A NOTE ON THE LEVERAGE EFFECT ON PORTFOLIO PERFORMANCE MEASURES. James S. Ang*

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1 JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS September 1978 A NOTE ON THE LEVERAGE EFFECT ON PORTFOLIO PERFORMANCE MEASURES James S. Ang* I. In a recent article, Modigliani and Pogue [2] raised the issue of "leverage bias" in portfolio performance measures. Specifically, they contended that the value of the Jensen's alpha (a) could be affected by borrowing or lending at the risk-free rate, while the Treynor index (TI) does not suffer from this shortcoming. They illustrated this effect through the use of a graphical example similar to the one in Exhibit I where A and B are two unlevered portfolios with the same a's but different TI's, Modigliani and Pogue argued that by leveraging, i.e., borrowing at R, the portfolio with the greater slope (TI), A, could attain a levered portfolio A which clearly dominates portfolio B. In other L words, the line with the higher TI will dominate the line with a lower TI regardless of a values. This seems to imply that, in general, TI is a better measure of ex post portfolio performance, and that ranking based on TI's is consistent and invariant to the leverage effect, while ranking based on a's is not. * Oklahoma State University, The Jensen's alpha [1] and the Treynor index [4], two portfolio measures derived from the capital asset pricing model, were originally developed to rank the past performance of managed portfolios and are defined as follows: a = (R. if - R.) - S. 1 (R m - R^) f TI = (R. - R^)/6. where R, R, R are, respectively, the average returns of the portfolio i, i m f market portfolio m, and the risk-free instrument, and 3. is the beta value of the portfolio i. Thus, a calculates the excess return after adjusting for risk at the market return-risk tradeoff, (R^ - Rf)/(6^ = 1.0), and TI calculates the distance between the point on the average return axis drawn from the line connecting portfolio i and parallel to the market line and R, while TI measures the slope of the line connecting portfolio i and R (see Exhibit I). 567

2 The purposes of this note are: (1) to demonstrate that when the leverage effect is properly adjusted for both portfolios, it does not affect portfolio ranking based on the a values (Section II); and (2) to clarify some ambiguities concerning the leverage effect on ranking portfolios (Section III). II. The issue at hand is whether portfolio ranking based on the Jensen a can be affected by the leverage effect. In particular, will the ranking of portfolios before the introduction of leverage, e.g., a = a ; a, a > a > ct etc., be the same after the introduction of an equal proportion of leverage into each portfolio, e.g., a. _ = ct ; a a > o'^ y ^ ^A T "^ere the a's and a 's J.,L ^,L 1.,Li Z,1J J, LI ifl L represent, respectively, the alpha values of the unlevered and the levered portfolios? First, let us consider the case where the a's of the unlevered portfolios are initially the same (a_ = a_), but the TI's are not (TI > TI ); this is A B A B 2 Modigliani and Pogue's example in Exhibit I. Since leverage was introduced to portfolio A, it would seem likely that leverage could also be introduced to portfolio B. Table I shows the values of a's for portfolios A and B if both 3 are allowed the same degree of leverage. The result indicates that, although leverage may change the absolute value of the a's, when leverage is properly accounted for, it will not change their relative values if these two portfolios had the same a's initially. Note that this result is true, in spite of the 4 fact that TI > TI. Thus, two portfolios with similar a's may have different TI's; the ranking based on a's would still be preserved if the leverage effect is properly adjusted. Again, following the same process, it is possible to show that in the case where a > a, and TI >^ TI, the ranking a > a will n D n D 1,L^,L 2 Another possibility is when TI = TI and a = a ; however, this would t\ o I\ a occur only when both portfolios A and B lie on the market line. In this case, the leverage effect is irrelevant since no amount of leverage could change the value of a, which is zero. The optimum amount of leverage for portfolio A and B is a moot point since- portfolio B has a higher return ard at the same time a higher risk than portfolio A. 4 It is possible to derive an analytical proof of this result by observing the relationships among the four triangles formed by R, the market line, and points A, B, A, and B, respectively. L L 568

3 EXHIBIT 1 The Leverage Effect on the Jensen's [Alpha) and Treynor's fti) Measures of Portfolio Performance B At The Market Line : a = 0 Tl = Slope = = = 0.05 At Point A: a TL A ** At Point B: a = =

4 still be preserved as long as equal proportional leverage is allowed. case, however, both the absolute and the relative a's will increase. In this III. Finally, we would like to offer some observations on the implications of the leverage effect and to help clarify some of its ambiguities. 1. The contribution of Modigliani and Pogue is to correctly identify two sources of risk-adjusted excess return. The first is the portfolio manager's ability to select underpriced securities, i.e., the a's of the unlevered portfolios, and the second is the leverage effect, i.e., the a's of the levered portfolio minus the a of the unlevered portfolio. 2. The second type of excess return, that due to the leverage effect, exists if and only if the first type, that due to selectivity, exists. As usual, leverage is a two-edged sword; it would amplify both excess positive and excess negative returns. 3. Since both a and TI indexes are used to measure the past performance of portfolios, homemade leverage is not available. In fact, it is physically impossible since the investor would not be able to tell which portfolio to lever unless he also had prior knowledge of which portfolio would have the highest TI's. To do so is tantamount to a situation where an individual, with borrowed money, attempts to bet on the announced winner of the previous horse race I 4. Given observations 2 and 3 and if the investor accepts the market return-risk tradeoff, he would still be better off with a higher a portfolio than with one with a lower a, regardless of the values of TI's. An instance where the Treynor index could give an inconsistent ranking is when the calculated TI is negative (R. < R ). Let us take two portfolios with the same TI values, i.e., that lie on the same negatively sloping TI line. As long as these two portfolios do not have identical average returns and betas, we would have one portfolio that has a lower return and, at the same time, a higher risk than the other. The a criteria would correctly rank the portfolio with a greater negative a as the less desirable one, but the TI criteria would not differentiate between the two. 5. Previous empirical studies of portfolio performance based on Jensen's alpha are still valid for the following reasons. First, most managed portfolios are either not able, e.g., mutual funds, or less willing, e.g., closed-end funds, to use leverage. Thus, the leverage effect is negligible. Secondly, even in the case when leverage is significant in the portfolio, there are two recourses. The first is to acknowledge the excess return due to leverage as 570

5 part of superior (or inferior) portfolio performance since, as pointed out in (2), leverage effect could not have occurred without positive (or negative) excess return. Secondly, if desired, the a's of the levered and unlevered portfolios could also be calculated separately. This could easily be done since both portfolios must lie on the same Tl line and the beta value of the unlevered portfolio, & by 3^ = 6 (1 + ), where I is the ratio of borrowed LI LI U fund to equity. 6. Although the leverage effect is of limited usefulness in ranking expost portfolio performance, it may have some significance in the ex-ante application of the ranking criteria, e.g., portfolio selection. An example is in the case of evaluating capital projects. The implications of the leverage effect are that for a project earning a positive net present value, i.e., unlevered a>0, further increase in the net present value due to leverage could be obtained (levered a > unlevered a) even when corporate tax is absent. The constraint is the borrowing cost that is expected to be positively related to increase in solvency risk as leverage increases. TABLE 1 THE EFFECT OF PROPORTIONAL CHANGE IN LEVERAGE RATIOS ON THE ALPHA VALUES Leverage Ratios Portfolios 0 1/3 1/2 2/3 1.0 A B REFERENCES [1] Jensen, Michael C. "The Performance of Mutual Funds in the Period " Journal of Finance (May 1968), pp [2] Modigliani, Franco, and Gerald Pogue. "An Introduction to Risk and Return: Concepts and Evidence, Part II." Financial Analyst Journal (May/June 1974), pp [3]. A Study of Investment Performance Fees. Heath-Lexington Books (1974), Chapter II. [4] Treynor, Jack L. "How to Rate the Management of Investment Funds." Harvard Business Review (January/February 1965), pp

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