Average Return and Relevant Volatility 25 Consequently, investors will all hold portfolios that contain only market volatility; we can ignore the ei component of a stock’s volatility. Each asset contributes to the volatility of the portfolio return in proportion to its own bi and this is the only part of the asset’s volatility that is relevant. If we relate the average return on assets and asset classes to their relevant volatility, we find a reasonably good fit. For asset classes which are already diversified, the relevant volatility is simply their own return volatility. For individual assets, the relevant volatility is that part of their volatility that cannot be diversified, that is their bi times the volatility of the market portfolio. An asset’s bi is called its beta (written b) and its average return in a market where assets are properly priced will be proportional to its beta risk. If we plot the betas of single assets against their average returns, they should form a straight line The model that describes this relationship is called the Capital Asset Pricing Model – the CAPM.

Volatility and Correlations • We have already seen that we can easily estimate the volatility and correlation using Excel functions STDEV and CORREL. The variance is defined as the square of the volatility (or standard deviation). • Similar to the case of the returns, it is conventional to express the volatility in an annual basis. • Annual Volatility = sqrt(12) [Monthly Volatility]. • Annual Volatility = sqrt(260)[Daily Volatility]. • For example, a daily volatility of 1% implies an annual volatility of about 16%. • Recently, we have been observing daily fluctuations of about 1.5% - what does that imply about the annual volatility? 34

A bad argument for hedging Since investors don’t like volatility, one argument that is often offered for hedging at the firm level is simply to reduce firm return volatility. If investors prefer lower volatility, shouldn’t it be a good thing if firms also try to reduce their cashflow volatility? The answer is twofold: One, some firm volatility is diversifiable, and will ultimately not show up in investors’ portfolio return volatility. Most firm level volatility that firm managers can control is of this nature. However, investors and portfolio managers can diversify idiosyncratic risk better and cheaper than firm managers. Two, except for extremely large firms (like GM in its heyday, perhaps), firms cannot affect economy-level non-diversifiable volatility in any appreciable manner. Firm manager focus should be on the primary activities of the firm, on increasing average returns on assets rather than reducing volatility per se. So if portfolio management can better manage investor return volatility, is there, then, a role for volatility reduction at the firm level?

Some Conclusions (6/6) • 2. As the correlation decreases, the more we can reduce the portfolio volatility. However, it takes more assets to bring down the portfolio volatility to its theoretical minimum. • Example: if the correlation is 0.9 and the average volatility of each stock in the portfolio is 40%, then the lowest portfolio volatility that is possible is about 37.95%. We can reach within 0.5% of this minimum volatility by creating a portfolio of only 4 assets. Suppose instead that the average correlation is 0.5. Then the lowest possible portfolio volatility is 28.28%; however, to reach within 0.5% of this value, we need as many as 30 stocks. 34

Some Conclusions (2/2) • 2. As the correlation decreases, the more we can reduce the portfolio volatility. However, it takes more assets to bring down the portfolio volatility to its theoretical minimum. – Suppose the average correlation is 0.9, and the average volatility of each stock in the portfolio is 40%, then the lowest portfolio volatility that is possible is about 37.95%. We can reach within 0.5% of this minimum volatility by creating a portfolio of only 4 assets. – Suppose instead that the average correlation is 0.5. Then the lowest possible portfolio volatility is 28.28%; However, to reach within 0.5% of this value, we need as many as 30 stocks.

The M Square [2/3] • The M square measures the difference in the return of the portfolio P and the benchmark M, when portfolio P is mixed with a riskfree asset to make the volatility of portfolio (P + riskfree) the same as the volatility of the benchmark. • The M2 answers the following question: if the investor wants the same volatility as the benchmark, then how much worse or better would the investor do by investing in the actively managed portfolio? • Recall that portfolio P has a volatility of 42% and benchmark’s volatility is 30%. • We create a portfolio of w=0.714 in P and 0.286 in the riskfree asset. This portfolio now has a volatility of (0.714) (42)=30%.

Classification of complexity Complexity Θ(1) Θ(log n) Θ(log n)c Θ(n) Θ(nc) Θ(bn) (b>1) Θ(n!) Terminology Constant complexity Logarithmic complexity Poly-logarithmic complexity Linear complexity Polynomial complexity Exponential complexity Factorial complexity We also use such terms when Θ is replaced by O (big-O)

Using a One-Dimensional Array Lesson A Objectives After completing this lesson, you will be able to: Declare and initialize a one-dimensional array Assign data to a one-dimensional array Display the contents of a one-dimensional array Access an element in a one-dimensional array Search a one-dimensional array Compute the average of a one-dimensional array’s contents Find the highest entry in a one-dimensional array Update the contents of a one-dimensional array Sort a one-dimensional array 2

COLT Conclusions The PAC framework provides a reasonable model for theoretically analyzing the effectiveness of learning algorithms. The sample complexity for any consistent learner using the hypothesis space, H, can be determined from a measure of H’s expressiveness (|H|, VC(H)) If the sample complexity is tractable, then the computational complexity of finding a consistent hypothesis governs the complexity of the problem. Sample complexity bounds given here are far from being tight, but separate learnable classes from non-learnable classes (and show what’s important). Computational complexity results exhibit cases where information theoretic learning is feasible, but finding good hypothesis is intractable. The theoretical framework allows for a concrete analysis of the complexity of learning as a function of various assumptions (e.g., relevant variables) COLT CS446 -SPRING ‘17

Query volatility • Query results volatility: spam-prone queries are likely to produce semantically incoherent results over time • Query impressions volatility: buzzy queries are less likely to be spam-prone • Query clicks volatility: click-through densities on different search results positions are more consistent for less spam-prone queries • Query sessions volatility: users are less likely to be satisfied with search results and click on them for spam-prone queries

Traditional volatility modeling Mainstream ARCH, GARCH, and similar models focus on fundamental/informational volatility. Statistically: volatility in the unit-root component of prices. Economically important for portfolio allocation, derivatives valuation and hedging. Quote volatility is non-informational Statistically: short-term, stationary, transient volatility Economically important for trading and market making. 10

Portfolio of Risky + Riskless Asset • To calculate the portfolio return and portfolio variance when we combine the risky asset and riskless asset, we can use the usual formulas, noting that the volatility of the riskfree rate is zero. • Portfolio Return = w1 Rf + w2 Rp. • Portfolio Variance = (w1)^2 (0) + (w2 )^2 (vol of risky asset)^2 + 2 (correlation) (w1 )(w2 ) (0)(vol of risky asset). • Portfolio Volatility = w2 *(vol of risky asset). • This simplification in the formula for the portfolio volatility occurs because the vol of the riskfree asset is zero. • To understand the tradeoff between risk and return, we can graph the portfolio return vs the the portfolio volatility. • The following graph shows this graph for the case when the mean return for the riskfree asset is 5%, the mean return for the risky asset is 12%, and the volatility of the risky asset is 15%. 6

Portfolio of Risky + Riskless Asset • To calculate the portfolio return and portfolio variance when we combine the risky asset and riskless asset, we can use the usual formulas, noting that the volatility of the riskfree rate is zero. • Portfolio Return = w1 Rf + w2 Rp. • Portfolio Variance = (w1)^2 (0) + (w2 )^2 (vol of risky asset)^2 + 2 (correlation) (w1 )(w2 ) (0)(vol of risky asset). • Portfolio Volatility = w2 * (vol of risky asset). • This simplification in the formula for the portfolio volatility occurs because the vol of the riskfree asset is zero. • To understand the tradeoff between risk and return, we can graph the portfolio return vs the the portfolio volatility. • The following graph shows this graph for the case when the mean return for the riskfree asset is 5%, the mean return for the risky asset is 12%, and the volatility of the risky asset is 15%.