Targeting Macroeconomic Exposures in Equity Portfolios: A Firm-Level Measurement Approach for Out-of-Sample Robustness

Figures & data

Figure 1. Innovations in Macro Variables Are Different from Levels: The Case of Breakeven Inflation

Table 1. Out-of-Sample Exposures of Mimicking Portfolios to Backward-Looking Variables

United States 30 June 1970–30 June 2022	Realised Macro Beta	t-Statistic
Mimicking portfolios to backward-looking variables using naïve estimation techniques
Industrial Production Growth	−0.01	−0.28
Consumer Price Index Growth (CPI)	0.23	0.97

Notes: The realised macro exposures are estimated in a bivariate regression that includes the market factor and the corresponding macro variable. Standard errors are adjusted for heteroskedasticity and autocorrelation following Newey and West (1987). We follow Andrews (1991) when determining data-dependent bandwidth/lag truncation parameters, which are estimated in a first-order autoregressive model (AR(1)) with a maximum likelihood method. Statistical significance is indicated by a single asterisk (*) for results that are significant on a 5% basis and double asterisks (**) for results that are significant on a 1% basis.

Table 2. Reliability of Macro Beta Estimates Using Different Regression Specifications: Out-of-Sample Betas with Targeted Macro Variables

Estimation Frequency	Estimation Method	Bayesian Shrinkage	Short Rates	Term Spread	Credit Spread	Breakeven Inflation	Average
Monthly	OLS	No	1.07**	1.20*	1.84	2.73*	1.71
			(4.08)	(2.56)	(1.44)	(2.04)	(2.53)
Weekly	OLS	No	0.83**	1.25**	2.16**	5.37**	2.40
			(5.23)	(4.33)	(3.90)	(3.23)	(4.17)
Weekly	WLS	No	0.83**	1.34**	2.21**	5.32**	2.43
			(5.62)	(4.51)	(3.98)	(3.36)	(4.37)
Weekly	WLS	Yes	0.81**	1.34**	2.28**	5.36**	2.45
			(5.49)	(4.49)	(4.12)	(3.45)	(4.39)

OLS: ordinary least-squares; WLS: weighted least-squares.

Notes: The analysis has been done from 19 June 1970 to 30 June 2022. The only exception is breakeven inflation, for which the analysis starts in 31 March 2002. The out-of-sample macro exposures are estimated in a bivariate regression that includes the market factor and innovations in corresponding macro variable. The estimation frequency and frequency when calculating realised exposures are the same. Standard errors are adjusted for heteroskedasticity and autocorrelation following Newey and West (1987). We follow Andrews (1991) when determining data-dependent bandwidth/lag truncation parameters, which are estimated in a first-order autoregressive model (AR(1)) with a maximum likelihood method. Statistical significance is indicated by a single asterisk (*) for results that are significant on a 5% basis and double asterisks (**) for results that are significant on a 1% basis.

Figure 2. Relation between Returns on Macro Mimicking Portfolios and Macro Variables

Table 3. Realised Macro Exposures of Different Equity Macro Strategies

United States 19 June 1970–30 June 2022	Short Rates	Term Spread	Credit Spread	Breakeven Inflation
Panel A: macro dedicated portfolios
Macro Exposure +	0.20*	1.07**	1.31**	4.56**
	(2.49)	(6.39)	(4.44)	(4.68)
Macro Exposure −	−0.60**	−0.26*	−0.97**	−0.81**
	(−6.49)	(−2.08)	(−3.13)	(−2.64)
Differences in Macro Exposures	0.81**	1.33**	2.28**	5.37**
	(5.25)	(4.96)	(3.94)	(4.22)
Panel B: factor allocation
Macro Exposure +	0.01	0.48**	0.35	1.71**
	(0.12)	(3.71)	(0.99)	(4.15)
Macro Exposure −	−0.28**	0.07	−0.48	0.20
	(−3.12)	(0.54)	(−1.67)	(0.93)
Differences in Macro Exposures	0.29*	0.41*	0.84	1.51**
	(2.32)	(2.27)	(1.82)	(3.23)
Panel C: sector allocation
Macro Exposure +	0.43**	0.84**	0.51	6.41**
	(3.69)	(3.63)	(0.97)	(5.33)
Macro Exposure −	−0.51**	−0.36	−0.88	0.55
	(−3.71)	(−1.45)	(−1.49)	(0.87)
Differences in Macro Exposures	0.94**	1.19**	1.39	5.86**
	(5.21)	(3.55)	(1.75)	(4.31)

Notes: The analysis has been done from 19 June 1970 to 30 June 2022. The only exception is breakeven inflation, for which the analysis starts in 31 March 2002. The out-of-sample macro exposures are estimated in a bivariate regression that includes the market factor and innovations in corresponding macro variable. Standard errors are adjusted for heteroskedasticity and autocorrelation following Newey and West (1987). We follow Andrews (1991) when determining data-dependent bandwidth/lag truncation parameters, which are estimated in a first-order autoregressive model (AR(1)) with a maximum likelihood method. The null hypothesis that the differences in macro exposures are zero is tested by Welch’s test. The Welch’s t-statistics may not equal to the t-statistics of the null hypothesis that the macro mimicking portfolio has zero exposure, reported in Table 2. Statistical significance is indicated by a single asterisk (*) for results that are significant on a 5% basis and by double asterisks (**) for results that are significant on a 1% basis.

Figure 3. Percentage of Stocks with Statistically Significant Macro Exposures across Time

Table 4. Reliability of Macro Beta Estimates Using an Alternative Method to Form Mimicking Portfolios

United States 19 June 1970–30 June 2022	Short-Rates	Term Spread	Credit Spread	Breakeven Inflation	Average
Panel A: macro dedicated portfolios—30% EW selection
Macro exposure +	0.20*	1.07**	1.31**	4.56**	1.78
	(2.49)	(6.39)	(4.44)	(4.68)	(4.50)
Macro exposure −	−0.60**	−0.26*	−0.97**	−0.81**	−0.66
	(−6.49)	(−2.08)	(−3.13)	(−2.64)	(−3.59)
Panel B: macro dedicated portfolios—EW significant exposures (min 5%)
Macro exposure +	0.38**	1.22**	1.73**	5.65**	2.25
	(2.60)	(4.78)	(3.12)	(5.23)	(3.93)
Macro exposure −	−0.72**	−0.60*	−0.81	−0.86	−0.75
	(−5.31)	(−2.37)	(−1.61)	(−0.77)	(−2.51)

Notes: The analysis has been done from 19 June 1970 to 30 June 2022. The only exception is breakeven inflation, for which the analysis starts in 31 March 2002. The out-of-sample macro exposures are estimated in a bivariate regression that includes the market factor and innovations in corresponding macro variable. The estimation frequency and frequency when calculating realised exposures are the same. Standard errors are adjusted for heteroskedasticity and autocorrelation following Newey and West (1987). We follow Andrews (1991) when determining data-dependent bandwidth/lag truncation parameters, which are estimated in a first-order autoregressive model (AR(1)) with a maximum likelihood method. Statistical significance is indicated by a single asterisk (*) for results that are significant on a 5% basis and double asterisks (**) for results that are significant on a 1% basis.

EW = equal weighted.

Table 5. Ex-Ante and Realised Exposures of Equity-Style Factors

19 June 1970–30 June 2022	Mid-Cap	Value	Momentum	Low Volatility	High Profitability	Low Investment	Average across 6 Factors	Macro Exposure (+)	Macro Exposure (−)
Exposures to short rates
Realised exposure	−0.16*	−0.28**	−0.06	−0.32**	0.07	−0.24**	−0.17	0.20*	−0.60**
Mean ex-ante exposure	−0.12	−0.17	−0.12	−0.22	0.09	−0.18	−0.12	1.45	−1.75
Min	−0.75	−0.64	−0.60	−0.76	−0.31	−0.68	−0.50	0.35	−3.84
25th percentile	−0.37	−0.39	−0.35	−0.40	−0.07	−0.40	−0.30	0.61	−2.41
Median	−0.13	−0.19	−0.18	−0.26	0.04	−0.22	−0.15	0.98	−1.45
75th percentile	0.00	−0.03	0.01	−0.02	0.28	−0.08	0.04	2.26	−1.05
Max	1.11	1.35	1.12	0.44	0.71	1.34	0.39	4.84	−0.81
Exposures to term spread
Realised exposure	0.42**	0.67**	0.06	0.26*	0.12	0.44**	0.33*	1.07**	−0.26*
Mean ex-ante exposure	0.21	0.28	0.18	0.10	0.17	0.22	0.25	2.04	−1.59
Min	−0.80	−0.85	−0.73	−0.75	−0.77	−0.77	−0.74	0.97	−4.75
25th percentile	−0.07	0.05	−0.13	−0.24	0.03	−0.02	0.05	1.26	−2.43
Median	0.19	0.27	0.17	0.08	0.14	0.15	0.23	2.05	−1.12
75th percentile	0.52	0.52	0.47	0.46	0.31	0.50	0.50	2.42	−0.67
Max	0.96	0.98	1.04	0.93	0.95	0.92	0.88	4.80	−0.27
Exposures to credit spread
Realised exposure	−0.22	0.04	0.14	0.58*	−0.61**	0.29	0.04	1.32**	−0.97**
Mean ex-ante exposure	0.11	0.30	0.22	0.46	−0.16	0.32	0.35	3.86	−3.43
Min	−1.77	−1.85	−1.78	−1.18	−1.93	−1.41	−1.55	2.21	−7.12
25th percentile	−0.64	−0.41	−0.41	−0.17	−0.53	−0.24	−0.22	2.92	−4.96
Median	0.29	0.51	0.29	0.54	−0.09	0.52	0.49	3.76	−3.40
75th percentile	0.65	0.80	0.77	0.96	0.24	0.81	0.74	4.73	−1.77
Max	2.30	2.24	2.28	2.47	1.27	2.00	1.87	6.52	−1.21
Exposures to breakeven inflation (31 March 2002–30 June 2022)
Realised exposure	0.23**	0.20*	0.36*	0.34	0.20**	0.42**	0.29**	4.54**	−0.82**
Mean ex-ante exposure	1.30	1.39	0.86	0.27	1.01	1.00	0.65	5.37	−2.58
Min	0.39	0.17	−0.51	−0.49	0.11	0.07	0.24	3.50	−4.19
25th percentile	0.76	0.62	0.30	−0.08	0.79	0.45	0.36	4.56	−2.93
Median	1.40	1.57	0.75	0.33	1.00	1.13	0.45	5.23	−2.53
75th percentile	1.75	1.91	1.40	0.63	1.31	1.28	0.91	5.82	−2.04
Max	2.29	3.02	2.46	1.04	1.83	2.59	1.63	8.67	−1.78

Notes: The equity-style factor portfolios select 30% of the stocks with the best factor scores and equally weights them. The equity universe consists of 500 largest stocks in the U.S. and is reviewed quarterly, on the third Friday of March, June, September, and December. The percentile values are taken from the time series of ex-ante beta estimates for a given equity factor and a given macro variable. Realised exposures were computed over the full sample. Realised macro exposures are estimated in a bivariate regression using weekly observations, where explanatory variables are the market excess return and innovations in a given macro variable. Ex-ante exposure measures are estimated following our proposed methodology. Statistical significance is indicated by a single asterisk (*) for results that are significant on a 5% basis and double asterisks (**) for results that are significant on a 1% basis. We test statistical significance only for full-sample realised exposures. Values set in bold are realised (i.e., ex-post) exposures. Realised exposures of dedicated macro portfolios are highlighted in yellow when they are unique in the sense that none of the equity style factors provide significant exposures with the same sign.

Figure 4. Conditional Relative Performance of Macro Dedicated Portfolios

The plot shows average annualised returns of macro dedicated portfolios after adjusting for their market exposure (CAPM beta). The analysis was done from 19 June 1970 to 30 June 2022 for all portfolios, except for the inflation portfolios, for which the analysis starts on 31 March 2002. Positive (negative) surprise periods are defined as calendar weeks (Friday to Friday) when the change in a given macro variable was amongst the top (bottom) 25% of historical observations. Returns that are statistically significant at a 5% level are presented in bars with solid fill, whilst returns that are not statistically significant at 5% are presented in bars filled with diagonal stripes.

Table 6. Unconditional Performance of Macro Dedicated Portfolios

United States 19 June 1970–30 June 2022	Cap-Weighted	Short Rates (+)	Short Rates (−)	Term Spread (+)	Term Spread (−)	Credit Spread (+)	Credit Spread (−)
Absolute return	10.70%	11.35%	11.87%	11.28%	11.93%	12.45%	11.22%
Volatility	17.08%	18.61%	16.69%	18.78%	16.51%	15.97%	18.97%
Sharpe ratio	0.35	0.36	0.43	0.35	0.44	0.48*	0.34
p value of the differences in Sharpe ratio	–	83.6%	22.6%	76.0%	10.8%	1.5%	60.2%
Relative return	–	0.66%	1.18%	0.59%	1.23%	1.76%	0.53%
p value of relative return	–	25.9%	29.7%	40.9%	19.4%	9.3%	32.5%
Tracking error	–	5.45%	6.73%	7.16%	5.52%	5.85%	5.90%
Information ratio	–	0.12	0.17	0.08	0.22	0.30	0.09

Notes: The table reports performance and risk measures for macro dedicated portfolios. The analysis was done using daily returns in USD, from 19 June 1970 to 30 June 2022. The annualised returns are computed as geometric average, whilst the statistical tests of differences in Sharpe ratio and relative returns use arithmetic average. The p values of the differences in Sharpe ratios are computed following the methodology proposed by Ledoit and Wolf (2008). Statistical significance of the differences in relative returns and Sharpe ratios is indicated by a single asterisk (*) for results that are significant on a 5% basis and double asterisks (**) for results that are significant on a 1% basis.

Table 7. Unconditional Performance of Inflation Targeting Portfolios

United States 31 March 2002 to 30 June 2022	Cap-Weighted	Inflation (+)	Inflation (−)
Absolute return	7.88%	6.31%	9.42%
Volatility	19.65%	26.12%	18.61%
Sharpe ratio	0.33	0.19	0.44
p value of the differences in Sharpe ratio	–	16.3%	17.3%
Relative return	–	−1.56%	1.55%
p value of relative return	–	89.1%	34.6%
Tracking error	–	10.37%	5.56%
Information ratio	–	−0.15	0.28

Notes: The table reports performance and risk measures for macro dedicated portfolios. The analysis was done using daily returns in USD, from 31 March 2002 to 30 June 2022. The annualised returns are computed as geometric average, whilst the statistical tests of differences in Sharpe Ratio and relative returns use arithmetic average. The p-values of the differences in Sharpe ratios are computed following the methodology proposed by Ledoit and Wolf (2008). Statistical significance of the differences in relative returns and Sharpe ratios is indicated by a single asterisk (*) for results that are significant on a 5% basis, and double asterisks (**) for results that are significant on a 1% basis.

Table 8. Factor Exposure Analysis of Macro Dedicated Portfolios

United States 19 June 1970^a–30 June 2022	Short Rates (+)	Short Rates (−)	Term Spread (+)	Term Spread (−)	Credit Spread (+)	Credit Spread (−)	Inflation (+)	Inflation (−)
Unexplained	1.14%	1.53%	1.45%	1.36%	2.00%*	1.13%	1.54%	0.37%
Market	1.05**	0.87**	0.94**	0.99**	0.88**	1.04**	1.06**	0.99**
Size	0.15**	0.37**	0.32**	0.15**	0.24**	0.19**	0.27**	0.17**
Value	0.08	0.06	0.26**	−0.05	0.19**	0.07	0.37**	−0.10**
Momentum	0.05**	−0.10**	0.07**	−0.10**	−0.05*	0.06**	0.14**	−0.12**
Volatility	−0.01	−0.05**	−0.28**	0.30**	0.00	−0.11**	−0.16**	0.22**
Profitability	−0.32**	0.15**	−0.12**	−0.14**	0.20**	−0.30**	−0.44**	0.12**
Investment	−0.08*	−0.01	0.15**	−0.24**	0.04	−0.20**	−0.10*	−0.10**
R-squared	90.95%	85.87%	89.79%	88.22%	87.24%	90.96%	92.26%	93.26%

Notes: The table reports factor exposures of macro dedicated portfolios. The analysis was done using weekly returns in USD, from 19 June 1970 to 30 June 2022. ^aThe only exception is portfolios targeting exposure to breakeven inflation, for which the analysis starts in 31 March 2002. Factor intensity is the sum of all factor exposures except for the market factor. The unexplained returns are annualised. Statistical significance is indicated by a single asterisk (*) for results that are significant on a 5% basis and double asterisks (**) for results that are significant on a 1% basis.

Table 9. Performance of Macro Dedicated Portfolios Net of Transaction Costs

31 December 1977–30 June 2022	Broad U.S. Cap-Weighted	Short Rates		Term Spread		Credit Spread
		(+)	(−)	(+)	(−)	(+)	(−)
Gross annual return	11.41%	12.09%	11.84%	11.63%	12.53%	13.13%	11.60%
Net annual return	11.39%	11.79%	11.60%	11.31%	12.21%	12.85%	11.26%
Annual transaction cost	1 bps	30 bps	24 bps	32 bps	32 bps	27 bps	34 bps
Annual turnover	2.90%	51.77%	42.67%	53.49%	52.79%	48.45%	56.22%
Net Sharpe ratio	0.40	0.40	0.43	0.37	0.49	0.53*	0.37
p value	–	80.6%	63.2%	43.7%	13.3%	2.6%	38.1%
Net multifactor alpha	0.00%	0.80%	0.81%	0.64%	1.18%	1.84%*	0.43%
p value	–	36.3%	39.3%	48.0%	19.1%	3.5%	62.1%

Notes: Transaction cost is estimated as a difference in gross annualised performance and annualised performance net of transaction costs. The stock-level transaction costs are estimated from daily prices. Prior to 1993, we use the methodology proposed by Corwin and Schultz (2012), whilst after 1993, we rely on the methodology of Chung and Zhang (2014). The reported turnover is a one-way turnover, computed quarterly and annualised by multiplied the average by 4. The p values of the differences in Sharpe ratios are computed following the methodology proposed by Ledoit and Wolf (2008). The analysis was done from 31 December 1977 to 30 June 2022. The analysis starts only in 1977 due to data quality and availability of intra-day high and low prices. Statistical significance of the differences in net Sharpe ratios and net multifactor alphas is indicated by a single asterisk (*) for results that are significant on a 5% basis and double asterisks (**) for results that are significant on a 1% basis.

Table 10. Performance of Inflation Portfolios Net of Transaction Costs

31 March 2002–31 December 2019	Broad U.S. Cap-Weighted	Breakeven Inflation
		(+)	(−)
Gross annual return	8.20%	7.71%	9.12%
Net annual return	8.20%	7.56%	8.99%
Annual Transaction cost	<1 bps	14 bps	13 bps
Annual turnover	3.50%	47.58%	43.29%
Net Sharpe ratio	0.36	0.25	0.42
p value	–	33.8%	32.6%
Net multifactor alpha	0.00%	1.41%	0.25%
p value	–	35.6%	80.2%

Notes: Transaction cost is estimated as a difference in gross annualised performance and annualised performance net of transaction costs. The stock-level transaction costs are estimated from daily prices following the methodology proposed by Chung and Zhang (2014). Statistical significance of the differences in net Sharpe ratios and net multifactor alphas is indicated by a single asterisk (*) for results that are significant on a 5% basis and double asterisks (**) for results that are significant on a 1% basis.

Newey, W. K., and K. D. West. 1987. “A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.” Econometrica 55 (3): 703–708. doi:10.2307/1913610.

 Web of Science ®Google Scholar

Andrews, D. W. K. 1991. “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation.” Econometrica: Journal of the Econometric Society 59 (3): 817–858. doi:10.2307/2938229.

 Web of Science ®Google Scholar

Ledoit, O., and M. Wolf. 2008. “ Robust Performance Hypothesis Testing With the Sharpe Ratio.” Journal of Empirical Finance 15 (5): 850–859. doi:10.1016/j.jempfin.2008.03.002.

 Web of Science ®Google Scholar

Corwin, Shane A., and Paul Schultz. 2012. “A Simple Way to Estimate Bid‐Ask Spreads from Daily High and Low Prices.” The Journal of Finance 67 (2): 719–760. doi:10.1111/j.1540-6261.2012.01729.x.

 Web of Science ®Google Scholar

Chung, Kee H., and Hao Zhang. 2014. “A Simple Approximation of Intraday Spreads Using Daily Data.” Journal of Financial Markets 17 : 94–120. doi:10.1016/j.finmar.2013.02.004.

 Web of Science ®Google Scholar