Tobin q Survey

Here are my initial findings about Tobin’s q using Smithers and Co. Ltd. data.

I am still screening. This analysis is almost entirely based on numbers alone.

Data

I used Excel to calculate regression equations between the real, annualized, total return of the S&P500 and several measures of valuation. In addition, I collected regression equations for 30-Year Historical Surviving Withdrawal Rates for portfolios HSWR50 (50% stocks and 50% commercial paper) and HSWR80 (80% stocks and 20% commercial paper). In most instances, I used 1923-1980 data, which has been best for calculating Safe Withdrawal Rates. I used 1900-1990 data for an excursion.

My measures of valuation were the percentage earnings yield 100E10/P of the S&P500 (where 100E10/P = 100/[P/E10] using Professor Robert Shiller’s P/E10), qefm, q where q=0.5 in 1871, q where q=2 in 1871, 1/qefm, 1/q where q=0.5 in 1871 and 1/q where q=2 in 1871. I got my values of q via the Smithers and Co. Ltd. web site.

Professor Shiller’s Web Site
Professor Shiller’s Online Data
Smithers and Co. Ltd. q and FAQs
Stephen Wright link from Smithers and Co. Ltd.

Results

1923-1980 Data: Returns

Returns at Year 5
If x=100E10/P: y=2.1172x-9.1683. R-squared=0.3717.
If x=qefm: y=-17.572x+17.246. R-squared=0.2348.
If x=q, q=0.5: y=-15.681x+14.785. R-squared=0.1554.
If x=q, q=2: y=-11.872x+14.67. R-squared=0.1344.
If x=1/qefm: y=6.2755x-5.0672. R-squared=0.2232.
If x=1/q, q=0.5: y=4.5087x-3.3243. R-squared=0.1874.
If x=1/q, q=2: y=5.0922x-1.888. R-squared=0.1265.

Returns at Year 10
If x=100E10/P: y=1.3564x-3.8223. R-squared=0.383.
If x=qefm: y=-14.568x+15.111. R-squared=0.4051.
If x=q, q=0.5: y=-14.799x+14.012. R-squared=0.3476.
If x=q, q=2: y=-11.221x+13.916. R-squared=0.3014.
If x=1/qefm: y=5.4397x-3.8278. R-squared=0.421.
If x=1/q, q=0.5: y=3.6463x-1.7419. R-squared=0.3077.
If x=1/q, q=2: y=4.8217x-1.7494. R-squared=0.2847.

Returns at Year 15
If x=100E10/P: y=1.3165x-3.4499. R-squared=0.5387.
If x=qefm: y=-14.804x+15.331. R-squared=0.6247.
If x=q, q=0.5: y=-16.762x+15.116. R-squared=0.6657.
If x=q, q=2: y=-13.273x+15.391. R-squared=0.6296.
If x=1/qefm: y=5.615x-4.0768. R-squared=0.6698.
If x=1/q, q=0.5: y=4.2952x-3.0905. R-squared=0.6375.
If x=1/q, q=2: y=5.7696x-3.2484. R-squared=0.6086.

Returns at Year 20
If x=100E10/P: y=1.0946x-1.5004. R-squared=0.5591.
If x=qefm: y=-10.719x+13.149. R-squared=0.4916.
If x=q, q=0.5: y=-14.163x+14.055. R-squared=0.7135.
If x=q, q=2: y=-11.356x+14.383. R-squared=0.6918.
If x=1/qefm: y=4.1801x-1.1157. R-squared=0.5572.
If x=1/q, q=0.5: y=3.6291x-1.3287. R-squared=0.6832.
If x=1/q, q=2: y=5.0404x-1.7373. R-squared=0.6973.

1900-1990 Data: Returns

Returns at Year 5
If x=qefm: y=-8.7305x+13.437. R-squared=0.449.
If x=1/qefm: y=5.1119x-3.0405. R-squared=0.208.

Returns at Year 10
If x=qefm: y=-14.684x+15.551. R-squared=0.2063.
If x=1/qefm: y=5.3816x-3.4002. R-squared=0.4656.

Returns at Year 15
If x=qefm: y=-15.304x+16.075. R-squared=0.4524.
If x=1/qefm: y=4.6589x-2.06. R-squared=0.5505.

1923-1980 Data

HSWR50 30-Year Historical Surviving Withdrawal Rates

If x=100E10/P: y=0.4087x+2.5821. R-squared=0.7025.
If x=qefm: y=-3.4163x+7.6956. R-squared=0.4502.
If x=q, q=0.5: y=-3.5556x+7.4824. R-squared=0.4054.
If x=q, q=2: y=-2.9214x+7.613. R-squared=0.4128.
If x=1/qefm: y=1.3516x+3.1134. R-squared=0.5252.
If x=1/q, q=0.5: y=0.9215x+3.5977. R-squared=0.3971.
If x=1/q, q=2: y=1.3428x+3.3893. R-squared=0.4462.

HSWR80 30-Year Historical Surviving Withdrawal Rates
If x=100E10/P: y=0.6892x+1.5239. R-squared=0.7491.
If x=qefm: y=-6.0969x+10.351. R-squared=0.5376.
If x=q, q=0.5: y=-6.9511x+10.288. R-squared=0.5809.
If x=q, q=2: y=-5.492x+10.394. R-squared=0.5469.
If x=1/qefm: y=2.3649x+2.2614. R-squared=0.6028.
If x=1/q, q=0.5: y=1.8171x+2.659. R-squared=0.579.
If x=1/q, q=2: y=2.4587x+2.5626. R-squared=0.5608.

Analysis

1923-1980 Data: Returns

The values of R-squared for q and 1/q returns are similar. This is consistent with an exponential representation: exp(q)=1+q approximately when q is small. This is consistent with plots that compare P/E10 with the natural logarithm of q.

The best choice of q varies with the time period being projected. At 5 years, none were especially good, but qefm and 1/qefm were best. At 10 years, qefm and 1/qefm were best. At 15 years, all values of q and 1/q were excellent. At 20 years, all values of q and 1/q except qefm and 1/qefm were excellent.

The percentage earnings yield 100E10/P was best at year 5, comparable to qefm and 1/qefm at year 10, a laggard at year 15 and a laggard in most cases at year 20. It did better than qefm and 1/qefm (slightly) at year 20.

1900-1990 Data: Returns

Extending the data to include 90 start years indicates a possible anomaly associated with Tobin’s q. If so, it would be similar to the P/E10 anomaly, which was most noticeable in the first decade of the twentieth century.

Judging from R-squared: qefm does much better than 1/qefm at 5 years. The opposite is true at 10. They do almost the same at 15 years.

Judging from R-squared: it is better to use the abbreviated 1923-1980 data set than 1900-1990.

1923-1980 Data

30-Year Historical Surviving Withdrawal Rates

The percentage earnings yield 100E10/P was the best choice for both portfolios (HSWR50 and HSWR80).

Follow-On

The next obvious steps are (a)to compare the percentage earnings yield 100E10/P and qefm and 1/qefm results in depth at years 10 and 15 and (b)to compare the percentage earnings yield 100E10/P and q and 1/q with q=0.5 in 1871 and with q=2 in 1871 at year 20.

One of the things to look for is unusual behavior, not typical of random effects, that R-squared fails to capture.

Another issue is finding updated data. The Smithers and Co. Ltd. data ends in 2002. They mention Federal Reserve flow of funds data as a reference, but I think that I will have to make adjustments.

Making updates for predictions should not be a problem. The value of q is similar to a price to book value, modified by using replacement value. Scaling from a pair of references, S&P500 reference and q reference, should be as simple as: scaled q (for making predictions) = ([q reference]/[S&P500 reference])*(current value of the S&P500 index).

Have fun.

John Walter Russell
August 10, 2006