Alex Kane

Measuring risk aversion from excess returns on a stock index

Abstract Measuring risk aversion from excess returns on a stock index presents two obstacles: 1. the time path of the stock-index variance needs to be modeled and estimated, and 2. other components of wealth must be accounted for. We distinguish two measures that relate the risk premium to variance: 1. the measure of risk aversion which, by the single-factor CAPM, would be the slope coefficient in the linear relation between the mean excess return and the variance of the overall risky portfolio of the representative investor, and 2. the slope coefficient in the linear relationship between the mean excess return on a stock index and its variance. Even when risk aversion is constant, the latter can vary significantly with the relative share of stocks in the risky wealth portfolio, and with the beta of unobserved wealth on stocks. We introduce a statistical model with ARCH disturbances and a time-varying parameter in the mean (TVP ARCH-M). The model decomposes the predictable component in stock returns into two parts: the time-varying price of volatility and the time-varying volatility of returns. The relative share of stocks and the beta of the excluded components of wealth on stocks are instrumented by macroeconomic variables. The ratio of corporate profit over national income and the inflation rate are found to be important forces in the dynamics of stock price volatility.

Forecasting Volatility and Option Prices of the S&P 500 Index

Models using implied volatility derived from observed option prices are commonly employed to forecast future option prices. The ARCH model proposed by Engle, however, models the dynamic behavior in volatilitr, forecastinghture volatility using only an asset’s return series. We assess the performance of these two volatility prediction models for SGP 500 index options over the April 1986 to December 199 1 period. We devise trading rules using the two forecast methods ,to trade at-the-money straddles. Straddle trading is used because a n at-the-money straddle is deltaneutral. Each rule prices options according to its method of forecast, buying (selling) a straddle when its price forecast for tomorrow is higher (lower) than today’s market closing price; a t the end of each day the rates of return are computed. W e j n d that the rule using the GARCHforecast method returns a greater pro@ than the rule based on an implied volatility regression model. I n particular, the GARCH rule earns a p r o j t in excess of the assumed transaction cost of

Skewness Preference and Portfolio Choice

0.25 per straddle with near-the-money straddles.

Geometric or Arithmetic Mean: A Reconsideration

One of the virtues of parameter preference models (presented in general form in Rubinstein [23]) is their empirical content. Applied models of financial theory rely heavily on the mean variance (MV) version of parameter preference. As spelled out in Samuelson [25], MV models are adequate with compact distributions of returns and when portfolio decisions are made frequently so that the risk parameter becomes sufficiently small.

Index-Option Pricing with Stochastic Volatility and the Value of Accurate Variance Forecasts

An unbiased forecast of the terminal value of a portfolio requires compounding of its initial value at its arithmetic mean return for the length of the investment period. Compounding at the arithmetic average historical return, however, results in an upwardly biased forecast. This bias does not necessarily disappear even if the sample average return is itself an unbiased estimator of the true mean, the average is computed from a long data series, and returns are generated according to a stable distribution. In contrast, forecasts obtained by compounding at the geometric average will generally be biased downward. The biases are empirically significant. For investment horizons of 40 years, the difference in forecasts of cumulative performance can easily exceed a factor of 2. And the percentage difference in forecasts grows with the investment horizon, as well as with the imprecision in the estimate of the mean return. For typical investment horizons, the proper compounding rate is in between the arithmetic and geometric values. An unbiased forecast of the terminal value of a portfolio requires the initial value to be compounded at the arithmetic mean rate of return for the length of the investment period. An upward bias in forecasted values results, however, if one estimates the mean return with the sample average and uses that average to compound forward. This bias arises because cumulative performance is a nonlinear function of average return and the sample average is necessarily a noisy estimate of the population mean. Surprisingly, the bias does not necessarily disappear asymptotically, even if the sample average is computed from long data series and returns come from a stable distribution with no serial correlation. Instead, the bias depends on the ratio of the length of the historical estimation period to that of the forecast period. Forecasts obtained by compounding at the geometric average will generally be downwardly biased. Therefore, for typical investment horizons, the proper compounding rate is in between the arithmetic and geometric rates. Specifically, unbiased estimates of future portfolio value require that the current value be compounded forward at a weighted average of the two rates. The proper weight on the geometric average equals the ratio of the investment horizon to the sample estimation period. Thus, for short investment horizons, the arithmetic average will be close to the “unbiased compounding rate.” As the horizon approaches the length of the estimation period, however, the weight on the geometric average approaches 1. For even longer horizons, both the geometric and arithmetic average forecasts will be upwardly biased. The implications of these findings are sobering. A consensus is already emerging that the 1926–2002 historical average return on broad market indexes, such as the S&P 500 Index, is probably higher than likely future performance. Our results imply that the best forecasts of compound growth rates for future investments are even lower than the estimates emerging from the research behind this consensus. The choice of compounding rate can have a dramatic impact on forecasts of future portfolio value. Compounding at the arithmetic average return calculated from sample periods of either the most recent 77 or 52 years results in forecasts of future value for a sample of countries that are roughly double the corresponding unbiased forecasts based on the same data periods. Indeed, for reasonable risk and return parameters, at investment horizons of 40 years, the differences in forecasts of total return generally exceed a factor of 2. The percentage differences between unbiased forecasts versus forecasts obtained by compounding arithmetic or geometric average returns increase with the ratio of the investment horizon to the sample estimation period as well as with the imprecision in the estimate of the mean return. For this reason, emerging markets present the greatest forecasting problem. These markets have particularly short historical estimation periods and return histories that are particularly noisy. For these markets, therefore, the biases we analyzed can be especially acute. Even for developed economies, however, with their longer histories, bias can be significant if one disregards data from very early periods.

The Delivery Option on Forward Contracts: A Note

In pricing primary-market options and in making secondary markets, financial intermediaries depend on the quality of forecasts of the variance of the underlying assets. Hence, pricing of options provides the appropriate test of forecasts of asset volatility. NYSE index returns over the period of 1968–1991 suggest that pricing index options of up to 90-days maturity would be more accurate when: (1) using ARCH specifications in place of a moving average of squared returns; (2) using Hull and Whites (1987) adjustment for stochastic variance in the Black and Scholes formula; (3) accounting explicitly for weekends and the slowdown of variance whenever the market is closed. (JEL C22, C53, C10, G11, G12)

Coins: Anatomy of a fad asset

A number of futures contracts conveys to the short position various delivery options regarding the quality and exact timing of delivery. Moreover, the compensation to the long position is not solely determined by the market value of the delivered asset at the time of delivery. Sometimes, the long position can hedge this delivery risk by holding an appropriate portfolio of the underlying asset. It often has been stated that whenever the long position can form a dynamic hedge against the delivery risk, the delivery option has a zero value. This paper demonstrates the implication of such erroneous intuition to the pricing of options. It is shown that the root of the issue is the property of diffusion processes whereas, within a given time interval, a random variable either will never cross a given boundary or else, cross it an infinite number of times.

Trading cost premiums in capital asset returns--a closed form solution

T his study emerged from my acquaintance with Charles J. Lidman, President of New England Rare Coin, who offered to support an academic study of the rates of return on investments in rare coins, one of a group of exotic assets that until recently had caught the fancy of investors and portfolio managers. Indeed, the study uses data from the 1970s, a decade that was generally a profitable period for investors in coins, and that bred many investment enthusiasts. Had I completed and published this study sometime during 1980, few would have paid much attention to yet another analysis recommending that one hold coins in ones portfolio. As the wheels of research were slowly crunching the data, however, the year 1981 wrought disaster on coin investors, while 1982, as of this writing, has brought more of the same. Nevertheless, these events are not inconsistent with results from the 1970s, presented below. While we find that the standard Capital Asset Pricing Model (CAPM) cannot explain the success of coins during the 1970s, neither can it explain their failure in the last two years! We can, however, explain both the boom and the bust in coins by an expanded CAPM that accounts for the effects of inflation on asset returns. We show that coin returns behaved similarly to commodity futures contracts, providing a potent inflation hedge. Thus, the returns of both coins and futures during the 1970s and the 1980s can be explained by our experience with inflation in those periods.

Performance Evaluation of Market Timers: Theory and Evidence

Abstract This paper presents a closed form solution to the Amihud-Mendelson model that incorporates bid-ask spreads into the required rate of return on capital assets. Return premiums are shown to be concave in the cost of trading. Two types of liquid assets are defined; quoted and non-quoted, and the applicability of the model to the two types is discussed.

International interest rates and inflationary expectations

Previous investigators have shown that the Sharpe measure of the performance of a managed portfolio may be flawed when the portfolio manager has market timing ability. Herein we develop the exact conditions under which the Sharpe measure will completely and correctly order market timers according to ability. The derived conditions are necessary, sufficient, and observable. We compare these derived conditions to empirical estimates of actual market conditions and find that, under typical market conditions, the practice of using quarterly portfolio return data will frequently result in a failure of the Sharpe measure to order timers according to ability. We show, however, that such failures can be greatly reduced by more frequent sampling of managed portfolio returns.