Duration of Equity 2008

8/14/2019 Duration of Equity 2008

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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

Philip Murphy, CFA

[email protected]

In early 2004, we published a paper describing a simple model of assetallocation 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 measuresthe 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, andasset allocation.

We developed a simple model of equity duration that uses the dividenddiscount model and incorporates the sensitivity of growth to rates. Based on

our empirical model, duration (or interest-rate sensitivity) is higher for high-growth stocks, stocks whose dividend growth is not sensitive to interest

rates, and in lowdiscount rate environments.

Standard & Poors publishes, on an annual basis, a current report and a 30-year history of duration for the S&P 500. We acknowledge that equityduration 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 middleof 2008. It has risen markedly in recent years, to record levels, suggesting

that the market has become much more rate-sensitive.
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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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 ratechanges.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. Dividendpayments 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 refersnot 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 equitiesdo 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 equityprices 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
http://www.standardandpoors.com/http://www.standardandpoors.com/


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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 (P/k) = -1/(k-g) (1-g/k) (4)

Where P is the price of the stock, kis the equity discount rate, and g is the dividend

growth rate. This is a simple flow-through model, where dg/dkmeasures 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 ahigher duration and, therefore, greater sensitivity to interest rates.

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

3. Low sensitivity of growth opportunities to the discount rate increases the duration ofa portfolio and therefore increases the sensitivity of a portfolios value to changes ininterest 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 Moodys 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 long-

term (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 ofg to k

is difficult to estimate. Following some prior literature, we take this factor as thecorrelation 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.

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Equity Duration

Exhibit 1: Duration of the U.S. Equity Market

10

15

20

25

30

35

40

45

50

55

60

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

Source: Standard & Poors. 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 importantcomponent 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.

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Equity Duration

Exhibit 2: Model Inputs through Time

-10%

-5%

0%

5%

10%

15%

20%

Mar-73

Mar-74

Mar-75

Mar-76

Mar-77

Mar-78

Mar-79

Mar-80

Mar-81

Mar-82

Mar-83

Mar-84

Mar-85

Mar-86

Mar-87

Mar-88

Mar-89

Mar-90

Mar-91

Mar-92

Mar-93

Mar-94

Mar-95

Mar-96

Mar-97

Mar-98

Mar-99

Mar-00

Mar-01

Mar-02

Mar-03

Mar-04

Mar-05

Mar-06

Mar-07

Mar-08

Corporate Baa Rates (k) Dividend Growth (g)

Source: Moodys, Standard & Poors.

The shaded area shows the raw data inputs over the past 10 years. Considering that the

period included differences between kand 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

0%

1%

2%

3%

4%

5%

6%

7%

Mar

-73

Mar

-74

Mar

-75

Mar

-76

Mar

-77

Mar

-78

Mar

-79

Mar

-80

Mar

-81

Mar

-82

Mar

-83

Mar

-84

Mar

-85

Mar

-86

Mar

-87

Mar

-88

Mar

-89

Mar

-90

Mar

-91

Mar

-92

Mar

-93

Mar

-94

Mar

-95

Mar

-96

Mar

-97

Mar

-98

Mar

-99

Mar

-00

Mar

-01

Mar

-02

Mar

-03

Mar

-04

Mar

-05

Mar

-06

Mar

-07

Mar

-08

Source: Moodys, Standard & Poors.

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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.

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Equity Duration

Appendix 1: Annual Duration of S&P 500

Duration of U.S. Equity Market 12 Quarter Moving Average of Duration

1973 36.4

1974 30.6

1975 23.9

1976 17.8 26.01977 22.9 22.2

1978 30.2 22.7

1979 33.8 27.1

1980 31.5 30.8

1981 39.0 33.8

1982 39.5 36.2

1983 29.1 36.4

1984 21.9 32.4

1985 21.2 26.2

1986 21.4 22.5

1987 16.0 20.4

1988 13.3 17.9

1989 12.8 15.1

1990 14.9 13.7

1991 14.2 13.81992 14.2 14.2

1993 17.2 14.9

1994 19.9 16.3

1995 17.1 17.3

1996 19.6 18.2

1997 25.0 19.7

1998 24.2 21.9

1999 23.4 23.3

2000 18.5 22.5

2001 15.0 19.7

2002 16.0 16.9

2003 15.2 15.4

2004 17.5 15.8

2005 24.9 18.1

2006 28.6 20.42007 36.9 22.1

2008 44.9 24.2

Source: Standard & Poors. 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-g/k). We take annualized quarterly dividend growth

of the S&P 500 for g. For k, we choose to use the Moodys 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 g/kterm, we use the correlation of change

in g to change in kfor the previous 40 quarters for 1973 2005 and the previous 80

quarters for 2006 - 2008.

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Duration of Equity 2008

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