Abstract
This chapter is a continuation and extension of modern portfolio theory presented in Chapter 3, with an emphasis on risk measures and risk management of a portfolio. It introduces the capital asset pricing model (CAPM), linear factor models, and several approaches to portfolio risk measures such as value-at-risk, conditional value-at-risk and the concept of coherent risk measures, as well as a variety of portfolio evaluation techniques such as the alpha and beta, the Sharpe ratio, the Sortino ratio and maximum drawdown. The introduction to factor models is brief and from intuitive perspective.
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Notes
- 1.
For example, REITs, which is an acronym for Real Estate Investment Trusts.
- 2.
The return can be applied either in a simple or compounded context.
- 3.
Pronounced “CAP-M.”
- 4.
Harry Markowitz, Merton Miller, and William F. Sharpe shared the 1990 Nobel Prize in Economic Sciences. Sharpe won for his contributions to the Capital Asset Pricing Model. See the Press Release at Novelprize.org.
- 5.
See Section 3.1
- 6.
Under normal circumstances, inflation constitutes a major portion of the risk-free rate. The only problem with this is when the inflation is far above the risk-free rate due to a central bank intervention.
- 7.
A risk-free security is, of course, a theoretical concept. In reality, any investment carries a certain amount of risk. In this context, by risk-free securities we mean US T-bonds or FDIC insured bank accounts to which we can lend out our money and obtain a sufficient credit line from which we can borrow money. Under our assumption in the last section, the interest rate for lending is equal to that for borrowing and occurs at the risk-free rate.
- 8.
At this stage, N is arbitrary. Eventually we need to consider only a sufficiently large N for which we have a diversified portfolio.
- 9.
Recall that the feasible set for N = 2 is a curve, while for N ≥ 3, it is a region (see page 126).
- 10.
The proof to be provided here is for a one-period portfolio. The proof for n-period self-rebalancing portfolios is similar. The details are left as an exercise for the reader.
- 11.
If \(\mathbb{E}(X) = \mathbb{E}(Y )\), then \(X = Y +\varepsilon\), where \(\varepsilon\) is a random variable with \(\mathbb{E}(\varepsilon ) = 0\), called an error term. Typically, an error term \(\varepsilon\) is assumed to be normal (with mean zero).
- 12.
Recall that US Treasuries can be classified into bills, notes, and bonds according to their initial maturities (in years) in terms of time intervals: (0, 1], (1, 10], and \((10,\infty )\), respectively. We consider only T-bills here since, the longer the maturities, the bigger the risk of inflation, and consequently, the less reliable.
- 13.
Named after William Sharpe.
- 14.
Two basic ways of achieving leverage are (a) to borrow money for investment and (b) to use financial instruments such as futures and options (see Chapter 7).
- 15.
The quantity r 0 was originally known as the minimum acceptable return (or hurdle rate) and is often taken to be \(\mathsf{r}\).
- 16.
A distortion risk measure is a type of risk measure related to the cumulative distribution function of a financial portfolio return. CVaR (to be introduced shortly) is an example of a distortion risk measures.
- 17.
One of those versions is given as follows:
Let X be (random) returns of a portfolio in P/L form, the VaR of X at confidence level (1 − p)100% is defined by
$$\displaystyle{\mathop{\mathrm{VaR}}\nolimits _{p}(X) = -\min \{x\vert P(X \leq x) \geq p\}.}$$That is, the VaR at tail probability p is the negative of the lower p-quantile of the return distribution.
- 18.
Mark-to-market accounting is an accounting process by which the price of an asset held in an account is valued each day to reflect the daily closing price of the asset.
- 19.
Annualizing returns with compounding would make these factor models almost useless because of the linearity of the model. As we pointed out in Remark 4.1, using the logarithmic return in factor models can avoid this shortcoming and take advantage of time-additivity and statistical tools in studying properties of individual securities.
- 20.
Error terms are usually assumed to be normal with mean zero.
- 21.
Observable variables, as a statistical term, are those that can be directly measured.
- 22.
Latent variables, as opposed to observable variables in statistics, are those inferred through mathematical models and cannot be directly observed.
- 23.
A system of linear equations is called overdetermined if there are more equations than unknowns.
- 24.
The solution to this system can only be minimum point(s) since L does not have maximum point.
- 25.
The best line to fit the data is in the sense of the least Euclidean distance.
- 26.
For example, if the data space is ℝ 2, and the unknown space is ℝ 3, say \(y =\alpha +\beta x +\gamma x^{2}\), then the best fit graph is a curve in the data space. If both the data space and unknown space are ℝ 3, say \(y =\alpha +\beta x^{2} +\gamma z^{2}\), then the best fit graph is a surface in the data space.
- 27.
We assume that error terms are normal with mean zero.
- 28.
A balance sheet gives a snapshot of the financial position of a company at a given time. Most accounting balance sheets consist of two sides: the left side indicates ASSETS (things that have value), and the right indicates LIABILITIES (things that are owed to third parties) and STOCKHOLDERS’ EQUITY (which is the value of the remaining assets if the company were to go out of business immediately). For an oversimplified example, consider equity in your home = what you paid - what you owe (loan remaining). It is called balance sheet because it has to balance between both sides: Assets = Liabilities + Stockholder’s Equity.
- 29.
- 30.
Liquidity risk is the risk that an investor cannot execute a buy/sell order in the market due to the lack of anticipated/reasonable bid/ask spread or sufficient volume.
- 31.
Regulatory changes or governmental policy changes may have significant impact on asset values. Such risk can be either systematic risk or market risk.
- 32.
Such risk is often associated with unexpected and unfavorable volatility.
- 33.
Counterparty here means the other party in a financial transaction.
- 34.
SPDR (Spiders) is a short form of Standard & Poor’s depositary receipt, an exchange-traded fund (ETF) that tracks the Standard & Poor’s 500 Index (S&P 500). Each share of SPY contains one-tenth of the S&P index and trades at approximately one-tenth of the dollar value of the S&P 500. Thus, the rate of daily returns of SPY and S&P 500 index are basically the same.
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Petters, A.O., Dong, X. (2016). Capital Market Theory and Portfolio Risk Measures. In: An Introduction to Mathematical Finance with Applications. Springer Undergraduate Texts in Mathematics and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3783-7_4
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