Last updated: March 2026
This code updates the measure of equity risk premium.
I use the dividend-price ratio, cay and the three-month T-bill to predict future excess returns
- Haddad Valentin, Erik Loualiche, and Matthew Plosser: Buyout Activity: the Impact of Aggregate Discount Rates; Journal of Finance, February 2017, 72:1
- Download the paper
- Download the data
For the exact replication of the published paper, use v1.0.
That release uses the original R pipeline with Lettau's cay downloaded directly from his website (1952Q1–2019Q3). CRSP data through 2017Q4. Regression coefficients: D/P = 3.370, cay = 1.814, T-bill = −1.246, R² = 0.344 (264 obs).
For updated estimates, use v2-202603.
This release rewrites the pipeline in Julia and constructs cay from publicly available FRED data, removing the dependency on Lettau's website. CRSP data through 2024Q4. DOLS coefficients: β_a = 0.195, β_y = 0.863 (estimation sample: 1951Q4–2019Q3). Regression coefficients: D/P = 2.779, cay = 0.398, T-bill = −1.232, R² = 0.234 (285 obs). See the construction of cay section below for validation details. Future data updates follow the v2-YYYYMM convention.
- Dividend-price ratio from CRSP Monthly Stock Market Index (
crsp.msi), available on WRDS- See the calculations to account for reinvested dividends in this note
- Risk-free rate from the H15 release: 3-month T-bill (
TB3MS) downloaded from FRED - cay (consumption-wealth ratio) constructed from FRED data following Lettau and Ludvigson (2001). See details below.
In v1.0, cay was downloaded directly from Martin Lettau's website, last updated 2019Q3. That series is no longer maintained.
In v2.0, I construct cay from publicly available FRED data. The cointegrating vector is estimated on the pre-COVID sample (1951Q4–2019Q3), the same endpoint as Lettau's last update. I deliberately exclude the pandemic period from estimation because COVID produced large, transitory swings in transfers (stimulus payments), consumption (lockdowns), and asset values that would distort the long-run cointegrating relationship. The estimated coefficients are then applied out of sample to compute cay through the latest available quarter.1
cay is the residual from a cointegrating regression of log real per-capita consumption on log real per-capita asset wealth and log real per-capita labor income:
c_t = α + β_a · a_t + β_y · y_t + cay_t
The cointegrating vector is estimated by Stock-Watson Dynamic OLS (DOLS) with 8 leads and lags of the first-differenced regressors.
| Variable | FRED series | Description |
|---|---|---|
| Consumption (c) | PCEC |
Personal Consumption Expenditures, quarterly, SAAR |
| Asset wealth (a) | TNWBSHNO |
Households and Nonprofits Net Worth (Z.1) |
| Labor income (y) | composite | See construction below |
| Price deflator | PCECTPI |
PCE Chain-Type Price Index (2017=100) |
| Population | B230RC0Q173SBEA |
BEA Midperiod Population |
All nominal series are deflated by PCECTPI and divided by population to obtain real per-capita values.
Labor income construction (Lettau and Ludvigson 2001, Appendix):
y = wages + transfers + other labor income − social insurance − (labor share × taxes)
where labor share = (wages + transfers + other labor income − social insurance) / personal income.
| Component | FRED series | Description |
|---|---|---|
| Wages and salaries | WASCUR |
Compensation of Employees: Wages and Salary Accruals |
| Transfer payments | A577RC1Q027SBEA |
Personal Current Transfer Receipts |
| Other labor income | B040RC1Q027SBEA |
Employer Contributions for Employee Pension and Insurance |
| Social insurance | A061RC1Q027SBEA |
Contributions for Government Social Insurance |
| Personal income | PINCOME |
Personal Income |
| Personal taxes | W055RC1Q027SBEA |
Personal Current Taxes |
I compare the constructed cay to Lettau's published series (1952Q1–2019Q3, 271 quarterly observations) on the same estimation sample. Current FRED vintages differ from Lettau's due to the 2023 NIPA comprehensive revision, which rescaled the PCE deflator by approximately 6%.
Component-level accuracy:
| Correlation of levels | Max first-difference error | |
|---|---|---|
| Consumption (c) | 0.99998 | 0.0046 |
| Asset wealth (a) | 0.99993 | 0.0083 |
| Labor income (y) | 0.99988 | 0.0075 |
DOLS coefficients (estimation sample: 1951Q4–2019Q3):
| Lettau | v2.0 | |
|---|---|---|
| β_a (wealth) | 0.218 | 0.195 |
| β_y (income) | 0.801 | 0.863 |
cay series correlation (demeaned): 0.986
Impact on predicted risk premium (264 common observations, 1952Q1–2017Q4):
| Lettau cay (v1.0) | Constructed cay (v2.0) | |
|---|---|---|
| D/P coefficient | 3.370 | 3.385 |
| cay coefficient | 1.814 | 1.564 |
| T-bill coefficient | −1.246 | −1.336 |
| R² | 0.344 | 0.332 |
The predicted risk premium from both specifications has correlation 0.997. The mean absolute difference is 0.35 percentage points; 77% of observations differ by less than 0.5pp and 98% by less than 1pp.
===========================================================
Future Excess Returns
-----------------------------------------------------------
D/P ratio 2.779***
(1.021)
cay 0.398
(0.36)
T-bill (three-month) -1.232***
(0.306)
Constant 0.034
(0.035)
Observations 285
R2 0.234
-----------------------------------------------------------
Notes: ***Significant at the 1 percent level.
**Significant at the 5 percent level.
*Significant at the 10 percent level. Footnotes
-
Estimating DOLS on the full sample (including 2020–2025) shifts the cointegrating vector substantially: β_a falls from 0.195 to 0.070 and β_y rises from 0.863 to 0.991. The predictive R² is nearly unchanged (0.234 vs 0.231), but the pre-COVID coefficients are closer to Lettau's published values (β_a = 0.218, β_y = 0.801) and more economically interpretable. ↩