The Capital Asset Pricing Model (CAPM) represented a significant step in the development of modern finance theory. It rested on Markowitz's concept of risk as the volatility of price, and on the ideas of mean-variance portfolio optimization, but it went a step further: It made a statement about the market, under the assumption that investors optimize their individual holdings. By postulating that the part of stock volatilities that are not correlated with the market portfolio (the specific risks) are mutually sufficiently independent to be diversified away, the model derives its powerful tenet: The risk of a security consists of two parts: the systematic risk and the specific risk. Systematic risk, measured by the covariance of the security price with the market portfolio price, carries a compensation in terms of expected return premium. The specific risk is not compensated for, since it can be reduced by diversification. Quantitatively, this is expressed by the equation for the capital market line,

where is the expected rate of return on the i–th security, is the expected rate of return on the market portfolio, is the regression coefficient of the security price on the market portfolio price, and r is the risk-free rate of return. A review of the model is provided in the paper “The Efficient Market Model” (Chapter 22) originally written in 1972 with John A. McQuown.

An interest in liability funding, such as fixed liabilities in pension plans, generated research in portfolio immunization strategies. Portfolio immunization is a technique of balancing long and short bonds in such a way that the portfolio value at a given horizon date (the date a liability is due) is guaranteed. Typically, this involves maintaining the Macaulay duration of the portfolio to be equal to the remaining horizon length. Such strategies are based on the assumption that interest rates of all maturities move up and down by the same amount (parallel shifts).

The paper “A Risk Minimizing Strategy for Portfolio Immunization” (Chapter 23), co-authored in 1984 with H. Gifford Fong, gives a lower bound on the portfolio value at the horizon date, should the assumption of parallel shifts be violated. The shortfall is bound by a quantity that is the product of two elements: the magnitude of a twist in the yield curve, as measured by the maximum value of the derivative with respect to term of the forward rate change; and a characteristic of the portfolio composition called M², which is a measure of dispersion of the portfolio cashflows around its duration. While the former quantity is outside the control of the bond investor, the latter can be minimized to assure a minimum risk of not meeting the fixed obligation on the horizon date.

This result is exploited in the 1983 paper “The Tradeoff Between Return and Risk in Immunized Portfolios” (Chapter 24), also written with H. G. Fong. It is argued that among portfolios with a fixed duration, some may offer higher return than others. An efficient frontier can be constructed that identifies the portfolio with the maximum expected return for a given level of risk, as measured by . Investors can choose a portfolio on the efficient frontier that represents the desired tradeoff between return and risk.

The paper “Bond Performance: Analyzing Sources of Return” (Chapter 25), written in 1983 with H. G. Fong and C. J. Pearson, identifies the principal sources of the gains or losses in performance measurement. It is argued that the total return on a bond portfolio can be meaningfully attributed to two primary components: the effect of external interest rate development, and the contribution of the management process. The interest rate effect can be further decomposed into that due to the initial yield curve level and shape, and that due to the changes in the forward rates over the performance measurement period. The contribution of the portfolio management consists of the following components: return from management of the portfolio maturity composition; return from the management of the industry and quality sectors; and return due to the selection of the specific issues.

The method proposed in the paper for measurement of these components of performance consists of repricing of the securities in the portfolio. By pricing each bond as if it were a Treasury issue and calculating the return on such portfolio (including the actual changes in the portfolio composition over the period due to purchases, sales, swaps, etc.), the maturity management component is equal to the difference between this return and the external interest rate component. The next step is repricing each bond as if it were exactly in line with its own sector and quality group—that is, with the same average yield spread over Treasuries. The difference between the return on the portfolio with such pricing, and the sum of the external component and the maturity management, is the part of the total return due to the management of sectors and qualities. Finally, the difference between the actual total return and the sum of the previous components is due to the choice and management of the individual bonds.

A desirable property of this methodology is that the timing of any changes in the portfolio composition over the performance measurement period is properly allocated to each component of the management process.

Portfolio insurance is a technique that guarantees a minimum return over a specified horizon period even if the target asset, usually the market portfolio, is down by more than the minimum. This is achieved by a strategy that simulates the performance of a call on the target asset. Buying a call, together with an investment in a fixed-return bond, has the required property of having a guaranteed minimum return (in case the call is not exercised) together with sharing the performance of the target asset (in case the call is worth exercising). Creating a synthetic call, by maintaining the proportion of funds invested in the target asset equal to the hedge ratio of the call, achieves the same result. The cost of the strategy, equal to the cost of buying the call, is realized by a fixed and known return difference between the target asset return and the return on the portfolio.

A generalization of portfolio insurance is the best return strategy. Portfolio insurance can be viewed as a strategy that promises the better of two returns, each less a fixed return differential. The two assets are the target portfolio and a bond maturing on the horizon date. In the best return strategy, the objective is to get the best of several returns on different assets, each less a fixed return differential. The return on the strategy is thus

The costs are subject to a single condition, which corresponds to the pricing of the multiple asset call that the strategy creates synthetically. A detailed description of the strategy and examples of the strategy costs are given in the 1987 paper “The Best-Return Strategy” (Chapter 26).

Changes in interest rates are not the only source of risk in fixed-income investment. A bond portfolio value also depends on changes in interest rate volatility. Nearly all fixed-income instruments contain embedded options— callable bonds, pass-throughs, futures, and so on. Just as fixed-income investors need to know how changes in interest rates affect portfolio value, they need to be also concerned about changes in interest rate volatility.

The 1992 paper “Volatility: Omission Impossible” (Chapter 27) with H. Gifford Fong and D. Yoo presents a term structure model with two stochastic factors: the short rate and the short rate volatility. This model allows for measurement of the exposure to the risk of volatility changes.

The article “A Multidimensional Framework for Risk Analysis” (Chapter 28) was co-authored in 1997 with H. Gifford Fong. It provides a methodology for quantitative analysis of portfolio exposures to different types of risks, and the calculation of VaR measures. These risks can run from market risks (such as changes in stock market index or changes in interest rate levels) to credit or operations risks. As long as a given source of risk can be described by a risk factor, such as the level of interest rates or the quality spread of corporate yields over Treasuries, the sensitivity of the portfolio to changes in the risk source can be measured. By determining the covariance matrix of the different risk sources, value-at-risk can be calculated.

A characteristic of some commodities, foremost of which is electricity, is that they cannot be easily stored. For a storable commodity, the forward price is equal to the current spot price divided by the price of a zero coupon bond with the same maturity as the forward contract. This is because the forward contract to buy a commodity at a future date can be duplicated by issuing a bond maturing on the contract date and buying the commodity now. For electricity, this relationship between the spot and forward prices breaks down.

“Plugging into Electricity” (Chapter 29) was written in 2001 with Helyette Geman. (The article was originally called “Forward and Futures Contracts on Non-Storable Commodities: The Case of Electricity,” but Risk likes to assign cute titles to its feature articles.) The paper investigates the relationship between spot and forward prices, and proposes a methodology for modeling electricity spot and futures prices.

The unpublished 2002 memorandum “Pricing of Energy Derivatives” (Chapter 30) goes further: It provides a general equation that represents a complete specification of the forward/spot process. The spot price behavior is fully described by the current forward curve and by forward contract volatilities, and it only includes processes whose stochastic properties under the martingale measure are known. Therefore, the prices of energy derivatives and contingent claims can be calculated without recourse to the market prices of risk, which are not directly observable.