Duration Of Equity 2008

July 2008

Equity Duration – Updated Duration of the S&P 500

David M. Blitzer, Ph.D [email protected] 1-212-438-3907 Srikant Dash, CFA, FRM [email protected] 1-212-438-3012

‰

In early 2004, we published a paper describing a simple model of asset allocation for pension plans that incorporates the concept of equity duration. We believe that a diversified portfolio of equities and bonds can be immunized and lowers the risk of deficits.

‰

Akin to the well-known concept of bond duration, equity duration measures the sensitivity of equities to interest rates. Although research on this subject is more recent and the concept is rarely used in practice, we believe equity duration is of significant importance in immunization, risk management, and asset allocation.

‰

We developed a simple model of equity duration that uses the dividend discount model and incorporates the sensitivity of growth to rates. Based on our empirical model, duration (or interest-rate sensitivity) is higher for highgrowth stocks, stocks whose dividend growth is not sensitive to interest rates, and in low–discount rate environments.

‰

Standard & Poor’s publishes, on an annual basis, a current report and a 30year history of duration for the S&P 500. We acknowledge that equity duration estimation is an evolving science. We also believe that a regularly available and updated source of equity duration data will make this important metric more accessible for further research and practitioner use.

‰

We estimate the duration of the S&P 500 index to be 45 years at the middle of 2008. It has risen markedly in recent years, to record levels, suggesting that the market has become much more rate-sensitive.

Philip Murphy, CFA [email protected] 1-212-438-1368

Equity Duration

Equity Duration In our earlier paper, we discussed various approaches to equity duration evaluation and described a rather simple model of asset allocation in pension funds.1 Duration is a standard and ubiquitous measure of the price sensitivity of a bond to interest rate changes in fixed income analytics. Equity duration measures the sensitivity of equity prices to rate changes.2 The extension of the duration concept to equities is more recent, with the earliest literature on the subject dating back just over 20 years and its use in investment management is far from widespread. The reasons for this are not hard to find: • Unlike plain bonds, the terminal value of equities is not fixed. • Interest payments of plain bonds are predetermined and known in advance. Dividend payments of equities are not as certain. We suggested that the difficulties in estimating equity duration do not detract from its importance in immunization, tactical asset allocation, and risk management. Immunization: Immunization refers to investment of assets in such a manner so as to enable matching of assets and liabilities regardless of changes in interest rates. It refers not only to matching the present value of assets with the present value of liabilities, but also to matching the interest rate sensitivities of assets with those of liabilities. Since the duration of any instrument varies with time and changes in rates, complete immunization is costly or impractical. Immunization in practice is often a tradeoff between cost and efficiency. As we mentioned in the previous section, a common example is a pension plan that not only has to match its present value of assets with its projected obligations, but also has to ensure that the duration of assets matches those of its obligations. Since equities account for nearly half of assets in most pension plans, an estimate of equity duration is important. Risk Management: Equities constitute a significant proportion of investor portfolios, and empirical evidence suggests that equities do react to changes in rates. Therefore, any risk management plan needs to factor in the sensitivity of the equity portfolio to rate changes. Tactical Asset Allocation: Tactical asset allocation makes opportunistic bets on changes in the external economic environment by shifting allocations among different asset classes. Since interest rate changes are one signal of the external economic environment, knowledge of equities’ rate sensitivity would be very important for plan managers considering shifts in asset allocations to take advantage of projected changes in interest rates. There are three distinct approaches to evaluate equity duration.3 The Dividend Discount Model Approach is the earliest and simplest approach. However, it gives high estimates of equity duration. More importantly, it does not take into account the “flow-through” effects of interest rates; that is, it does not consider the fact growth might be sensitive to rates. The Empirical Approach derives equity duration from historical changes in equity prices and interest rates, and yields much shorter duration estimates. While statistically 1

“Using Equity Duration In Pension Fund Asset Allocation - Introducing a new data series: The 30-year history of duration for the S&P 500,” January 27, 2004, www.standardandpoors.com. 2 It is important to note that, unlike in bonds, interest rates do not have significant explanatory power for equity returns; rather, the rate effect is transmitted to equity prices through other variables that have significant explanatory power. 3 See our previous paper for a fuller description of these approaches and historical estimates derived from them. 2

Equity Duration

appealing and direct, it suffers from biases that result in lower than expected estimates of duration. Flow-Through Duration Models follow from the Dividend Discount Model and factor in the sensitivity of growth to rates. In our previous paper, we derived our estimate of equity duration as 1/P (δP/δk) = -1/(k-g) (1-δg/δk)

(4)

Where P is the price of the stock, k is the equity discount rate, and g is the dividend growth rate. This is a simple flow-through model, where dg/dk measures the sensitivity of dividend growth to changes in the equity discount rate. Several properties of duration can be drawn from this approach. Ceteris paribus, 1.

Higher growth implies higher duration. That is, higher-growth portfolios will have a higher duration and, therefore, greater sensitivity to interest rates.

2.

If the dividend growth rate is steady, a higher equity discount rate implies a lower duration and, therefore, a lower sensitivity to changes in interest rates.

3.

Low sensitivity of growth opportunities to the discount rate increases the duration of a portfolio and therefore increases the sensitivity of a portfolio’s value to changes in interest rates.

In our calculations for evaluating the duration of the S&P 500, we take quarterly dividend growth of the S&P 500 for g. For k, we choose to use the Moody’s Baa yield series. The choice of a corporate bond yield series departs from literature, but we believe is more practical. Traditionally, the equity discount yield in this context has been taken as a longterm (10- or 20-year) treasury bond, with a constant equity risk premium added to it. However, because the equity risk premium varies from one time period to another, an average might not be appropriate — leaving aside the intricacies involved in computing the risk premium if one is not adding an average number. The corporate bond series gives a market-determined, risk-adjusted measure of the discount rate. The sensitivity of g to k is difficult to estimate. Following some prior literature, we take this factor as the correlation of change in g to change in k. Updated Duration Estimates The duration of the S&P 500 since 1973 is shown in Appendix 1 and plotted in Exhibit 1. The most striking feature is the recent strong upward trend to record levels. This is related to growth in dividend payments attributable to record levels of corporate profits and more beneficial tax treatment over the last several years, as well as generous liquidity and credit conditions that have lowered the sensitivity of growth opportunities to the equity discount rate. Though certain sectors of credit markets have become dislocated since the summer of 2007, the corporate sector has seen relatively stable required rates of return and continued to grow dividends at double-digit rates through the first quarter of 2008.

3

Equity Duration

Exhibit 1: Duration of the U.S. Equity Market 60 55 50 45 40 35 30 25 20 15 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

10

Source: Standard & Poor’s. Estimates are for the middle of each calendar year.

We estimate the duration of the S&P 500 index to be 45 years at the end of the second quarter of 20084. It has risen markedly from its level of 16 years at the end of 2003, suggesting that the market has become very much more rate-sensitive. Much of the increase took place in the last three years, with the 12-quarter moving average increasing from 18 as of June 2005 to 24 by June 2008. These figures far surpass those of the technology-driven bubble years and are the highest yet recorded. Several dynamics are behind the considerable shift in equity market rate sensitivity. Since the first quarter of 2004 through the first quarter of this year, shareholders enjoyed 17 consecutive quarters of double-digit dividend growth. This is the longest sustained period of such growth since the 18 quarters from June 1947 through September 1951. Since we measure our growth parameter, g, as the growth rate in dividends, this has a direct impact on our duration measure. The other factor which directly impacted equity duration of late is the sensitivity of growth to the equity discount rate. In recent years, the dividend growth exhibited little sensitivity to changes in the discount rate. Perhaps because of the relatively modest level of corporate rates throughout the period, S&P 500 companies robustly grew dividend payouts with little regard to changes in those rates, and our trailing 40-quarter sensitivity parameter decreased from .218 in June 2005 to .081 in June 2008. Because the denominator of the Gordon dividend discount model relies on these two inputs, growth (g) and the equity discount rate (k), it is useful to see how they have varied through time, which Exhibit 2 shows. One notable observation is that the data feeding our estimate of growth has been greater in magnitude than the data feeding our estimate of the discount rate for an unusually prolonged period. Keeping in mind that an important component of our model denominator is a difference (k-g), there are a few key things to understand. First, changing the smoothing, or averaging, period of the input variables can change the sign of the duration estimate. Second, changing the smoothing period of the input variables can change the magnitude of the duration estimate.

4

Estimates for 2006 – 2008 have been calculated using the modified smoothing approach discussed below. 4

Equity Duration

Exhibit 2: Model Inputs through Time 20%

15%

10%

5%

0%

-5%

Corporate Baa Rates (k)

Dividend Growth (g)

Source: Moody’s, Standard & Poor’s.

The shaded area shows the raw data inputs over the past 10 years. Considering that the period included differences between k and g of similar average magnitudes but different signs, it becomes clear that this component of the model denominator is currently a relatively small number. Exhibit 3 tracks this value through time, and shows that it is the smallest yet recorded. Exhibit 3: DDM Denominator (k-g); With 10-Yr Averaging 7%

6%

5%

4%

3%

2%

1%

M ar M 73 ar M 74 ar M 75 ar M 76 ar M 77 ar M 78 ar M 79 ar M 80 ar M 81 ar M 82 ar M 83 ar M 84 ar M 85 ar M 86 ar M 87 ar M 88 ar M 89 ar M 90 ar M 91 ar -9 M 2 ar M 93 ar M 94 ar M 95 ar M 96 ar M 97 ar M 98 ar M 99 ar M 00 ar M 01 ar M 02 ar M 03 ar M 04 ar M 05 ar M 06 ar M 07 ar -0 8

0%

Source: Moody’s, Standard & Poor’s.

5

Mar-08

Mar-07

Mar-06

Mar-05

Mar-04

Mar-03

Mar-02

Mar-01

Mar-00

Mar-99

Mar-98

Mar-97

Mar-96

Mar-95

Mar-94

Mar-93

Mar-92

Mar-91

Mar-90

Mar-89

Mar-88

Mar-87

Mar-86

Mar-85

Mar-84

Mar-83

Mar-82

Mar-81

Mar-80

Mar-79

Mar-78

Mar-77

Mar-76

Mar-75

Mar-74

Mar-73

-10%

Equity Duration

This has an important effect on our duration estimate. Since our model denominator is very small, our duration estimate is very high. Since our initial objective in applying an average of any length was to smooth out short-term variations in the raw data, we lengthened our smoothing function for the years 2006 – 2008 to 20 years to correct for the anomalous past 10 years. Had we not done this, our duration estimates for 2006 – 2008 would have been even higher. However, they would have been misleading because, over the long run, achieving growth higher than the required rate of return on equity is unsustainable. As market participants come to expect a given level of growth, shares are bid up or down accordingly, directly impacting the required rate of return on equity for the new level of expected growth. Our flow through duration estimate involves long-term parameters and is inappropriate for short-term market timing. It is intended to suit the purposes of long-term asset allocation involving rebalancing every three years or more. This is consistent with asset allocation review cycles of most pension plans. Further, the trend should be considered as important as the point estimate. Therefore, in Appendix 1, we have added a three-year moving average column. In light of this, it would be inaccurate to interpret the estimate as “based on June 2008 duration estimates, the S&P 500 would fall 45% for every 1% rise in rates.” Rather, a more appropriate way of describing the estimate is that based on June 2008 estimates, duration of the S&P 500 index is 45 years if it would have been a fixed income instrument discounted at it appropriate risk-adjusted rate, and therefore the market is more rate sensitive than it has been in a long time. If one is looking for more direct metrics of interest rate versus equity returns, our latest empirical results based on regression of S&P 500 returns versus 10-year rates over the previous 40 quarters suggests a sensitivity of 5.9, i.e., subject to model limitations, equity returns fall 5.9% for every 1% rise in the 10-year rate.5

5

Please refer to our previous paper on the limitations of the empirical estimate. 6

Equity Duration

Appendix 1: Annual Duration of S&P 500 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Duration of U.S. Equity Market 36.4 30.6 23.9 17.8 22.9 30.2 33.8 31.5 39.0 39.5 29.1 21.9 21.2 21.4 16.0 13.3 12.8 14.9 14.2 14.2 17.2 19.9 17.1 19.6 25.0 24.2 23.4 18.5 15.0 16.0 15.2 17.5 24.9 28.6 36.9 44.9

12 Quarter Moving Average of Duration

26.0 22.2 22.7 27.1 30.8 33.8 36.2 36.4 32.4 26.2 22.5 20.4 17.9 15.1 13.7 13.8 14.2 14.9 16.3 17.3 18.2 19.7 21.9 23.3 22.5 19.7 16.9 15.4 15.8 18.1 20.4 22.1 24.2

Source: Standard & Poor’s. Estimates are as of the middle of each year.

The duration estimate is obtained from the formula given in equation (4), with equity duration being equal to -1/(k-g) (1-δg/δk). We take annualized quarterly dividend growth of the S&P 500 for g. For k, we choose to use the Moody’s Baa yield series. We use averages for the past 40 quarters (10 years) for 1973 – 2005 and averages for the past 80 quarters (20 years) for 2006 – 2008. For the δg/δk term, we use the correlation of change in g to change in k for the previous 40 quarters for 1973 – 2005 and the previous 80 quarters for 2006 - 2008.

7