buy with that money now and in 10 years then there will be a difference due to inflation. The real interest rate is more important for the investors than the nominal one. But the real interest rate is not known in advance. Interest rates are a compensation for giving away the savers money, it can’t be used for consumption until it is paid back. If inflation is high, lenders may ask a compensation. M EAN SCENARIO OR SUBJECTIVE RETURNS Until now we worked with historical returns which we can simply read from the data. But some investments concern the future, you need to estimate those. Sometimes it is possible to define some scenarios/states for the future and assign probabilities to each of these scenarios. P ERCENTAGE VS. LOGARITHMIC RETURNS Gedownload door: hannahducatteeuw | [email protected] Wil jij5€76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Logarithmic returns offer some advantages: Easy calculation (temporal aggregation): Easily aggregate returns to lower frequencies by summarizing high frequency returns. This means we can use the monthly return as the sum of the daily logarithmic returns Symmetry (up and down): Logarithmic returns will give the initial price after a price increase of e.g. 10% followed by a decrease of 10%. See book p.27! Symmetry (exchange rates): Specifically, for exchange rates the return in price quotation and indirect quotation only differs by sign, not in magnitude. M EAN AND VARIANCE OF HISTORICAL RETURNS If we look at the past, there are no scenarios but there is however a lot of variation. Returns are not constant over time, they change almost every period. So how can you get some information about the probability distribution of returns based on historical data? You can simply assume the distribution is normal. In this case you would only need standard deviation/variance and the mean to describe the distribution! S HARPE RATIO Knowing the return and the variation of returns doesn’t help a lot, if we choose between investment alternatives. There is a trade-off between them: assets with higher risk (undesired) will provide higher returns (desired)… Sharpe ratio A simple measure that provides information on how much the = reward-to-variability additional return “costs” in terms of risk. It gives you the additional ratio return the market has paid for an additional unit of standard deviation. risk premium Sharp ratio = SD of the excess return Sharp ratio: the price of the risk or the reward to variability, a premium that you earn for taking risk. Risks may be due to volatility but it can also be a default risk or something… *Risk premium = return of the risk-free asset – return that you earned NOTE: the sharpe ratio is a meaningful performance measure only for an investor’s whole portfolio, whereas you should not use it for evaluating particular assets that you want to add to your portfolio. Gedownload door: hannahducatteeuw | [email protected] Wil jij6€76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Chapter 3: Distribution of financial terms We are going to look at one particular type of risk: price risk We are not going to look at the other two:  Default risk = the risk that someone is not able to pay  Liquidity risk = the risk that you don’t find a counterpart if you want to sell your assets Normal distribution For assessing the risk of an investment, one should consider the whole distribution and not only the variance. It is useful to first introduce higher moments of distribution. We already know the mean (first moment) and the variance (second moment). The third moment is the (sample) skewness. S KEWNESS The skewness provides information about a distribution’s symmetry. Skewness of 0 = symmetric distribution Negative skewness = left-skewed or skewed to the left. The tail on the left side of the distribution is longer than the right side and the majority of the probability mass is located to the right of the mean. Positive skewness = right-skewed or skewed to the right. Realization to the right tend to be larger. If a return distribution is left-skewed, the investment provides frequent small gains and few extreme losses. A right-skewed return distribution provides frequent small losses and few large gains. Investors will be more attracted by positively skewed return distributions, because they provide large gains (although they occur rarely). Left-skewed distributions are bad for investors because the rare extreme losses may jeopardize the investor’s wealth. Gedownload door: hannahducatteeuw | [email protected] Wil jij7€76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen K URTOSIS The fourth moment is the kurtosis. A normal distribution has a kurtosis of 0. Positive kurtosis = leptokurtic Negative kurtosis = platykurtic A higher kurtosis means more of the variance is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations. As a result, the leptokurtic distribution has a higher peak around the mean compared to the normal distribution, i.e. the probability density is concentrated around the mean and at the tails. The distribution is therefore said to have fat tails. Leptokurtic return distributions are DANGEROUS, because the fat (negative) tails make extreme losses occur more frequently. Neglecting them may lead to a dangerous underestimation of (tail) risk. We have too many extreme changes, too many changes that are close to zero and but a few moderate changes… The extreme changes are very important for risk management, because these changes can determine whether you go bankrupt or not. If you work on a normal distribution (the red one) you might underestimate these risks because, when in fact, the blue one is the true distribution then the risk is much higher and that is a problem. With the red line you underestimate the risk of an extreme event. (links en rechts onderaan, die verschillen) See p.36 book for example Stylized fact 1: The distribution of high-frequency financial market returns is leptokurtic Stylized fact 2: The kurtosis of financial returns declines with temporal aggregation. Stylized fact 3: Almost symmetric return distributions. The skewness of financial returns does not systematically differ from zero. Stylized fact 4: the volatility of financial time series clusters. “Large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes.” Gedownload door: hannahducatteeuw | [email protected] Wil jij8€76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Portfolios and investments section 1: historical volatility The problem of estimating volatility Volatility is important for: 1. Investment decisions (->next section) 2. Performance evaluation (->last section) 3. Risk management (not in this course, sorry…) There are two ways to get a volatility forecast: 1. Implied volatility: the volatility that is consistent with the observed option prices.  Look at what other people on average believe about the future. But markets can be wrong, so it isn’t guaranteed that markets in the future will actually look like that. 2. Historical volatility: one can use historical price series to estimate historical volatility and use it as a tool to forecast future volatility.  Look at the past and hope that you can draw some conclusions from the past about the future. The problem is that volatility is a latent factor and not observable. You cannot look into parallel universes and see all the different outcomes, you can only draw conclusions from the one sample that you get. Therefore, we have to use estimators. There is not a single “correct” way to estimate the historical volatility. The concrete value depends on a couple of decisions we have to make. Sample length: How far do we look back? Sample frequency: How frequent do we collect/use data? Daily, weekly, monthly, … Measure to use: Which assumptions on the price process? S AMPLE LENGTH The choice of the sample length is a trade-off between two goals: accuracy and currentness.  Long samples increase accuracy if the samples are stable. But we know that there are volatility clusters so that they aren’t stable. There may be structural changes as well or the central bank may decide to peck your currency against another currency.  Short samples better capture time-variations: it is better to look at the past at a similar length then as you want to look in the future. If you want to forecast for 1 year then it would be good to also look back for 1 year. Looking only a couple of days back would be useless… Gedownload door: hannahducatteeuw | [email protected] Wil jij9€76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Estimating the volatility over a long sample period would rather measure the long-term average volatility. On the other hand, a sample period that is too short will introduce estimation errors and put too much weight on the very recent past. It is good to use multiples of three months because that is in accordance with the quarterly reporting periods of macro announcements. This is to ensure a constant number of volatility increasing quarterly reporting periods, when the rolling window moves. S AMPLING FREQUENCY Most commonly used are daily and weekly sampling frequencies. DAILY DATA: has to huge advantage of almost five times as many observations, which should increase the accuracy of the estimation. If you use daily data, you have much more data in 1 month. If you work with monthly data you only have 1 observation, with weekly data you have 4 observations per month and with daily data you have 21 observations per month. On the other hand, the use of weekly or monthly data decreases the impact of holidays or vacations, in daily data these would be missing observations. For monthly data you should always be able to make a calculation. Rule of thumb: weekly data is better for longer forecasting horizons and several markets are considered, otherwise you use daily data. T HE CHOICE OF VOLATILITY MEASURE : PRICE RANGES The concrete choice of the volatility measure has a huge impact on the volatility estimate. Therefore, this is a very difficult question! Often the standard deviation of the past returns is referred to as historical volatility: However, there are many more ways to measure volatility. Efficiency is measured as the deviation of the accuracy of a volatility estimator compared to the volatility of a benchmark estimator. The benchmark estimator that is traditionally used is the classical estimator or the close-to-close range- based estimator or squared return. All range-based estimators assume that the price P t follows the geometric Brownian motion: μ = the drift term σ = the constant volatility Bt = standard Brownian motion If only closing prices are available and if the mean return is 0, this is the average of the close-to-close return or classical estimator. The classical estimator for σ is: Gedownload door: hannahducatteeuw | [email protected] Wil10 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen The first formula gives the normal standard deviation. If you would take the average of all the daily standard deviations (last formula) then you would get approximately the same as the top formula. 1 Except you don’t get the mean, so you get *∑(rt)². N The classical estimator is unbiased if and only if μ* = 0. The estimator is quite noisy, its error is comparatively large. It can be used as a benchmark, but also as an input for constructing new measures. The average of all the squared returns of the day would give you a measure of the volatility. Is that a good measure? Well if one thing, it is straightforward. If the security is not continuously traded, so if the exchange is closed during certain hours per day, we can also incorporate the opening prices and get the following measure: The market closes and when it opens again, there is a sudden price jump. This means that the value of a stock may change, even if the stock isn’t traded at the moment. So if the price represents the value, then the price should move as well even when the market is closed. We would have price jumps overnight, we will never be able to observe that, but we need to remember that these prices are constantly moving. Ot = opening time Ct = closing time F = a fraction of the day, e.g. if the market is open from 8 to 5, the fraction is 9/24 So if we only consider opening and closing prices, we will tend to underestimate the volatility, because the distance between opening and closing prices is always smaller (or equal) to the real range. For this reason Parkinson suggest a range-based estimator based on high (Ht) and low (Lt) prices. High and low prices are the highest and lowest prices that appear during the trading day. Gedownload door: hannahducatteeuw | [email protected] Wil11 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen But a logical next question is: if we know that there are price jumps overnight and that the price changes when the market is closed, then why don’t we simply include that? If we have opening and closing value and the highest and lowest value then why don’t we simply use everything that we know? Garman and Klass then included opening and closing prices as well in the formula: (You don’t need to know the formulas by hard, what you should need to know is which points you can use to estimate the volatility) Lastly, Garman and Klass suggested to combine the classical estimator and the estimator by Parkinson: And this is the best we can do, if there is NO PRICE DRIFT i.e. μ*=0 !! What is left is to find a solution for non-zero drifts. A suitable estimator has been suggested by Rogers and Satchell. This if for continuous trading: And Yang and Zhang come with an alternative for non-continuous trade: OVERVIEW: If we look at the efficiency and take the classical estimator as a benchmark then we see that Parkinson for example gets an efficiency of 5,2. This means that, to get the same accuracy as you would get with the Parkinson estimator, you would need 5,2 times more data / information for the classical estimator. The more data you use (C, O, H, L), the less data you need and the less you are exposed to structural changes. Which these estimator you can calculate the values for one day, if you want to go from 1 day to a longer period you need to calculate the average. However, there is one thing that the estimators cannot deliver and that are volatility clusters, because they just measure volatility over an interval and there is no time dimension in that. Gedownload door: hannahducatteeuw | [email protected] Wil12 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen The main advantage of the range-based measures is that they do not require lots of data. In the worst case the only data you need are open, close, high and low prices together with information about the times when the market closed. GARCH models We already mentioned that volatility is heteroscedastic and clusters. Heteroscedastic With non-constant volatility Homoscedastic With constant volatility Now, let us assume a very simple time series model: Rt = a + εt Rt = return, a = average and ε = error term εt N(0, σ²) the return of an asset is linked to the market’s return plus an error term. In this simple model the variance of the error term is not-time varying, the volatility σ² is constant (homoscedastic). The question arises: how can we incorporate the observed volatility clusters into the this model?  A first important step was taken by Engle who let the error term’s volatility depend on the squared previous period’s disturbance. Use yesterday’s volatility to estimate today’s volatility! If yesterday’s price movements already have a little bit of an indication of today’s price movement then that is actually a good indicator. If yesterday’s price change was large, then today’s price change will probably also be large. We don’t know which sign it will have, but we know it will be large. However, we cannot observe volatility. So what can we observe? We can observe yesterday’s innovation, thus εt-1. So why not use εt-1²? Rt = a + εt εt N(0, σ²) σt² = ω + β.εt-1² This model provides us with a simple parameterization of the observed volatility clusters: the larger yesterday’s disturbance was, the more likely it is to observe a large disturbance today, of either sign. Volatility σ² is now conditional and autoregressive, thus time-dependent and depending on its own history! Engle called this model the ARCH model = AutoRegressive Conditional Heteroscedasticity σt² = ω + β.εt-1² is an arch-model of order 1 because the volatility is determined by the disturbance with lag 1. So we only look 1 day into the past and we know that because the formula uses t-1. Gedownload door: hannahducatteeuw | [email protected] Wil13 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen p 2 This model can also be generalized to any order: σt² = ω + ∑ β i. ε t −i i=1 ω p And the unconditional variance is: σt² = 1−∑ β i i=1 In practical applications, it turns out that for modelling financial time series, high orders of the ARCH process are necessary. Bollerlev asked himself how to avoid the huge order of ARCH models. For this reason he developed the generalized ARCH model of GARCH. p q 2 2 σt² = ω + ∑ ai. ε t−i + ∑ b j. σ t − j i+ 1 j+1 In contrast to Engle’s original ARCH model, the conditional variance now does not only depend on yesterday’s disturbance term, but also on yesterday’s conditional variance. In practise it turns out that for almost every financial time series a GARCH(1,1) model is sufficient to capture all ARCH effects: σ 2t =ω+ a. ε 2t−1 +b. σ 2t −1 σ 2t−1 this captures additional disturbances. You only need to look at yesterday’s price change and yesterday’s volatility, so only look back one day into the past. This means the lowest order model is actually almost always sufficient because in fact σ 2t−1 already includes a whole history about the volatility. Therefore you only need to include omega, a and b. 2 2 The current volatility σ t is therefore a function of the whole history of price evolution. σ t is the CONDITIONAL variance of the garch process. ω σ 2= p q The UNCONDITIONAL variance is: 1−∑ ai −∑ b j i=1 j =1 a + b should NEVER exceed 1.The closer the sum is too one, the longer the volatility shocks last, so the longer it takes for them to die out. If a + b is close to 1 it will take a very long time before the volatility goes back to the normal level after a volatility shock. a + b < 1 : you expect that it will go back to the normal level very fast. a + b = 1 : volatility shocks would never die out, therefore it is not allowed for a + b to equal 1. a = b = 0 : we have a constant volatility model, yesterday’s volatility will not have any impact on today’s volatility. In practical applications we see that the sum is always very close to one, which implies that financial markets are characterized by a high degree of volatility persistence. Gedownload door: hannahducatteeuw | [email protected] Wil14 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen The unconditional variances captures very well volatility clusters and also the size of the kurtosis. The unconditional kurtosis of a GARCH(1,1) process is: 6 a2 K = 3+ 1−b 2−2 ab−3 a2 The kurtosis always exceeds the normal distribution’s kurtosis of 3, if a > 0. The unconditional distribution of GARCH(1,1) is therefore leptokurtic (while the conditional one is normal). The GARCH model is able to model the two most important stylized facts regarding series of financial returns: volatility clusters leptokurtic unconditional distributions GARCH-in-mean The GARCH-in-mean captures the leverage effect, the observation that declining share prices often go along with higher volatility. Rt = α + β.RM + θ.σt² + εt σt² = ω + a.εt-1² + b.σt-1² GJR GARCH This model makes the volatility process asymmetric by adding an additional term to the variance equation that only occurs if the shock εt-1 in the previous period was negative. Rt = α + β.RM + εt σt² = ω + a.εt-1² + d.I a large cap portfolio bs = 1 --> a small cap portfolio bv = 0 --> a portfolio with a low book-to-market ratio bv = 1 --> a portfolio with a high book-to-market ratio Evolution of SMB and HML over time: You take all the stocks from one market and sort them based on their market capitalization. Capitalization is smallest on the bottom of the list and larger at the top of the list, so it goes up. You make a portfolio of that, RL would be the return of the portfolio of the large stocks in the market. Rs would be the return on the small stocks in the market. SMB = Rs – RL RA Gedownload door: hannahducatteeuw | [email protected] Wil35 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen RE RL =30% hoogste (vrij te kiezen hoeveel procent je wil) RF... RZ RV RS =30% laagste (moet hetzelfde percentage zijn als het hoogste) RL Fama and French still see the return as a reward for risk: if the return of one security Is higher than the return of another security, it means that the investment in the first security is more risky. If you believe in efficient markets, there must be a good reason if a stock is cheap why it is cheap. Namely there must be some risk related to the size or book-to-market ratio… SIZE: small companies are more vulnerable to shocks, they are on average younger and their cash flows are more volatile. BOOK-TO-MARKET RATIO: this appears in companies of which the future is doubtful or in distress or in capital intensive industries. Companies in capital intensive industries are more risky because they may be vulnerable to low earnings during recessions but also because their capital needs may have led to a higher dependence on debt, making the exposure to interest-rate risks higher. So the Fama-French-model captures a variety of situations and work well in practise. But another explanation why it performs well is because the additional factors “repair” the shortcoming of standard stock market indices as proxies for market returns.  These standard stock market indices weight the stocks according to their market capitalization, therefore they are size-biased and valuation blind! The Fama-French model is able to explain more than 90% of a portfolio’s returns and is used as a benchmark in practise (single factor models can only explain 80% of the returns). C ARHART ’ S FOUR - FACTOR MODEL Carhart extended the model by adding a fourth factor: momentum. This fourth factor is a zero-cost portfolio that goes long in previous winners and short in previous losers. Ri,t =  + ·RRM,t + bS·RSMB + bV·RHML + bM·RWML +εt Fama-French Three factor model Added by Carhart WML Winner minus loser 7.4. Liquidity as an additional factor Liquidity The ability to trade large quantities quickly at low cost and with little impact. Stocks with a higher liquidity should have a higher return. Because you often get a liquidity premium. Liquidity is a bit more difficult to include in your model, because it has many perspectives… “What is liquidity?”. That is why it isn’t often included… Gedownload door: hannahducatteeuw | [email protected] Wil36 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Liquidity captures how easy it is to convert an asset into cash, and is a key variable when investigating financial markets. Ri,t =  + ·RRM,t + bS·RSMB + bV·RHML + bM·RWML + bL·RLIQ+ εt Liquidity has at least four dimensions:  Depth and breadth: The market should be deep and broad o Deep: the number of price levels in the book. You have all the potential buy orders and sell orders and they are waiting for execution. If the market is deep, that means that there are many levels behind each other. Someone would be ready to sell at €1,08, someone at €1,085, someone at €1,09 etc. So there are many levels and the market is deep. o Broad: the volume per price level is very large.  Tightness: The costs at which an asset can be traded, usually measured as the bid-ask spread. The more liquid the market, the cheaper the trade and the lower or tighter the spread.  Resiliency: the ability to absorb shocks coming after large trades. Higher liquidity means higher resilience, the market quickly recovers from a liquidity shock.  Immediacy: how long does it take to trade, is it easy to find a trading partner? Traders that don’t want to wait need to place a trading order for which they need to pay extra, how much extra depends on the spread. The more liquid a market, the shorter the waiting time. Moreover liquidity also measures the speed at which information about an asset can be processed and it as well affects the asset’s expected return. Generally it is seen as a positive aspect of an asset and there higher liquidity = higher price = lower required return. 7.5. Components of risk Total risk = systematic + unsystematic risk Gedownload door: hannahducatteeuw | [email protected] Wil37 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen  Market risk or systematic risk: comes from the factors or market index  Unsystematic or firm specific risk: comes from the error terms As already mentioned two assets can only be connected through a common factor and their covariance is determined by their exposure to the common factor and its volatility. Covariance = product of betas * market index risk (or volatility) Cov(ri,rj) = i j M2 We can also express this using the correlation coefficient instead of the covariance, which simplifies the calculation even more: Correlation = correlations * market index Corr(ri,rj) = ij /(ij) = i.M2*j.M2/(iM*jM) = Corr(ri,rm) Corr(rj,rM) M 2 The previous 2 equations are of importance because they show that we do not need to calculate all the bivariate correlations of the available assets but we can simply derive them from the betas. Another advantage of the factor model is that we can easily learn about the characteristics of a portfolio by looking at the individual assets: For each security: Ri = i + i.F + εi For the portfolio: RP = P + P.F + εP With: F = (rM,t – rf) n 1 α P= ∑α The coefficients for the portfolio are simply the n i=1 i n 1 βP= ∑β unweighted averages of the coefficients for n i=1 i n 1 ε P= ∑ e i asset-specific equations. n i=1 This equation plus the equations for the coefficients allow us to describe the effect of arbitrarily adding additional assets, to which we refer as simple diversification. What happens if we diversify? You can write for each of the stocks in the portfolio a regression or one regression for the entire portfolio. The average alpha and beta would be the average of all the alpha’s and betas from all the stocks which are included. α P  0. Alpha represents the excess returns, these will go to zero if you keep increasing the number of stocks in your portfolio. Simply because in the end you will have the entire market in your portfolio and the market cannot perform better than the market! β P  1. If you keep adding stocks beta will become the beta of the whole market and that beta is by definition 1. Gedownload door: hannahducatteeuw | [email protected] Wil38 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen ε P 0. The error term will converge to the population mean which is 0 due to the law of large numbers. The terms are all independent from each other. In the end our portfolio will get closer and closer to the market portfolio by purely increasing the number of stocks, thus naïve diversification. We can also derive some information on the variance of the portfolio: n 1 σp² = βp². σM² + σ².(εp) = βp². σM² + ∑ σ ² ε i n ² i=1 = σM² for n  ∞ = 0 for n  ∞ σp² → σM² for n → ∞ So the risk of a portfolio equals more or less the market if the number of assets is sufficiently large. The idiosyncratic error will on average get closer to zero but you cannot make all your risk disappear because you will always have the market risk! If you have 20 stocks you are already close to the market risk so you don’t need that many, like 100 or something stocks in your portfolio, to come close to it. SO for a portfolio of 100 stocks (=n) we only need: n=100 estimates of i n=100 estimates of I n=100 estimates of the securities residual variances 1 estimate of the market excess return 1 estimate of the market volatility (3n+2)=302 estimates in total (compared to 5,250 with the Markowitz procedure) Gedownload door: hannahducatteeuw | [email protected] Wil39 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen  You can also construct an efficient frontier with factor models. 7.6. Single factor models and active diversification Up to now we always talked about passive or naïve diversification which meant that we just randomly selected stocks for in our portfolio. But you can also actively chose which stocks you put into your portfolio, that is active diversification. In that case the equations for the weights change. Let us assume that we cannot only chose which assets we want to include in the portfolio but also their weights, with the only restriction that the weights have to sum up to 1. There are several goals you may follow when actively constructing a portfolio: Increase alpha: That is the most difficult part, a high positive αp will increase the overall expected portfolio return. Try to achieve an αp > 0, then you have excess returns. Decrease beta: this leads to a reduction of systematic risk, if you find a portfolio with βp < 1. Thus making it less exposed to market risk because due to more diversification you are less vulnerable to market fluctuations. High number of assets: keep your number of assets sufficiently high and their wrights sufficiently low to ensure good diversification. n 2 The last two points will affect the variance equation or risk: σ P = 2 [∑ ]i=1 wi β i σ M 2 + σ 2 (ε p ) n 2 n ¿[⏟ ∑ ] i=1 w i βi σ M + 2 ∑ wiεi ⏟ i=1 1 n → 0 forn → ∞ ? ≤ ∑β? n i=1 i Gedownload door: hannahducatteeuw | [email protected] Wil40 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Active management shifts the efficient frontier upwards and makes it possible to realize a steeper CAL. How far we can shift the frontier depends on the stock picking abilities of the fund manager. But the frontier can also shift downwards if the manager’s stock picking abilities are poor! 7.7. Further multifactor models Fama-French was a microeconomic multifactor model because the factors SMB and HML refer to certain characteristics of the underlying assets. Macroeconomic based models use risk factors that reflect macroeconomic risk. You can also combine macro- and microeconomic risk factors. C HEN , R OLL AND R OSS ri,t = αi + β1·RF1,t+...+β6·RF6,t+εi,t  They add a couple of macro factors: market risk, output, inflation change, unexpected inflation, credit spread and term structure. Slide 16 or p.132 states the evolution of factor models. People started with simple multi-factor models and in some cases you had factors of special interest: e.g. if you want to know which impact a change in the exchange rate has on an exporting company, you can do that by using a multi-factor model where you only have the exchange rate as extra factor. The two up-to-date standard models are the Fama-French Three factor model and the Carhart Four Factor model. People are working on the five factor model with liquidity, but this isn’t standard yet and it isn’t in use either. Chapter 8: The Capital Asset Pricing Model 8.1. The basic setting CAPM is a market oriented and objective approach that tries to explain which return may be expected for a security if the market is in equilibrium. So the CAPM is an equilibrium model.  The market is in equilibrium, if the prices are such that supply for securities equals demand. This means that all investors hold their desired portfolio. CAPM is based on a couple of simplifying assumptions: Perfect competition: large numbers of investors, each of them is “small”  The decision of one individual investor has no effect, they are price takers. Gedownload door: hannahducatteeuw | [email protected] Wil41 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen All securities are highly divisible into small parcels. All investors are rational mean-variance optimizers: all investors base their decisions on the mean-variance criterion. So they all follow the Markowitz theory perspective. Homogeneous expectations: all investors have the same expectations on future returns, variances and correlations. E.g.: because they all use the same historical data. No taxes and transaction costs. All investors have the same investment horizon. Information is costless and available to all investors. Investors can borrow and lend at a risk-free rate. Not a single of those assumptions if realistic. However if you follow these assumptions and they are all fulfilled then:  All investors face the identical efficient frontier. This is because they all use the same input, consider the same information, use the same evaluation criterion and so on.  Only differences: personal risk averseness (preferences), amount of wealth to be invested What is the difference with the mean-variance analysis from the previous chapter? The CAL we worked with there was for an individual investor, based on his personal investment universe. We dealt with his particular decision problem and his personal set of investment opportunities. CAPM is about ALL investors and ALL available assets, about what happens in the whole market if it is in equilibrium. Based on CAPM we can draw conclusions for a representative investor. Resulting equilibrium conditions Based on these considerations we can draw a number of important conclusions that are valid under strict conditions which we formulated above. These considerations or resulting equilibrium conditions are: What are the equilibrium conditions? All investors will hold the same portfolio for risky assets – the market portfolio. This is because everybody relies on the same input, is effected by the same restrictions and processes the same information in the same way.  So there exists a universal, market-wide CAL = capital market line (CML). The market portfolio contains all risky assets in the world. So not only stocks or bonds but also real-estate holdings, commodities, human capital etc. The proportion of each security is its market value as a percentage of total market value. The risk premium on the market depends on the average risk aversion of ALL market participants because we refer to a representative investor. The market risk premium may Gedownload door: hannahducatteeuw | [email protected] Wil42 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen change, during crises it will definitely be higher as investors will be more reluctant to taking risk then. The risk premium on an individual security is a function of its covariance with the market. There is NO need to short-sell assets in the market portfolio. Some investors will however go short but this is per definition not possible on an aggregate level. As all investors face the same riskless rate and the same set of risky assets, the CML is the same for all investors. The separation principle of the Markowitz portfolio selection procedure still holds. The investment process for each investor is: 1) finding a market portfolio 2) determining the proportions of the riskless asset and the market portfolio However, the investor does not need to find “his’ optimal risky portfolio, he simply needs to buy the market portfolio (if the market is in equilibrium). Again an asset’s risk is determined by its contribution to the risk of the entire portfolio. 8.2. The security characteristics line and the security market line The pervious section deals with the market portfolio BUT the CAPM also allows to learn about single securities! The variance of the market portfolio was: m2 = is wi ·R ws ·R i,s The contribution of an asset to the total risk is given by its covariance with the market portfolio. An asset that is only weakly or negatively correlated is more valuable for investors and the required return for this asset must be lower! i,m = E[rirm] – E[ri] ·R E[rm] Beta In practise the covariance is usually normalized with the market returns’ variance: i = cov(Ri, Rm)/ m2 = i,m / m2 This beta refers solely to the non-diversifiable or systematic risk. In other words, it gives you the market risk and therefore it is also called the market beta. Beta for the market portfolio itself: m = cov(Rm, Rm)/ m2 = m² / m2 = 1 Beta for the risk-free asset is: f = cov(Rf, Rm)/ m2 = 0 / m2 = 0 But in general securities can be divided into three main groups based on their market beta: Gedownload door: hannahducatteeuw | [email protected] Wil43 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Aggressive securities: β > 1 Neutral securities: β = 1 Defensive securities: β < 1 The higher the beta, the more a securities return will fluctuate with the market. So beta is a measure of the security’s exposure to market fluctuations. The weighted average of these 3 should be equal to the market because they form the market all together. If you have a portfolio with a lot of stocks with a beta of less than 1, you have a defensive market portfolio. You can also make your portfolio more aggressive. CAPM says that we all hold the market portfolio, nothing else. Security characteristics line (SCL) The relation between Ri and Rm, with beta being the slope coefficient. » This relation can be formulated as: Ri = i + i Rm + ei Ri = excess returns of the asset Rm = market return Security market line (SML) It provides the return for a security I, depending on its security- specific beta. All beta-expected return combinations of securities must be located on the line. N OTE Capital market line The relation between risk and return Security characteristics line The relation between return of the security and market return Security market line The relationship between the mean and beta The SCL relates the return of a security to the return of the market portfolio. But the SML expresses a security’s return as a function of the security’s beta. The beta becomes the independent variable of the regression, while the market’s expected excess return is now the slope coefficient. If we replace returns in the SCL by expected returns we get the SML. Ri = i + i Rm + ei Gedownload door: hannahducatteeuw | [email protected] Wil44 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen  E[Ri] = E[i + i Rm + ei ]  E[Ri] = E[i ] + i E[ Rm] + E[ei ] Or: E[ri] = rf + i E[rm – rf ] Security Market Line Point (1,rm) represents the market portfolio or a neutral security. Defensive securities can be found to the left of the market portfolio. They offer a lower return than the average. Offensive or aggressive securities offer a higher return than the average and can be found to the right of the market portfolio. Expected excess return of the market E[rm – r ]. Any neutral stocks should have the same expected return as the market. The only determinant of a stock’s expected return is beta. Beta resembles the co-movement of the stocks with the market. Stocks that co-move with the market (beta > 1), should have higher expected returns because they have more risk. Defensive stocks are desired because they protect you from market movements but you need to pay a sort of “insurance premium” for them and this lowers your expected return. Some securities are not located on the SML: ABOVE: expected return above the SML. The expected return is too high given the security- specific beta. Accordingly, the price of the security is too low, it is under-priced.  Alpha is higher! UNDER: expected return is below the SML. The security is overpriced. What will happen? If the market participants realize that a security is under-priced they will start buying it, attracted by the high expected return compared to the risk. As a result the price will increase and the expected return will decline. This well continue until the expected return equals the value predicted by the SML. In case the security is overpriced, investors will start selling their security. The price will decrease and the expected return will increase. This until the expected return equals the value predicted by the SML. E[Rt] = E[αi] + β(E[RM]) Gedownload door: hannahducatteeuw | [email protected] Wil45 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen  In CAPM (so in an efficient market): people will realize that if alpha is positive they need to buy the stock, and if alpha is negative they need to sell their stocks. So CAPM expects that investors adjust their movements and realize all these things. How to get to alpha and beta? In general it should be possible to estimate the α and β from historical data. What is needed? ri – rf = αi + βi.(rm – rf) + ei SLIDE 2 On slide 2 you see the SCL of Microsoft. The beta is 1,5 which means that if the market increases by 1% the respective stock would on average increase with 1,5%. So it reacts stronger than the whole market = aggressive stock. This also happens if the market falls, then the stock of Microsoft falls more on average. Chapter 9: Arbitrage pricing theory (APT) SIMILARITY BETWEEN CAPM & APT Both approaches assume that the market is in equilibrium. DIFFERENCE BETWEEN CAPM & APT The way the equilibrium is understood is different. APT says that prices are in equilibrium if no arbitrage opportunities exist (this does not imply that prices are correct!). The equilibrium conditions are mild in the APT. In CAPM prices are in equilibrium if the supply of securities equals demand. 9.1. What does arbitrage mean? Arbitrage based on the law of one price Two items that are the same cannot sell at different prices. Items that are the same also include (a combination of) items that provide the same cashflows. If we can replicate an assets cashflow by the cashflows of a portfolio of other assets, we know what the price of this security must be. If the law of one price is violated, arbitrage is possible if no institutional barriers prevent agents from exploiting arbitrage opportunities. Agents exploiting arbitrage opportunities are called arbitrageurs. Arbitrage is an investment strategy that makes a 1) positive expected return 2) without the risk of loss and with 3) a zero initial investment. So you have found a “money machine”. EXAMPLES E XAMPLE 1: B ANKS You borrow money at bank A at an interest rate of 5% and deposit the money in a fully insured bank B at 6%. For each €100 that you borrow at bank A and deposit in bank B you have to pay €5 interest and get €6 interest. Thus you earned €1 without initial investment. The bank where you deposit must be fully insured, this means there is no risk. If there would be risk, the €1 that we can earn is nothing more than a compensation for taking the risk. Gedownload door: hannahducatteeuw | [email protected] Wil46 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen E XAMPLE 2: T RIANGLE ARBITRAGE Triangle arbitrage Exploits the state of imbalance between three foreign exchange rates. You observe the following exchange rates: 1.30 USD/EUR in New York 100 JPY/USD in Tokyo 129.9 JPY/EUR in Tokyo The crossrate: JPY/EUR = 100 JPY/USD * 1.30 USD/EUR = 130 JPY/EUR We see that there is a difference between the crossrate and the JPY/EUR, albeit small, but can we exploit it?  We should sell JPY to Euros and buy the Yen back at the crossrate. The USD serves as a vehicle currency. 1. Change 1,000,000 EUR into 1,300,000 USD (NY) 2. Change 1,300,000 USD into 130,000,000 JPY (T) 3. Change 130,000,000 JPY into 1,000,769.82 EUR (T) Your profit is €769,82 although we didn’t need to invest money. We could do this again and again. As an effect the Yen will depreciate versus the Euro, while it appreciates versus the Dollar until the arbitrage opportunity has disappeared. E XAMPLE 3: SHORT SELLING / GOING SHORT Short selling / going short / shorting Selling assets which the seller does not own but that are borrowed from a third party. One may decide to short sell the stock of company A because he expects the prices of the stock to fall. He sells the stock and once the prices have fallen he buys it back at a lower price and delivers it to the lender. This is NOT an arbitrage strategy because the short seller faces an enormous risk, it is far from sure that the prices will move in the desired direction. It is however possible to use short sales in CONSTRUCTING arbitrage strategies. Suppose you have this available: State Probability Security A Security B Security C R(ecession) 0,25 -6 -10 -2 S(able) 0,50 10 16 6 B(oom) 0,25 20 34 6 Security C shows the lowest variability but also the lowest returns in states S and B. As a reward it will suffer less in case of a recession. The prices however are in a disequilibrium! To show this, consider the following investment strategy: State Short 2 x A Purchase 1 x B Purchase 1 x C Portfolio B + C Total net return R 2*(6) = €12 - €10 - €2 - €12 €0 S 2*(-10) = -€20 €16 €6 €22 €2 B 2*(-20) = -€40 €34 €6 €40 €0 With the arbitrage strategy you never suffer a loss but you have the chance to earn a profit in the stable state. Your expected return, without initial investment and without the risk of a loss is the €2 * 0,5 = €1. Gedownload door: hannahducatteeuw | [email protected] Wil47 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen If the investors realize the arbitrage opportunity they will all start short selling A and purchasing B and C. As a result the price of A will fall and the prices of B and C will increase. At the same time, the return for A will rise and the returns of B and C will fall. This will go on until the arbitrage opportunity has disappeared. Does every investor have to exploit the mispricing? No, not every single investor needs to be aware of the disequilibrium. It is only important that the number of market participants that know about it are able and willing to exploit it, is sufficiently large. Otherwise the prices will not be affected. Are these examples likely to happen in reality? EX 1: No because it is hard to find such a pair of banks EX 2: in reality this arbitrage does not work as suggested for two reasons. o On the exchange market there is a bid-ask spread for currencies that are traded. This means that the dealer will face transaction costs in terms of paying the spread. This spread is likely to make the deal unprofitable. o Prices on the foreign exchange are permanent. The dealer faces a huge risk that the prices will have changed before he could perform the entire process. He cannot perform all the deals simultaneously. 9.2. The arbitrage pricing theory APT does not need all the strong assumptions that have been made by CAPM. We do not longer need to assume that the investors are mean-variance optimizers and/or use quadratic utility functions. ASSUMPTIONS FOR APT: 1. Perfect competition: Large number of investors, each with a small wealth compared to the market and therefore being unable to affect prices individually  price takers. 2. No imperfections as transaction costs, taxes,.. 3. Homogeneous expectations about future returns, risks and covariances. 4. The return distribution is characterized by an index model (or more general by a factor model) 5. The number of assets is sufficiently high to create portfolios with no idiosyncratic (unsystematic) risk and any desired values for the beta. 6. Short sales are possible. An arbitrage portfolio Has three key properties: » It required no investment » It has no systematic risk » It is well diversified so it has no unsystematic risk either APT is based on an arbitrage portfolio. The portfolio’s excess return was: RP = P + PRi + eP Gedownload door: hannahducatteeuw | [email protected] Wil48 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen  With: P = I wi.P and P = i wi.i The expected return and risk were: E[RP] = P + PE[Ri] and P2 = P2i2 + (eP)2 For a large portfolio we can assume that (eP)2 = 0. Because if you have a large portfolio with a lot of securities it will probably be well diversified and you don’t have unsystematic risk. Now because APT is based on an arbitrage portfolio and an arbitrage portfolio means that you have no risk and no initial investment: i wii =0 and i wi = 0. What about the excess returns of the arbitrage portfolio and what if there is a risk-free asset? The excess return of any arbitrage portfolio must be zero in equilibrium, otherwise the arbitrage opportunity exists. If you have an excess return your return is higher than the average market return and this is not possible if the market is in equilibrium and there are no arbitrage opportunities. If there is a risk-free asset, the return must equal the return of the risk-free asset. An arbitrage portfolio is a riskless investment so it replicates the risk-free asset and therefore it cannot sell at a different price due to the law of one price. A linear risk-return relation The model again leads to a linear risk-return relation for security i: E[ri] = E[rZ] + (E[ri] – E[rZ])i rz = return of a zero-beta portfolio. This is the risk-free asset if there exists one. E[ri] – E[rz] = the risk premium So the expected return of a security i is determined by the expected return of a zero-beta portfolio. This is a portfolio without systematic risk. What can we do with the model? We can use the model for PRICING. In pricing you will not restrict yourself to only one factor. Identifying the relevant factors for a particular security is the job of the investor or analyst. He has to identify the following three essential things: 1. The set of factors affecting a particular security 2. The expected returns for each of these factors 3. The sensitivity or betas of the stock to each of these factors E XAMPLE : USE THE MODEL FOR PRICING ASSETS Portfolio A B C E[Ri] 8% 13% Beta βi 1 2 3 Gedownload door: hannahducatteeuw | [email protected] Wil49 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Unsystematic risk σ(ei)² 0 0 0 We can simply read what the return of portfolio C is because they should all lie on the regression line. The expected returns the assets are determined by their beta. We know that the expected return of the asset with a β of 3 equals 18! So imagine you have 3 well-diversified portfolios A, B and C. they are well-diversified in the sense that their idiosyncratic risk is zero. Therefore it is possible to characterize each of the portfolios by just their expected return and the factor exposures βi,1 and βi,2. Every triple of points in a three-dimensional space spans a plane. All portfolios that we can construct as a linear combination of the three portfolios A, B and C lie on this plane. The plane span by A, B and C is the grey region. Image you also have a portfolio D, but in relation to its risk exposures βD,1 and βD,2 the expected returns are too high. This point will then lie above the plane defined by A, B and C. How can APT help us in practise? It allows us to price an asset or portfolio and helps to build a synthetic portfolio that replicates the risk structure of another asset of portfolio. It helps asset managers who seek to track an index. You can say that the best way to track an index is to simply buy all the stocks included in the index. However, for many reasons, asset managers might not want to do just that. What they can do instead is replicate the benchmark’s risk structure by a number of securities that exceeds the number of factors by 1. It is also useful in active portfolio management to identify under- or overvalued securities. What is the main advantage and disadvantage of the APT? The main advantage is at the same time the main disadvantage. The model is very general. While it states that expected returns can be described by a factor model, it does not say anything about the type or the number of factors. Gedownload door: hannahducatteeuw | [email protected] Wil50 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Another problem is that the number, the expected returns and the sensitivity towards the relevant factors are likely to change over time. Therefore Ross and others identified the following macroeconomic factors which they felt played a significant role in explaining the return on the stock: » Inflation » Output (say GNP or GDP) » Investor confidence » Shifts in the yield curve 9.3. CAPM and ATP CAPM APT Equilibrium model Yes Yes Condition of market equilibrium Supply = demand No arbitrage possibility Perfect competition Yes Yes No market imperfections Yes Yes Homogenous expectations Yes Yes Investors are mean-variance optimizers Yes No Quadratic utility function or normal Yes No distribution of returns Number of assets traded No statement High Factor model Yes (result) Yes (assumption) Investors are risk averse Yes Not necessarily Short sales No Yes The CAPM is rooted and based on the Markowitz portfolio theory, while the APT can be seen as a model based on factor models. CAPM can be seen as a special case of the APT, with additional and more restrictive assumptions. APT is a more realistic model than CAPM because it allows the investors to have individual preferences, occasional mispricing, speculators and a financial industry that deals with information and information processing. APT can be seen as a supply-side model. In contrast, the CAPM can be seen as a demand-side model. The central point CAPM uses generally more restrictive assumptions than APT but the results are similar: there is a linear relation between risk and return. Even the use of multi-factor index models as a basis is not in contradiction with the CAPM: While the CAPM states that the relevant risk factor is the market portfolio, the APT doesn‘t tell exactly what the risk factors are. Practical applications of CAPM use a broad market index as a single factor, which is a simplification of the original model relying on the market portfolio. An index can at best serve as a proxy for the market portfolio. Now it is possible that APT just gives a better approximation of the ‘market’ than a single index as used in the empirical applications of CAPM. But CAPM does NOT assume that you cannot use more than one risk factor, as long as they are related to the market risk! Gedownload door: hannahducatteeuw | [email protected] Wil51 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Chapter 10: The efficient market hypothesis 10.1. Origins of the efficient market hypothesis GIBSON: “Shares become publicly known in an open market, the value which they acquire may be regarded as the judgment of the best intelligence concerning them.” BACHELIER: price changes are a fair game and follow a stochastic process. KENDALL: a series of British share prices “looks like a wandering one, almost as of one week a Demon of Change drew a random number and added it to the current price to determine next week’s price.” FAMA picked up these results and incorporated them into a theory. He formulated the concept as: “I take the market efficiency hypothesis to be the simple statement that security prices fully (add: and properly) reflect all available information.” Fama’s definition means that one cannot gain profits by exploiting new information since all information is already incorporated in the security’s price. What does this mean? » Price formation means a form of information processing. If the market is efficient, this information processing works well, which means that all the relevant information has been transformed correctly into prices. Fama’s definition raises four questions : - What is the set of available information? - What does fully (or correctly) reflected mean? - Do efficient markets make portfolio analysis irrelevant? - Are markets efficient in reality? What is the set of available information? Fama distinguished several degrees of informational efficiency, depending on the particular subset of information incorporated in the securities’ prices. WEAK FORM EFFICIENCY: prices reflect all information from past prices. Past prices are the most obvious and widely available set of information. Because the information about past prices is already incorporated in the price of the security, analysis of past prices does not generate any profits.  This implies that the use of technical analysis would be useless and unprofitable. Weak form efficiency means that the price process should follow a random walk. Why? Because prices are based on news and news occurs randomly. It is by definition unpredictable. If news would not occur randomly, it would not be news but known information. The process has ‘no memory’. Since prices follow a random walk, past prices do not display information. A random walk A time series where subsequent price changes represent random deviations from the past ones. Gedownload door: hannahducatteeuw | [email protected] Wil52 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen SEMI-STRONG FORM EFFICIENCY: information in past prices + all publicly available information is incorporated into the prices. Publicly available information includes: trading volumes, fundamental news, earnings announcements, sales figures, management news, interest rate changes, policy decision,... STRONG FORM EFFICIENCY: no information can generate extra profits. Don’t worry about information, it won’t help you. Not even private information can generate profits because it is already incorporated (but using it would be illegal anyway because that would be insider trading). What does fully (or correctly) reflected mean? EMPIRICAL PROBLEM If we want to see whether prices are correct, we need to know the correct price. To know the correct price, we usually apply a valuation model. Now if we find a deviation of the observed price from the price predicted by the model, we can conclude that the asset is not correctly priced by the market. However, an alternative explanation can be that the used valuation model is not correct. Or even both at the same time. It may even be that the asset isn’t priced correctly but there is a model in line with the mispriced asset. THEORETICAL PROBLEM There exists an information paradox, formulated by Grossman and Stiglitz. If markets are efficient, an analysis of the data does not provide any extra profit, because all the information is already incorporated in the prices. But how does information get priced? By people who do research and interpret these information. However, that’s costly. So, why should people still do this if they have no incentive (there are no profit opportunities).  Thus, no profit, no research, no information processing, no efficiency. This information paradox was a form of criticism on the EMH and it lead to a weaker formulation of the EMH by Fama himself: “Prices reflect information (add: properly) to the point where the marginal benefits of acting on information (the profits to be made) do not exceed the marginal costs.” Gedownload door: hannahducatteeuw | [email protected] Wil53 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen As a result the EMH no longer holds an absolute concept. Both extremes – the absolutely efficient market where no information is of any use, as well as the completely inefficient market, where every piece of information is useful and provides profits – are probably irrelevant in practice. Do efficient markets make portfolio analysis irrelevant? Given that efficient markets do not allow to lake a profit from the analysis of securities, active portfolio management will not add value. Since there already is accounted for costs and risk. However there is still a benefit from passive diversification. How an investor diversifies risk depends on his risk tolerance, tax considerations, liquidity needs and institutional restrictions. All these considerations require portfolio management. 10.2. Event studies Event study An analysis that estimates the impact of a particular type of event on the prices of assets. ‘Event’ can mean a wide range of things: firm-specific events, sector-based events or market-wide events. It may be economical, political or legal events. The studies may be applied to equity or any other financial prices, such as bonds, exchange rates, gold etc. When testing for semi-strong form efficiency event studies have gained popularity. BUT event studies cannot prove the EMH. This is because they tell us whether there is a reaction on events but they don’t tell us if this reaction is the fundamentally justified one. B ASIC STEPS FOR AN EVENT STUDY 1. Define the events that have to be analysed and collect a number of events, i.e. a list of stocks and dates. 2. Define the event window and collect stock price changes for those firms in periods around the event that we want to study, collect also changes in a market-wide index in the same periods (e.g., collecting this data from databases such as data stream), 3. Evaluate whether event-period price changes for the list of firms are abnormally large, compared to normal or expected returns for those firms and control for market-wide effects on all firms' returns during the event periods. 4. Optionally: run additional regressions to explain the abnormal returns in (3). Event window The period around the event we wish to examine. This allow us to capture over- or underreactions following the event that we are studying. So event studies are going to look if the return differs from what was expected (the ‘normal’ return).  They analyse the abnormal return = actual (realized) return – normal (expected) return. BUT how should one define the normal return? STATISTICAL MODELS Gedownload door: hannahducatteeuw | [email protected] Wil54 jij €76 per Dit document is auteursrechtelijk beschermd, het verspreiden van dit document is strafbaar. maand verdienen?  Stuvia - Koop en Verkoop de Beste Samenvattingen Take the average of past returns = constant mean return model.  this is a poor approach because why should the return always be the same? Correct expected returns for movements of the whole market = market model.  This is better but still problematic. What if the whole market is affected by the event? What if the stocks under considerations show in general a stronger reaction than the market? ECONOMIC MODELS CAPM or APT or other index models.  this is theoretically the best model, but again the joint hypotheses-problem occurs…. Joint hypothesis means that markets are efficient and the underlying models correct. But we have seen that there was an empirical problem and we aren’t sure of these joint hypothesis. E STIMATING THE NORMAL / EXPECTED RETURN If you use the index model, you can find the expected return of a stock i on day t as: Ri,t = i + i.Rm,t + et It is however important that you estimate the index model for a time period that does NOT include the event window or similar events! Otherwise the data may be contaminated and then it isn’t a good reflection anymore of the ‘normal’ return. E STIMATING THE ABNORMAL RETURN The abnormal return is nothing else than the difference between the actual and the expected return. ARi,t = Ri,t – (αi + βi.RM,t) Furthermore, we know that the distribution of the error term (and therefore the abnormal return) is: ARi,t N(0, var(ARi,t)) With : σ²(ARi,t) = σε² + {1 + (RM,t – μM)/σM²} / T Variance of the sample abnormal return Sampling error in the estimation of the population parameters Thus the variance of the sample abnormal return equals the variance of the error term plus an additional component due to the sampling error in estimating αi and βi. Since ARi,t only measures the abnormal return at observation t, we will in practise prefer the cumulative abnormal return CARi,t,τ over the period [t, t+τ]. The CAR is simply the sum over all AR but]. The CAR is simply the sum over all AR but it measures the impact of an event over a period of time and not at one specific moment. Similarly as for ATi,t we get for CARi,t,τ]. The CAR is simply the sum over all AR but : CARi,t,τ]. The CAR is simply the sum over all AR but = ∑j = t, …, t+τ]. The CAR is simply the sum over all AR but ARi,t The sample distribution is: CARi,t,τ]. The CAR is simply the sum over all AR but N(0, var(CARi,t,τ]. The CAR is simply the sum