```
library(tidyverse)
library(lubridate)
library(RSQLite)
```

# Parametric Portfolio Policies

In this chapter, we apply different portfolio performance measures to evaluate and compare portfolio allocation strategies. For this purpose, we introduce a direct way to estimate optimal portfolio weights for large-scale cross-sectional applications. More precisely, the approach of Brandt, Santa-Clara, and Valkanov (2009) proposes to parametrize the optimal portfolio weights as a function of stock characteristics instead of estimating the stock’s expected return, variance, and covariances with other stocks in a prior step. We choose weights as a function of the characteristics, which maximize the expected utility of the investor. This approach is feasible for large portfolio dimensions (such as the entire CRSP universe) and has been proposed by Brandt, Santa-Clara, and Valkanov (2009). See the review paper Brandt (2010) for an excellent treatment of related portfolio choice methods.

The current chapter relies on the following set of packages:

## Data Preparation

To get started, we load the monthly CRSP file, which forms our investment universe. We load the data from our `SQLite`

-database introduced in Chapters 2-4.

```
<- dbConnect(
tidy_finance SQLite(), "data/tidy_finance.sqlite",
extended_types = TRUE
)
<- tbl(tidy_finance, "crsp_monthly") |>
crsp_monthly collect()
```

To evaluate the performance of portfolios, we further use monthly market returns as a benchmark to compute CAPM alphas.

```
<- tbl(tidy_finance, "factors_ff_monthly") |>
factors_ff_monthly collect()
```

Next, we retrieve some stock characteristics that have been shown to have an effect on the expected returns or expected variances (or even higher moments) of the return distribution. In particular, we record the lagged one-year return momentum (`momentum_lag`

), defined as the compounded return between months \(t-13\) and \(t-2\) for each firm. In finance, momentum is the empirically observed tendency for rising asset prices to rise further, and falling prices to keep falling (Jegadeesh and Titman 1993). The second characteristic is the firm’s market equity (`size_lag`

), defined as the log of the price per share times the number of shares outstanding (Banz 1981). To construct the correct lagged values, we use the approach introduced in Chapter 3.

```
<- crsp_monthly |>
crsp_monthly_lags transmute(permno,
month_13 = month %m+% months(13),
mktcap
)
<- crsp_monthly |>
crsp_monthly inner_join(crsp_monthly_lags,
by = c("permno", "month" = "month_13"),
suffix = c("", "_13")
)
<- crsp_monthly |>
data_portfolios mutate(
momentum_lag = mktcap_lag / mktcap_13,
size_lag = log(mktcap_lag)
|>
) drop_na(contains("lag"))
```

## Parametric Portfolio Policies

The basic idea of parametric portfolio weights is as follows. Suppose that at each date \(t\) we have \(N_t\) stocks in the investment universe, where each stock \(i\) has a return of \(r_{i, t+1}\) and is associated with a vector of firm characteristics \(x_{i, t}\) such as time-series momentum or the market capitalization. The investor’s problem is to choose portfolio weights \(w_{i,t}\) to maximize the expected utility of the portfolio return: \[\begin{aligned} \max_{w} E_t\left(u(r_{p, t+1})\right) = E_t\left[u\left(\sum\limits_{i=1}^{N_t}w_{i,t}r_{i,t+1}\right)\right] \end{aligned}\] where \(u(\cdot)\) denotes the utility function.

Where do the stock characteristics show up? We parameterize the optimal portfolio weights as a function of the stock characteristic \(x_{i,t}\) with the following linear specification for the portfolio weights: \[w_{i,t} = \bar{w}_{i,t} + \frac{1}{N_t}\theta'\hat{x}_{i,t},\] where \(\bar{w}_{i,t}\) is a stock’s weight in a benchmark portfolio (we use the value-weighted or naive portfolio in the application below), \(\theta\) is a vector of coefficients which we are going to estimate, and \(\hat{x}_{i,t}\) are the characteristics of stock \(i\), cross-sectionally standardized to have zero mean and unit standard deviation.

Intuitively, the portfolio strategy is a form of active portfolio management relative to a performance benchmark. Deviations from the benchmark portfolio are derived from the individual stock characteristics. Note that by construction the weights sum up to one as \(\sum_{i=1}^{N_t}\hat{x}_{i,t} = 0\) due to the standardization. Moreover, the coefficients are constant across assets and over time. The implicit assumption is that the characteristics fully capture all aspects of the joint distribution of returns that are relevant for forming optimal portfolios.

We first implement cross-sectional standardization for the entire CRSP universe. We also keep track of (lagged) relative market capitalization `relative_mktcap`

, which will represent the value-weighted benchmark portfolio, while `n`

denotes the number of traded assets \(N_t\), which we use to construct the naive portfolio benchmark.

```
<- data_portfolios |>
data_portfolios group_by(month) |>
mutate(
n = n(),
relative_mktcap = mktcap_lag / sum(mktcap_lag),
across(contains("lag"), ~ (. - mean(.)) / sd(.)),
|>
) ungroup() |>
select(-mktcap_lag, -altprc)
```

## Computing Portfolio Weights

Next, we move on to identify optimal choices of \(\theta\). We rewrite the optimization problem together with the weight parametrization and can then estimate \(\theta\) to maximize the objective function based on our sample \[\begin{aligned}
E_t\left(u(r_{p, t+1})\right) = \frac{1}{T}\sum\limits_{t=0}^{T-1}u\left(\sum\limits_{i=1}^{N_t}\left(\bar{w}_{i,t} + \frac{1}{N_t}\theta'\hat{x}_{i,t}\right)r_{i,t+1}\right).
\end{aligned}\] The allocation strategy is straightforward because the number of parameters to estimate is small. Instead of a tedious specification of the \(N_t\) dimensional vector of expected returns and the \(N_t(N_t+1)/2\) free elements of the covariance matrix, all we need to focus on in our application is the vector \(\theta\). \(\theta\) contains only two elements in our application - the relative deviation from the benchmark due to *size* and *momentum*.

To get a feeling for the performance of such an allocation strategy, we start with an arbitrary initial vector \(\theta_0\). The next step is to choose \(\theta\) optimally to maximize the objective function. We automatically detect the number of parameters by counting the number of columns with lagged values.

```
<- sum(str_detect(
n_parameters colnames(data_portfolios), "lag"
))
<- rep(1.5, n_parameters)
theta
names(theta) <- colnames(data_portfolios)[str_detect(
colnames(data_portfolios), "lag"
)]
```

The function `compute_portfolio_weights()`

below computes the portfolio weights \(\bar{w}_{i,t} + \frac{1}{N_t}\theta'\hat{x}_{i,t}\) according to our parametrization for a given value \(\theta_0\). Everything happens within a single pipeline. Hence, we provide a short walk-through.

We first compute `characteristic_tilt`

, the tilting values \(\frac{1}{N_t}\theta'\hat{x}_{i, t}\) which resemble the deviation from the benchmark portfolio. Next, we compute the benchmark portfolio `weight_benchmark`

, which can be any reasonable set of portfolio weights. In our case, we choose either the value or equal-weighted allocation. `weight_tilt`

completes the picture and contains the final portfolio weights `weight_tilt = weight_benchmark + characteristic_tilt`

which deviate from the benchmark portfolio depending on the stock characteristics.

The final few lines go a bit further and implement a simple version of a no-short sale constraint. While it is generally not straightforward to ensure portfolio weight constraints via parameterization, we simply normalize the portfolio weights such that they are enforced to be positive. Finally, we make sure that the normalized weights sum up to one again: \[w_{i,t}^+ = \frac{\max(0, w_{i,t})}{\sum_{j=1}^{N_t}\max(0, w_{i,t})}.\]

The following function computes the optimal portfolio weights in the way just described.

```
<- function(theta,
compute_portfolio_weights
data,value_weighting = TRUE,
allow_short_selling = TRUE) {
|>
data group_by(month) |>
bind_cols(
characteristic_tilt = data |>
transmute(across(contains("lag"), ~ . / n)) |>
as.matrix() %*% theta |> as.numeric()
|>
) mutate(
# Definition of benchmark weight
weight_benchmark = case_when(
== TRUE ~ relative_mktcap,
value_weighting == FALSE ~ 1 / n
value_weighting
),# Parametric portfolio weights
weight_tilt = weight_benchmark + characteristic_tilt,
# Short-sell constraint
weight_tilt = case_when(
== TRUE ~ weight_tilt,
allow_short_selling == FALSE ~ pmax(0, weight_tilt)
allow_short_selling
),# Weights sum up to 1
weight_tilt = weight_tilt / sum(weight_tilt)
|>
) ungroup()
}
```

In the next step, we compute the portfolio weights for the arbitrary vector \(\theta_0\). In the example below, we use the value-weighted portfolio as a benchmark and allow negative portfolio weights.

```
<- compute_portfolio_weights(
weights_crsp
theta,
data_portfolios,value_weighting = TRUE,
allow_short_selling = TRUE
)
```

## Portfolio Performance

Are the computed weights optimal in any way? Most likely not, as we picked \(\theta_0\) arbitrarily. To evaluate the performance of an allocation strategy, one can think of many different approaches. In their original paper, Brandt, Santa-Clara, and Valkanov (2009) focus on a simple evaluation of the hypothetical utility of an agent equipped with a power utility function \(u_\gamma(r) = \frac{(1 + r)^{(1-\gamma)}}{1-\gamma}\), where \(\gamma\) is the risk aversion factor.

```
<- function(r, gamma = 5) {
power_utility 1 + r)^(1 - gamma) / (1 - gamma)
( }
```

We want to note that Gehrig, Sögner, and Westerkamp (2020) warn that, in the leading case of constant relative risk aversion (CRRA), strong assumptions on the properties of the returns, the variables used to implement the parametric portfolio policy, and the parameter space are necessary to obtain a well-defined optimization problem.

No doubt, there are many other ways to evaluate a portfolio. The function below provides a summary of all kinds of interesting measures that can be considered relevant. Do we need all these evaluation measures? It depends: the original paper Brandt, Santa-Clara, and Valkanov (2009) only cares about the expected utility to choose \(\theta\). However, if you want to choose optimal values that achieve the highest performance while putting some constraints on your portfolio weights, it is helpful to have everything in one function.

```
<- function(weights_crsp,
evaluate_portfolio full_evaluation = TRUE) {
<- weights_crsp |>
evaluation group_by(month) |>
summarize(
return_tilt = weighted.mean(ret_excess, weight_tilt),
return_benchmark = weighted.mean(ret_excess, weight_benchmark)
|>
) pivot_longer(-month,
values_to = "portfolio_return",
names_to = "model"
|>
) group_by(model) |>
left_join(factors_ff_monthly, by = "month") |>
summarize(tibble(
"Expected utility" = mean(power_utility(portfolio_return)),
"Average return" = 100 * mean(12 * portfolio_return),
"SD return" = 100 * sqrt(12) * sd(portfolio_return),
"Sharpe ratio" = sqrt(12) * mean(portfolio_return) / sd(portfolio_return),
"CAPM alpha" = coefficients(lm(portfolio_return ~ mkt_excess))[1],
"Market beta" = coefficients(lm(portfolio_return ~ mkt_excess))[2]
|>
)) mutate(model = str_remove(model, "return_")) |>
pivot_longer(-model, names_to = "measure") |>
pivot_wider(names_from = model, values_from = value)
if (full_evaluation) {
<- weights_crsp |>
weight_evaluation select(month, contains("weight")) |>
pivot_longer(-month, values_to = "weight", names_to = "model") |>
group_by(model, month) |>
transmute(tibble(
"Absolute weight" = abs(weight),
"Max. weight" = max(weight),
"Min. weight" = min(weight),
"Avg. sum of negative weights" = -sum(weight[weight < 0]),
"Avg. fraction of negative weights" = sum(weight < 0) / n()
|>
)) group_by(model) |>
summarize(across(-month, ~ 100 * mean(.))) |>
mutate(model = str_remove(model, "weight_")) |>
pivot_longer(-model, names_to = "measure") |>
pivot_wider(names_from = model, values_from = value)
<- bind_rows(evaluation, weight_evaluation)
evaluation
}return(evaluation)
}
```

Let us take a look at the different portfolio strategies and evaluation measures.

```
evaluate_portfolio(weights_crsp) |>
print(n = Inf)
```

```
# A tibble: 11 × 3
measure benchmark tilt
<chr> <dbl> <dbl>
1 Expected utility -0.249 -0.261
2 Average return 7.08 0.166
3 SD return 15.3 20.9
4 Sharpe ratio 0.463 0.00793
5 CAPM alpha 0.000129 -0.00538
6 Market beta 0.992 0.950
7 Absolute weight 0.0249 0.0637
8 Max. weight 3.55 3.68
9 Min. weight 0.0000277 -0.145
10 Avg. sum of negative weights 0 77.9
11 Avg. fraction of negative weights 0 49.5
```

The value-weighted portfolio delivers an annualized return of more than 6 percent and clearly outperforms the tilted portfolio, irrespective of whether we evaluate expected utility, the Sharpe ratio, or the CAPM alpha. We can conclude the market beta is close to one for both strategies (naturally almost identically 1 for the value-weighted benchmark portfolio). When it comes to the distribution of the portfolio weights, we see that the benchmark portfolio weight takes less extreme positions (lower average absolute weights and lower maximum weight). By definition, the value-weighted benchmark does not take any negative positions, while the tilted portfolio also takes short positions.

## Optimal Parameter Choice

Next, we move to a choice of \(\theta\) that actually aims to improve some (or all) of the performance measures. We first define a helper function `compute_objective_function()`

, which we then pass to an optimizer.

```
<- function(theta,
compute_objective_function
data,objective_measure = "Expected utility",
value_weighting = TRUE,
allow_short_selling = TRUE) {
<- compute_portfolio_weights(
processed_data
theta,
data,
value_weighting,
allow_short_selling
)
<- evaluate_portfolio(processed_data,
objective_function full_evaluation = FALSE
|>
) filter(measure == objective_measure) |>
pull(tilt)
return(-objective_function)
}
```

You may wonder why we return the negative value of the objective function. This is simply due to the common convention for optimization procedures to search for minima as a default. By minimizing the negative value of the objective function, we get the maximum value as a result. In its most basic form, R optimization relies on the function `optim()`

. As main inputs, the function requires an initial guess of the parameters and the objective function to minimize. Now, we are fully equipped to compute the optimal values of \(\hat\theta\), which maximize the hypothetical expected utility of the investor.

```
<- optim(
optimal_theta par = theta,
compute_objective_function,objective_measure = "Expected utility",
data = data_portfolios,
value_weighting = TRUE,
allow_short_selling = TRUE
)
$par optimal_theta
```

```
momentum_lag size_lag
0.31 -1.99
```

The resulting values of \(\hat\theta\) are easy to interpret: intuitively, expected utility increases by tilting weights from the value-weighted portfolio toward smaller stocks (negative coefficient for size) and toward past winners (positive value for momentum). Both findings are in line with the well-documented size effect (Banz 1981) and the momentum anomaly (Jegadeesh and Titman 1993).

## More Model Specifications

How does the portfolio perform for different model specifications? For this purpose, we compute the performance of a number of different modeling choices based on the entire CRSP sample. The next code chunk performs all the heavy lifting.

```
<- expand_grid(
full_model_grid value_weighting = c(TRUE, FALSE),
allow_short_selling = c(TRUE, FALSE),
data = list(data_portfolios)
|>
) mutate(optimal_theta = pmap(
.l = list(
data,
value_weighting,
allow_short_selling
),.f = ~ optim(
par = theta,
compute_objective_function,data = ..1,
objective_measure = "Expected utility",
value_weighting = ..2,
allow_short_selling = ..3
$par
) ))
```

Finally, we can compare the results. The table below shows summary statistics for all possible combinations: equal- or value-weighted benchmark portfolio, with or without short-selling constraints, and tilted toward maximizing expected utility.

```
<- full_model_grid |>
performance_table mutate(
processed_data = pmap(
.l = list(
optimal_theta,
data,
value_weighting,
allow_short_selling
),.f = ~ compute_portfolio_weights(..1, ..2, ..3, ..4)
),portfolio_evaluation = map(processed_data,
evaluate_portfolio,full_evaluation = TRUE
)|>
) select(
value_weighting,
allow_short_selling,
portfolio_evaluation|>
) unnest(portfolio_evaluation)
|>
performance_table rename(
" " = benchmark,
Optimal = tilt
|>
) mutate(
value_weighting = case_when(
== TRUE ~ "VW",
value_weighting == FALSE ~ "EW"
value_weighting
),allow_short_selling = case_when(
== TRUE ~ "",
allow_short_selling == FALSE ~ "(no s.)"
allow_short_selling
)|>
) pivot_wider(
names_from = value_weighting:allow_short_selling,
values_from = " ":Optimal,
names_glue = "{value_weighting} {allow_short_selling} {.value} "
|>
) select(
measure,`EW `,
`VW `,
sort(contains("Optimal"))
|>
) print(n = 11)
```

```
# A tibble: 11 × 7
measure `EW ` `VW ` VW Op…¹ VW (no…² EW Op…³ EW (no…⁴
<chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 Expected ut… -0.250 -2.49e-1 -0.246 -0.247 -0.250 -2.50e-1
2 Average ret… 10.7 7.08e+0 14.9 13.6 13.1 8.09e+0
3 SD return 20.3 1.53e+1 20.5 19.5 22.5 1.70e+1
4 Sharpe ratio 0.525 4.63e-1 0.727 0.698 0.584 4.76e-1
5 CAPM alpha 0.00231 1.29e-4 0.00659 0.00531 0.00437 4.76e-4
6 Market beta 1.13 9.92e-1 1.01 1.04 1.13 1.08e+0
7 Absolute we… 0.0249 2.49e-2 0.0381 0.0249 0.0260 2.49e-2
8 Max. weight 0.0249 3.55e+0 3.37 2.69 0.157 2.21e-1
9 Min. weight 0.0249 2.77e-5 -0.0352 0 -0.0331 0
10 Avg. sum of… 0 0 27.4 0 2.27 0
11 Avg. fracti… 0 0 38.6 0 7.40 0
# … with abbreviated variable names ¹`VW Optimal `,
# ²`VW (no s.) Optimal `, ³`EW Optimal `, ⁴`EW (no s.) Optimal `
```

The results indicate that the average annualized Sharpe ratio of the equal-weighted portfolio exceeds the Sharpe ratio of the value-weighted benchmark portfolio. Nevertheless, starting with the weighted value portfolio as a benchmark and tilting optimally with respect to momentum and small stocks yields the highest Sharpe ratio across all specifications. Finally, imposing no short-sale constraints does not improve the performance of the portfolios in our application.

## Exercises

- How do the estimated parameters \(\hat\theta\) and the portfolio performance change if your objective is to maximize the Sharpe ratio instead of the hypothetical expected utility?
- The code above is very flexible in the sense that you can easily add new firm characteristics. Construct a new characteristic of your choice and evaluate the corresponding coefficient \(\hat\theta_i\).
- Tweak the function
`optimal_theta()`

such that you can impose additional performance constraints in order to determine \(\hat\theta\), which maximizes expected utility under the constraint that the market beta is below 1. - Does the portfolio performance resemble a realistic out-of-sample backtesting procedure? Verify the robustness of the results by first estimating \(\hat\theta\) based on
*past data*only. Then, use more recent periods to evaluate the actual portfolio performance. - By formulating the portfolio problem as a statistical estimation problem, you can easily obtain standard errors for the coefficients of the weight function. Brandt, Santa-Clara, and Valkanov (2009) provide the relevant derivations in their paper in Equation (10). Implement a small function that computes standard errors for \(\hat\theta\).

## References

*Journal of Financial Economics*9 (1): 3–18. https://doi.org/10.1016/0304-405X(81)90018-0.

*Handbook of Financial Econometrics: Tools and Techniques*, edited by Yacine Ait-Sahalia and Lars Peter Hansen, 1:269–336. Handbooks in Finance. North-Holland. https://doi.org/10.1016/B978-0-444-50897-3.50008-0.

*Review of Financial Studies*22 (9): 3411–47. https://doi.org/10.1093/rfs/hhp003.

*Working Paper*. http://dx.doi.org/10.2139/ssrn.3081100.

*The Journal of Finance*48 (1): 65–91. https://doi.org/10.1111/j.1540-6261.1993.tb04702.x.