Var And Downside Risk
This chapter extends the discussion of Chapter 2 regarding value at risk VaR where we had implicitly assumed that the distribution of returns has the same time horizon as the banker or investor. For example, the artificial data example assumed that the time horizon t was only one day. Extending the time horizon to the case where t gt 1 requires a forecast of its volatility. Now that we have discussed GARCH volatility forecasts in Chapter 4 we are ready to relax the assumption of t 1. Recall...
Stable Pareto and ParetoLevy Densities
The stability property of distributions often refers to a shape parameter, which stays the same regardless of the scale. Rachev and Mittnik 2000 is a huge book devoted to a discussion of the properties of distributions include the estimation of stable distributions in finance. We discussed the simple Pareto density in Section 4.4.1. Here we consider its extensions, which are of interest in finance because they have various desirable properties of fat tails, excess kurtosis, and skewness, for...
Empirical cdf and QuantileQuantile QQ Plots
Rather than disregard the normal distribution in all applications, we will give it the benefit of the doubt and let the numbers tell us if an approximation is good. A test of the goodness of fit is usually made for discrete pdf's like the binomial and Poisson by preparing a table of observed Obs and fitted Fit values with say j 1,2, , k rows. The Pearson goodness of fit statistic is simply where we have inserted an additional subscript j for the jth row. We reject the null hypothesis of a good...
Jump Diffusion
The jump diffusion process recognizes the fact that not all stock movements follow a continuous smooth process. Natural disasters, revelation of new information, and other shock can cause a massive, instantaneous revaluation of stock prices. To account for these large shocks, the normal diffusion is augmented with a third term representing these jumps m-1k di sdz dq, 1.4.4 where l is the average number of jumps per unit of time, k is the average proportionate change of the jump the variance of...
Overoptimistic Consensus Forecasts by Analysts
Any company's data on past performance involve many factual series including gross earnings, EBITDA earnings before interest, taxes, depreciation, and amortization , net profits, and its price-to-earnings ratio PIE ratio . Stock market research services, such as Value Line, publish reports on individual companies available at most public libraries. Yahoo finance and other places also provide such information on line. The S amp P 500 stock index is often regarded as a leading indicator of market...
Using CAPM for Capital Investment Decisions
Since CAPM beta is widely reported and commonly known by investors in the stock market, the management of a corporation knows the value of their firm's own beta. It also determines the equity cost of capital to the firm. Hence it is recommended that the management decisions regarding the choice among investment projects should check whether the project yields a rate of return that exceeds the cost of capital. If the cost of debt financing is assumed to be equal to that of equity financing, a...
Nonparametric Value at Risk VaR Calculation from Low Percentiles
Normal distribution N m, o2 is parametric because it depends on estimating m and o2. The data driven distributions represented by the solid lines in Figure 2.1.2 are called nonparametric or empirical because they do not use moments or any other shape parameters. Suppose that a portfolio manager invests K 100 million in AAAYX mutual fund for t 1 month based on the data for 132 months. The one percentile of -8.708737 from the raw data means that she may lose VaR a' 0.01 8,708,737 with a chance of...
Patterns Of Downside Risk
In the preceding section, we argued that downside risk should receive a higher risk premium than upside risk. In this section we will look at the source and magnitude of this difference. The first thing to observe is how close the traditional measures of downside risk are to the standard deviation proxy among our recommended measures of downside risk. There will, of course, be some correlation because traditional measures of risk include the downside as well as the upside. The degree of...
Empirical Checking of Stochastic Dominance Using Matrix Multiplications and
Recall the 4DPs satisfied by the transformation 6.2.2 .That distribution helps us find the weights that map the cumulative probability p e 0,1 for perfectly EUT-compliant investor on the weighted p for less compliant individuals also in the same closed interval 0,1 . In this subsection we discuss a numerical algorithm for checking stochastic dominance of orders 1 through 4. The CDF represents the area under the PDF. A trapezoidal algorithm was suggested in Vinod 1985 for computing areas. Here...
FourthOrder Stochastic Dominance 4SD
The fund A dominates fund B in 4SD that is, A lt 4 B for the class D4 utility functions if in addition to 6.3.3 we have Hi Fab w dwdzdy 0 for all x e x , x . 6.3.4 As with other stochastic dominance checks, the fourth order has an empirical counterpart involving cumulative sums of cumulative sums. It is possible to consider fifth and higher orders of stochastic dominance with conditions on derivatives of U. However, these conditions cannot be justified on economic grounds. Similarly conditions...
Lognormal Distribution
The lognormal distribution is one of the most important distributions in finance. If returns are rt, gross returns are 1 rt . Let us consider log of gross returns, log 1 rt . Note that the Taylor series for log 1 x x - x2 2 x3 3. If we retain only the first term, we have log 1 x x. Hence in our case log 1 rt rt . The lognormal model says that log of gross returns is normally distributed. It has the advantage that it does not violate limited liability Campbell et al. 1997 . The expressions for...
Pearson Type IV Distribution for Our Mutual Fund Data
We can see from the estimates in Table 2.1.2 that the normal distribution is not appropriate. Kurtosis is high, and the returns are negatively skewed. If we plot the observed point in a diagram with Pearson's skewness p1 on the horizontal and p2 on the vertical axis, as in Figure 2.1.1, we note that AAAYX mutual fund falls in the region of Pearson's type IV curve. This is not good news. Type IV is the hardest to work with since it involves imaginary roots of the quadratic in 2.1.6 , so we...