**An SWR Application for Tobin's q**

I can now identify one way in which Tobin's q can improve our estimates of Safe Withdrawal Rates. It is a minor improvement. I have not developed it fully.

I use large differences between predictions based on Tobin's q and those based on PE10 to identify a few special cases. I make my estimated based on PE10 except in those special cases. In those special cases I substitute an estimate based on the product PE10 and q. Based upon comments by wanderer, I believe that the substitute can be refined further by adding a suitable constant to q. That is, we would make adjustments based upon PE10 *(q + a suitable constant).

**The tables**

The relevant tables provide the orders of PE10, q, PE10*q and HDBR along with HDBR80 percentages. By PE10, I am referring to Professor Shiller's P/E10, which is the current index value (or price) of the S&P 500 divided by the average of the trailing decade of earnings. At this point I am using Tobin's q in its basic form. The Historical Database Rates (HDBR) have been calculated in 0.2% increments on the FIRECalc. HDBR80 refers to the Historical Database Rate of an 80% stock (and 20% commercial paper) retirement portfolio that is designed to last 30 years and no longer. It is calculated based upon actual historical sequences beginning in the years shown.

This is the referenced thread:

**Ordered Tables (PE10, q, HDBR)**dated Sat, Aug 23, 2003 at 12:01 pm CDT

http://nofeeboards.com/boards/viewtopic.php?t=1305

These are the tables:

**The Orders of Different Indicators**

Differences of Orders

Differences of Orders

Tables 1, 2 and 3

http://nofeeboards.com/boards/viewtopic ... 415#p10415

http://nofeeboards.com/boards/viewtopic ... 416#p10416

http://nofeeboards.com/boards/viewtopic ... 418#p10418

**How to use the tables**

The tables are listed according to the order of the Historical Database Rates, starting with the highest percentage (or rate) and ending with the smallest. Bigger values of the order (which are written as O(HDBR)) corresponds to smaller Historical Database Rates. The order of PE10 (which is written as O(P)) ranges from the most favorable valuations (low PE10 ratios) to the least favorable valuations (highest prices). The order of q varies in a similar manner. It is written as O(q). I have also taken the product PE10*q, using q as-is, and ordered it. That order is O(P*q). A smaller order corresponds to a smaller product and it indicates more favorable valuations (i.e., stocks are cheap). A higher number corresponds to less favorable valuations and higher prices. The product is one way to make a compromise between the two indicators, PE10 and q, but it is not the only way.

I have treated the HDBR80 FIRECalc results as if they are truth data. In fact, they only approximate the truth. They represent past historical sequences. There are many ways in which future results will differ, not all of which are known.

Consider the 1924 row in the table. I have used the orders of PE10, q and PE10*q to predict HDBR80 percentages. Value of PE10 was 8.0 in 1924. It is the second entry. The order of PE10 is listed under O(P). It is 3. I did not include the value of q in the table, but its order O(q) is 23. I did not include the value of the product PE10*q in the table. Its order O(P*q) is 8. The Historical Database Rate HDBR is 8. The difference of O(q) and O(P) is 23 - 3 = 20. The difference of O(P*q) and O(P) is 8 - 3 = 5. I have listed both of those differences. I will only use the first ((O(P) - O(q) = 20) in my actual analysis. The Historical Database Rate for the 80% stock portfolio HDBR80 is 9.4%.

Ideally, the orders of PE10, q, PE10*q and HDBR would all be the same. Since I have treated the HDBR80 results as truth data, every one of them should be in the same order as HDBR. That is, the right answer for the year 1924 is 8 and the right withdrawal rate is 9.4%.

Since the actual order of PE10 in 1924 was 3, PE10 is in error. It predicted a withdrawal rate of 10.8%. (Look under O(HDBR) and find the row in which it equals 3. Read the percentage at the end of that row. It is 10.8%.) Now look at the order of q O(q) in 1924. It was 23. Looking under the order of the Historical Database Rate HDBR and locating 23 (in the row with the year 1932), we find that q predicted a withdrawal rate of 8.0%. Continuing, we find that the order of the product PE10*q, which is listed as O(P*q), is 8. That is the same as for the Historical Database Rate HDBR. Thus, it has the right answer and the correct prediction is a withdrawal rate of 9.4%.

**The thresholds**

I have looked at a variety of possible thresholds. I have ended up examining +15 and -15 in detail. I have applied them to the difference O(q) - O(P). When that difference is greater than or equal to +15, it means that Tobin's q is pessimistic and PE10 is optimistic (relative to each other). Remember that higher orders mean less favorable valuations and higher prices. Smaller orders mean bargain prices. A big difference, exceeding the +15 threshold, signals danger. It turns out that this danger is worth heeding. The threshold was met in 1924, 1933, 1925, 1926, 1935, 1927 and 1936.

At the other end are comparisons equal to or below the -15 threshold. That happened in 1953, 1941, 1946, 1974, 1931 and 1937. That means that Tobin's q is optimistic and PE10 is pessimistic. Alas, we should not heed this message from Tobin's q. It turns out that PE10 is more likely to be right. I have not reported those results. My algorithm sometimes makes things worse when Tobin's q is highly optimistic (i.e., when q is small and PE10 is large).

**The algorithm**

I have used a very simple algorithm. Use predictions from PE10 unless the threshold is met (O(q)-O(P) is +15 or bigger). If the threshold is met, substitute the product PE10*q, determine its order and use it (O(P*q)) to predict the withdrawal rate.

It turns out that this algorithm results in an over-correction. Something closer to splitting the difference between the original number based on PE10 alone and the new number based on the product PE10*q (or O(P*q)) would do better. That suggests that adding a constant to q might improve the results. This is not quite the same thing as splitting the difference but it should have a similar effect. Simply multiplying q by a (single) scale factor accomplishes nothing because the product term PE10*q (or P*q) would be multiplied by the same number.

**The numbers**

This is a list of the years when the difference O(q)-O(P) equaled or exceeded +15 along with that difference, the orders of PE10, q, PE10*q and HDBR80 (the historical database rate with the 80% stock portfolio) grouped inside of a pair of parentheses (separated by commas) and three relevant percentages. The first percentage is the HDBR80 rate determined from the order of PE10. The second is the HDBR80 rate determined from the order of PE10*q. The third is the HDBR80 rate listed for the year in which the threshold was met or exceeded. This third percentage rate is taken to be the truth data.

Code: Select all

```
Year Difference [O(P),O(q),O(P*q),O(HDBR)] [%(P),%(P*q),HDBR80]
1924 +20 [ 3,23, 8, 8] [10.8%, 9.4%, 9.4%]
1933 +31 [ 5,36,23,19] [10.2%, 8.0%, 8.4%]
1925 +20 [11,31,25,20] [ 9.2%, 7.6%, 8.4%]
1926 +16 [19,35,29,24] [ 8.4%, 7.0%, 7.6%]
1935 +27 [20,47,36,25] [ 8.4%, 6.0%, 7.6%]
1927 +15 [29,44,38,28] [ 7.0%, 6.0%, 7.2%]
1936 +17 [41,58,52,43] [ 5.8%, 4.6%, 5.4%]
```

We can calculate the adjustments needed to end up with the truth data by subtracting the first percentage number from the third and by subtracting the second percentage number from the third. In the first case, we have the adjustment that we should make to our PE10 based answer (the first percentage) to end up with the truth (the third percentage). In the second case, we have the adjustment that we should make to our PE10*q based answer (the second percentage) to end up with the truth (the third percentage).

The first adjustment tells us about the errors from using PE10 alone. The second tells us about the errors when we use our algorithm and switch to an answer based upon the product PE10*q. Ideally, the second number would be very close to zero. That gives us a before and after comparison.

For 1924 the adjustments are -1.4% and 0.0%. For 1933 the adjustments are -1.8% and +0.4%. For 1925 the adjustments are -0.8% and +0.8%. For 1926 the adjustments are -0.8% and +0.6%. For 1935 the adjustments are -0.8% and +1.6%. For 1927 the adjustments are +0.2% to +1.2%. For 1936 the adjustments are -0.4% to +0.8%.

**Summary**

High levels of Tobin's q relative to PE10 can warn you against excessive optimism. My algorithm will generally cause you to over-correct your withdrawal rate downward. This correction applies most often when withdrawal rates are already favorable. It reinforces a natural tendency to be cautious when withdrawing from retirement portfolios.

Have fun.

John R.