Tidy analysis of stock and portfolio return performance with

`PerformanceAnalytics`

Financial asset (individual stocks, securities, etc) and portfolio
(groups of stocks, securities, etc) performance analysis is a deep field
with a wide range of theories and methods for analyzing risk versus
reward. The `PerformanceAnalytics`

package consolidates
functions to compute many of the most widely used performance metrics.
`tidquant`

integrates this functionality so it can be used at
scale using the split, apply, combine framework within the
`tidyverse`

. Two primary functions integrate the performance
analysis functionality:

`tq_performance`

implements the performance analysis functions in a tidy way, enabling scaling analysis using the split, apply, combine framework.`tq_portfolio`

provides a useful tool set for aggregating a group of individual asset returns into one or many portfolios.

This vignette aims to cover three aspects of performance analysis:

The general workflow to go from start to finish on both an asset and a portfolio level

Some of the available techniques to implement once the workflow is implemented

How to customize

`tq_portfolio`

and`tq_performance`

using the`...`

parameter

An important concept is that performance analysis is based on the
statistical properties of **returns** (not prices). As a
result, this package uses inputs of **time-based returns as
opposed to stock prices**. The arguments change to
`Ra`

for the asset returns and `Rb`

for the
baseline returns. We’ll go over how to get returns in the Workflow section.

Another important concept is the **baseline**. The
baseline is what you are measuring performance against. A baseline can
be anything, but in many cases it’s a representative average of how an
investment might perform with little or no effort. Often indexes such as
the S&P500 are used for general market performance. Other times more
specific Exchange Traded Funds (ETFs) are used such as the SPDR
Technology ETF (XLK). The important concept here is that you measure the
asset performance (`Ra`

) against the baseline
(`Rb`

).

Now for a quick tutorial to show off the
`PerformanceAnalytics`

package integration.

One of the most widely used risk to return metrics is the Capital Asset Pricing Model (CAPM). According to Investopedia:

The capital asset pricing model (CAPM) is a model that describes the relationship between systematic risk and expected return for assets, particularly stocks. CAPM is widely used throughout finance for the pricing of risky securities, generating expected returns for assets given the risk of those assets and calculating costs of capital.

We’ll use the `PerformanceAnalytics`

function,
`table.CAPM`

, to evaluate the returns of several technology
stocks against the SPDR Technology ETF (XLK).

First, load the `tidyquant`

package.

```
library(tidyverse)
library(tidyquant)
```

Second, get the stock returns for the stocks we wish to evaluate. We
use `tq_get`

to get stock prices from Yahoo Finance,
`group_by`

to group the stock prices related to each symbol,
and `tq_transmute`

to retrieve period returns in a monthly
periodicity using the “adjusted” stock prices (adjusted for stock
splits, which can throw off returns, affecting the performance
analysis). Review the output and see that there are three groups of
symbols indicating the data has been grouped appropriately.

```
<- c("AAPL", "GOOG", "NFLX") %>%
Ra tq_get(get = "stock.prices",
from = "2010-01-01",
to = "2015-12-31") %>%
group_by(symbol) %>%
tq_transmute(select = adjusted,
mutate_fun = periodReturn,
period = "monthly",
col_rename = "Ra")
Ra
```

```
## # A tibble: 216 × 3
## # Groups: symbol [3]
## symbol date Ra
## <chr> <date> <dbl>
## 1 AAPL 2010-01-29 -0.103
## 2 AAPL 2010-02-26 0.0654
## 3 AAPL 2010-03-31 0.148
## 4 AAPL 2010-04-30 0.111
## 5 AAPL 2010-05-28 -0.0161
## 6 AAPL 2010-06-30 -0.0208
## 7 AAPL 2010-07-30 0.0227
## 8 AAPL 2010-08-31 -0.0550
## 9 AAPL 2010-09-30 0.167
## 10 AAPL 2010-10-29 0.0607
## # … with 206 more rows
```

Next, we get the baseline prices. We’ll use the XLK. Note that there is no need to group because we are just getting one data set.

```
<- "XLK" %>%
Rb tq_get(get = "stock.prices",
from = "2010-01-01",
to = "2015-12-31") %>%
tq_transmute(select = adjusted,
mutate_fun = periodReturn,
period = "monthly",
col_rename = "Rb")
Rb
```

```
## # A tibble: 72 × 2
## date Rb
## <date> <dbl>
## 1 2010-01-29 -0.0993
## 2 2010-02-26 0.0348
## 3 2010-03-31 0.0684
## 4 2010-04-30 0.0126
## 5 2010-05-28 -0.0748
## 6 2010-06-30 -0.0540
## 7 2010-07-30 0.0745
## 8 2010-08-31 -0.0561
## 9 2010-09-30 0.117
## 10 2010-10-29 0.0578
## # … with 62 more rows
```

Now, we combine the two data sets using the “date” field using
`left_join`

from the `dplyr`

package. Review the
results and see that we still have three groups of returns, and columns
“Ra” and “Rb” are side-by-side.

```
<- left_join(Ra, Rb, by = c("date" = "date"))
RaRb RaRb
```

```
## # A tibble: 216 × 4
## # Groups: symbol [3]
## symbol date Ra Rb
## <chr> <date> <dbl> <dbl>
## 1 AAPL 2010-01-29 -0.103 -0.0993
## 2 AAPL 2010-02-26 0.0654 0.0348
## 3 AAPL 2010-03-31 0.148 0.0684
## 4 AAPL 2010-04-30 0.111 0.0126
## 5 AAPL 2010-05-28 -0.0161 -0.0748
## 6 AAPL 2010-06-30 -0.0208 -0.0540
## 7 AAPL 2010-07-30 0.0227 0.0745
## 8 AAPL 2010-08-31 -0.0550 -0.0561
## 9 AAPL 2010-09-30 0.167 0.117
## 10 AAPL 2010-10-29 0.0607 0.0578
## # … with 206 more rows
```

Finally, we can retrieve the performance metrics using
`tq_performance()`

. You can use
`tq_performance_fun_options()`

to see the full list of
compatible performance functions.

```
<- RaRb %>%
RaRb_capm tq_performance(Ra = Ra,
Rb = Rb,
performance_fun = table.CAPM)
RaRb_capm
```

```
## # A tibble: 3 × 13
## # Groups: symbol [3]
## symbol ActivePr…¹ Alpha Annua…² Beta `Beta+` `Beta-` Corre…³ Corre…⁴ Infor…⁵
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 AAPL 0.119 0.0089 0.112 1.11 1.04 0.578 0.659 0 0.628
## 2 GOOG 0.034 0.0028 0.034 1.14 1.16 1.39 0.644 0 0.168
## 3 NFLX 0.447 0.053 0.859 0.384 0.0045 -1.52 0.0817 0.495 0.637
## # … with 3 more variables: `R-squared` <dbl>, TrackingError <dbl>,
## # TreynorRatio <dbl>, and abbreviated variable names ¹ActivePremium,
## # ²AnnualizedAlpha, ³Correlation, ⁴`Correlationp-value`, ⁵InformationRatio
```

We can quickly isolate attributes, such as alpha, the measure of growth, and beta, the measure of risk.

`%>% select(symbol, Alpha, Beta) RaRb_capm `

```
## # A tibble: 3 × 3
## # Groups: symbol [3]
## symbol Alpha Beta
## <chr> <dbl> <dbl>
## 1 AAPL 0.0089 1.11
## 2 GOOG 0.0028 1.14
## 3 NFLX 0.053 0.384
```

With `tidyquant`

it’s efficient and easy to get the CAPM
information! And, that’s just one of 129 available functions to analyze
stock and portfolio return performance. Just use
`tq_performance_fun_options()`

to see the full list.

The general workflow is shown in the diagram below. We’ll step through the workflow first with a group of individual assets (stocks) and then with portfolios of stocks.

Individual assets are the simplest form of analysis because there is
no portfolio aggregation (Step 3A). We’ll re-do the “Quick Example” this
time getting the **Sharpe Ratio**, a measure of
reward-to-risk.

Before we get started let’s find the performance function we want to
use from `PerformanceAnalytics`

. Searching
`tq_performance_fun_options`

, we can see that
`SharpeRatio`

is available. Type `?SharpeRatio`

,
and we can see that the arguments are:

`args(SharpeRatio)`

```
## function (R, Rf = 0, p = 0.95, FUN = c("StdDev", "VaR", "ES"),
## weights = NULL, annualize = FALSE, SE = FALSE, SE.control = NULL,
## ...)
## NULL
```

We can actually skip the baseline path because the function does not
require `Rb`

. The function takes `R`

, which is
passed using `Ra`

in
`tq_performance(Ra, Rb, performance_fun, ...)`

. A little bit
of foresight saves us some work.

Use `tq_get()`

to get stock prices.

```
<- c("AAPL", "GOOG", "NFLX") %>%
stock_prices tq_get(get = "stock.prices",
from = "2010-01-01",
to = "2015-12-31")
stock_prices
```

```
## # A tibble: 4,527 × 8
## symbol date open high low close volume adjusted
## <chr> <date> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 AAPL 2010-01-04 7.62 7.66 7.58 7.64 493729600 6.52
## 2 AAPL 2010-01-05 7.66 7.70 7.62 7.66 601904800 6.53
## 3 AAPL 2010-01-06 7.66 7.69 7.53 7.53 552160000 6.42
## 4 AAPL 2010-01-07 7.56 7.57 7.47 7.52 477131200 6.41
## 5 AAPL 2010-01-08 7.51 7.57 7.47 7.57 447610800 6.45
## 6 AAPL 2010-01-11 7.6 7.61 7.44 7.50 462229600 6.40
## 7 AAPL 2010-01-12 7.47 7.49 7.37 7.42 594459600 6.32
## 8 AAPL 2010-01-13 7.42 7.53 7.29 7.52 605892000 6.41
## 9 AAPL 2010-01-14 7.50 7.52 7.46 7.48 432894000 6.38
## 10 AAPL 2010-01-15 7.53 7.56 7.35 7.35 594067600 6.27
## # … with 4,517 more rows
```

Using the `tidyverse`

split, apply, combine framework, we
can mutate groups of stocks by first “grouping” with
`group_by`

and then applying a mutating function using
`tq_transmute`

. We use the `quantmod`

function
`periodReturn`

as the mutating function. We pass along the
arguments `period = "monthly"`

to return the results in
monthly periodicity. Last, we use the `col_rename`

argument
to rename the output column.

```
<- stock_prices %>%
stock_returns_monthly group_by(symbol) %>%
tq_transmute(select = adjusted,
mutate_fun = periodReturn,
period = "monthly",
col_rename = "Ra")
stock_returns_monthly
```

```
## # A tibble: 216 × 3
## # Groups: symbol [3]
## symbol date Ra
## <chr> <date> <dbl>
## 1 AAPL 2010-01-29 -0.103
## 2 AAPL 2010-02-26 0.0654
## 3 AAPL 2010-03-31 0.148
## 4 AAPL 2010-04-30 0.111
## 5 AAPL 2010-05-28 -0.0161
## 6 AAPL 2010-06-30 -0.0208
## 7 AAPL 2010-07-30 0.0227
## 8 AAPL 2010-08-31 -0.0550
## 9 AAPL 2010-09-30 0.167
## 10 AAPL 2010-10-29 0.0607
## # … with 206 more rows
```

Step 3A can be skipped because we are only interested in the Sharpe
Ratio for *individual stocks* (not a portfolio).

Step 3B can also be skipped because the `SharpeRatio`

function from `PerformanceAnalytics`

does not require a
baseline.

The last step is to apply the `SharpeRatio`

function to
our groups of stock returns. We do this using
`tq_performance()`

with the arguments `Ra = Ra`

,
`Rb = NULL`

(not required), and
`performance_fun = SharpeRatio`

. We can also pass other
arguments of the `SharpeRatio`

function such as
`Rf`

, `p`

, `FUN`

, and
`annualize`

. We will just use the defaults for this
example.

```
%>%
stock_returns_monthly tq_performance(
Ra = Ra,
Rb = NULL,
performance_fun = SharpeRatio
)
```

```
## # A tibble: 3 × 4
## # Groups: symbol [3]
## symbol `ESSharpe(Rf=0%,p=95%)` `StdDevSharpe(Rf=0%,p=95%)` VaRSharpe(Rf=0%,p…¹
## <chr> <dbl> <dbl> <dbl>
## 1 AAPL 0.173 0.292 0.218
## 2 GOOG 0.129 0.203 0.157
## 3 NFLX 0.237 0.284 0.272
## # … with abbreviated variable name ¹`VaRSharpe(Rf=0%,p=95%)`
```

Now we have the Sharpe Ratio for each of the three stocks. What if we want to adjust the parameters of the function? We can just add on the arguments of the underlying function.

```
%>%
stock_returns_monthly tq_performance(
Ra = Ra,
Rb = NULL,
performance_fun = SharpeRatio,
Rf = 0.03 / 12,
p = 0.99
)
```

```
## # A tibble: 3 × 4
## # Groups: symbol [3]
## symbol `ESSharpe(Rf=0.2%,p=99%)` `StdDevSharpe(Rf=0.2%,p=99%)` VaRSharpe(Rf=…¹
## <chr> <dbl> <dbl> <dbl>
## 1 AAPL 0.116 0.258 0.134
## 2 GOOG 0.0826 0.170 0.0998
## 3 NFLX 0.115 0.272 0.142
## # … with abbreviated variable name ¹`VaRSharpe(Rf=0.2%,p=99%)`
```

Portfolios are slightly more complicated because we are now dealing
with groups of assets versus individual stocks, and we need to aggregate
weighted returns. Fortunately, this is only one extra step with
`tidyquant`

using `tq_portfolio()`

.

Let’s recreate the CAPM analysis in the “Quick Example” this time comparing a portfolio of technology stocks to the SPDR Technology ETF (XLK).

This is the same as what we did previously to get the monthly returns
for groups of individual stock prices. We use the split, apply, combine
framework using the workflow of `tq_get`

,
`group_by`

, and `tq_transmute`

.

```
<- c("AAPL", "GOOG", "NFLX") %>%
stock_returns_monthly tq_get(get = "stock.prices",
from = "2010-01-01",
to = "2015-12-31") %>%
group_by(symbol) %>%
tq_transmute(select = adjusted,
mutate_fun = periodReturn,
period = "monthly",
col_rename = "Ra")
stock_returns_monthly
```

```
## # A tibble: 216 × 3
## # Groups: symbol [3]
## symbol date Ra
## <chr> <date> <dbl>
## 1 AAPL 2010-01-29 -0.103
## 2 AAPL 2010-02-26 0.0654
## 3 AAPL 2010-03-31 0.148
## 4 AAPL 2010-04-30 0.111
## 5 AAPL 2010-05-28 -0.0161
## 6 AAPL 2010-06-30 -0.0208
## 7 AAPL 2010-07-30 0.0227
## 8 AAPL 2010-08-31 -0.0550
## 9 AAPL 2010-09-30 0.167
## 10 AAPL 2010-10-29 0.0607
## # … with 206 more rows
```

This was also done previously.

```
<- "XLK" %>%
baseline_returns_monthly tq_get(get = "stock.prices",
from = "2010-01-01",
to = "2015-12-31") %>%
tq_transmute(select = adjusted,
mutate_fun = periodReturn,
period = "monthly",
col_rename = "Rb")
baseline_returns_monthly
```

```
## # A tibble: 72 × 2
## date Rb
## <date> <dbl>
## 1 2010-01-29 -0.0993
## 2 2010-02-26 0.0348
## 3 2010-03-31 0.0684
## 4 2010-04-30 0.0126
## 5 2010-05-28 -0.0748
## 6 2010-06-30 -0.0540
## 7 2010-07-30 0.0745
## 8 2010-08-31 -0.0561
## 9 2010-09-30 0.117
## 10 2010-10-29 0.0578
## # … with 62 more rows
```

The `tidyquant`

function, `tq_portfolio()`

aggregates a group of individual assets into a single return using a
weighted composition of the underlying assets. To do this we need to
first develop portfolio weights. There are two ways to do this for a
single portfolio:

- Supplying a vector of weights
- Supplying a two column tidy data frame (tibble) with stock symbols in the first column and weights to map in the second.

Suppose we want to split our portfolio evenly between AAPL and NFLX. We’ll show this using both methods.

We’ll use the weight vector, `c(0.5, 0, 0.5)`

. Two
important aspects to supplying a numeric vector of weights: First,
notice that the length (3) is equal to the number of assets (3). This is
a requirement. Second, notice that the sum of the weighting vector is
equal to 1. This is not “required”, but is best practice. If the sum is
not 1, the weights will be distributed accordingly by scaling the vector
to 1, and a warning message will appear.

```
<- c(0.5, 0.0, 0.5)
wts <- stock_returns_monthly %>%
portfolio_returns_monthly tq_portfolio(assets_col = symbol,
returns_col = Ra,
weights = wts,
col_rename = "Ra")
portfolio_returns_monthly
```

```
## # A tibble: 72 × 2
## date Ra
## <date> <dbl>
## 1 2010-01-29 0.0307
## 2 2010-02-26 0.0629
## 3 2010-03-31 0.130
## 4 2010-04-30 0.239
## 5 2010-05-28 0.0682
## 6 2010-06-30 -0.0219
## 7 2010-07-30 -0.0272
## 8 2010-08-31 0.116
## 9 2010-09-30 0.251
## 10 2010-10-29 0.0674
## # … with 62 more rows
```

We now have an aggregated portfolio that is a 50/50 blend of AAPL and NFLX.

You may be asking why didn’t we use GOOG? **The important thing
to understand is that all of the assets from the asset returns don’t
need to be used when creating the portfolio!** This enables us to
scale individual stock returns and then vary weights to optimize the
portfolio (this will be a further subject that we address in the
future!)

A possibly more useful method of aggregating returns is using a tibble of symbols and weights that are mapped to the portfolio. We’ll recreate the previous portfolio example using mapped weights.

```
<- tibble(
wts_map symbols = c("AAPL", "NFLX"),
weights = c(0.5, 0.5)
) wts_map
```

```
## # A tibble: 2 × 2
## symbols weights
## <chr> <dbl>
## 1 AAPL 0.5
## 2 NFLX 0.5
```

Next, supply this two column tibble, with symbols in the first column
and weights in the second, to the `weights`

argument in
`tq_performance()`

.

```
%>%
stock_returns_monthly tq_portfolio(assets_col = symbol,
returns_col = Ra,
weights = wts_map,
col_rename = "Ra_using_wts_map")
```

```
## # A tibble: 72 × 2
## date Ra_using_wts_map
## <date> <dbl>
## 1 2010-01-29 0.0307
## 2 2010-02-26 0.0629
## 3 2010-03-31 0.130
## 4 2010-04-30 0.239
## 5 2010-05-28 0.0682
## 6 2010-06-30 -0.0219
## 7 2010-07-30 -0.0272
## 8 2010-08-31 0.116
## 9 2010-09-30 0.251
## 10 2010-10-29 0.0674
## # … with 62 more rows
```

The aggregated returns are exactly the same. The advantage with this method is that not all symbols need to be specified. Any symbol not specified by default gets a weight of zero.

Now, imagine if you had an entire index, such as the Russell 2000, of 2000 individual stock returns in a nice tidy data frame. It would be very easy to adjust portfolios and compute blended returns, and you only need to supply the symbols that you want to blend. All other symbols default to zero!

Now that we have the aggregated portfolio returns (“Ra”) from Step 3A and the baseline returns (“Rb”) from Step 2B, we can merge to get our consolidated table of asset and baseline returns. Nothing new here.

```
<- left_join(portfolio_returns_monthly,
RaRb_single_portfolio
baseline_returns_monthly,by = "date")
RaRb_single_portfolio
```

```
## # A tibble: 72 × 3
## date Ra Rb
## <date> <dbl> <dbl>
## 1 2010-01-29 0.0307 -0.0993
## 2 2010-02-26 0.0629 0.0348
## 3 2010-03-31 0.130 0.0684
## 4 2010-04-30 0.239 0.0126
## 5 2010-05-28 0.0682 -0.0748
## 6 2010-06-30 -0.0219 -0.0540
## 7 2010-07-30 -0.0272 0.0745
## 8 2010-08-31 0.116 -0.0561
## 9 2010-09-30 0.251 0.117
## 10 2010-10-29 0.0674 0.0578
## # … with 62 more rows
```

The CAPM table is computed with the function `table.CAPM`

from `PerformanceAnalytics`

. We just perform the same task
that we performed in the “Quick Example”.

```
%>%
RaRb_single_portfolio tq_performance(Ra = Ra, Rb = Rb, performance_fun = table.CAPM)
```

```
## # A tibble: 1 × 12
## ActiveP…¹ Alpha Annua…² Beta `Beta+` `Beta-` Corre…³ Corre…⁴ Infor…⁵ R-squ…⁶
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.327 0.0299 0.425 0.754 0.503 -0.243 0.283 0.0162 0.858 0.0799
## # … with 2 more variables: TrackingError <dbl>, TreynorRatio <dbl>, and
## # abbreviated variable names ¹ActivePremium, ²AnnualizedAlpha, ³Correlation,
## # ⁴`Correlationp-value`, ⁵InformationRatio, ⁶`R-squared`
```

Now we have the CAPM performance metrics for a portfolio! While this is cool, it’s cooler to do multiple portfolios. Let’s see how.

Once you understand the process for a single portfolio using Step 3A, Method 2 (aggregating weights by mapping), scaling to multiple portfolios is just building on this concept. Let’s recreate the same example from the “Single Portfolio” Example this time with three portfolios:

- 50% AAPL, 25% GOOG, 25% NFLX
- 25% AAPL, 50% GOOG, 25% NFLX
- 25% AAPL, 25% GOOG, 50% NFLX

First, get individual asset returns grouped by asset, which is the exact same as Steps 1A and 1B from the Single Portfolio example.

```
<- c("AAPL", "GOOG", "NFLX") %>%
stock_returns_monthly tq_get(get = "stock.prices",
from = "2010-01-01",
to = "2015-12-31") %>%
group_by(symbol) %>%
tq_transmute(select = adjusted,
mutate_fun = periodReturn,
period = "monthly",
col_rename = "Ra")
```

Second, get baseline asset returns, which is the exact same as Steps 1B and 2B from the Single Portfolio example.

```
<- "XLK" %>%
baseline_returns_monthly tq_get(get = "stock.prices",
from = "2010-01-01",
to = "2015-12-31") %>%
tq_transmute(select = adjusted,
mutate_fun = periodReturn,
period = "monthly",
col_rename = "Rb")
```

This is where it gets fun. If you picked up on Single Portfolio, Step3A, Method 2 (mapping weights), this is just an extension for multiple portfolios.

First, we need to grow our portfolios. `tidyquant`

has a
handy, albeit simple, function, `tq_repeat_df()`

, for scaling
a single portfolio to many. It takes a data frame, and the number of
repeats, `n`

, and the `index_col_name`

, which adds
a sequential index. Let’s see how it works for our example. We need
three portfolios:

```
<- stock_returns_monthly %>%
stock_returns_monthly_multi tq_repeat_df(n = 3)
stock_returns_monthly_multi
```

```
## # A tibble: 648 × 4
## # Groups: portfolio [3]
## portfolio symbol date Ra
## <int> <chr> <date> <dbl>
## 1 1 AAPL 2010-01-29 -0.103
## 2 1 AAPL 2010-02-26 0.0654
## 3 1 AAPL 2010-03-31 0.148
## 4 1 AAPL 2010-04-30 0.111
## 5 1 AAPL 2010-05-28 -0.0161
## 6 1 AAPL 2010-06-30 -0.0208
## 7 1 AAPL 2010-07-30 0.0227
## 8 1 AAPL 2010-08-31 -0.0550
## 9 1 AAPL 2010-09-30 0.167
## 10 1 AAPL 2010-10-29 0.0607
## # … with 638 more rows
```

Examining the results, we can see that a few things happened:

- The length (number of rows) has tripled. This is the essence of
`tq_repeat_df`

: it grows the data frame length-wise, repeating the data frame`n`

times. In our case,`n = 3`

. - Our data frame, which was grouped by symbol, was ungrouped. This is
needed to prevent
`tq_portfolio`

from blending on the individual stocks.`tq_portfolio`

only works on groups of stocks. - We have a new column, named “portfolio”. The “portfolio” column name
is a key that tells
`tq_portfolio`

that multiple groups exist to analyze. Just note that for multiple portfolio analysis, the “portfolio” column name is required. - We have three groups of portfolios. This is what
`tq_portfolio`

will split, apply (aggregate), then combine on.

Now the tricky part: We need a new table of weights to map on. There’s a few requirements:

- We must supply a three column tibble with the following columns: “portfolio”, asset, and weight in that order.
- The “portfolio” column must be named “portfolio” since this is a key name for mapping.
- The tibble must be grouped by the portfolio column.

Here’s what the weights table should look like for our example:

```
<- c(
weights 0.50, 0.25, 0.25,
0.25, 0.50, 0.25,
0.25, 0.25, 0.50
)<- c("AAPL", "GOOG", "NFLX")
stocks <- tibble(stocks) %>%
weights_table tq_repeat_df(n = 3) %>%
bind_cols(tibble(weights)) %>%
group_by(portfolio)
weights_table
```

```
## # A tibble: 9 × 3
## # Groups: portfolio [3]
## portfolio stocks weights
## <int> <chr> <dbl>
## 1 1 AAPL 0.5
## 2 1 GOOG 0.25
## 3 1 NFLX 0.25
## 4 2 AAPL 0.25
## 5 2 GOOG 0.5
## 6 2 NFLX 0.25
## 7 3 AAPL 0.25
## 8 3 GOOG 0.25
## 9 3 NFLX 0.5
```

Now just pass the the expanded
`stock_returns_monthly_multi`

and the
`weights_table`

to `tq_portfolio`

for portfolio
aggregation.

```
<- stock_returns_monthly_multi %>%
portfolio_returns_monthly_multi tq_portfolio(assets_col = symbol,
returns_col = Ra,
weights = weights_table,
col_rename = "Ra")
portfolio_returns_monthly_multi
```

```
## # A tibble: 216 × 3
## # Groups: portfolio [3]
## portfolio date Ra
## <int> <date> <dbl>
## 1 1 2010-01-29 -0.0489
## 2 1 2010-02-26 0.0482
## 3 1 2010-03-31 0.123
## 4 1 2010-04-30 0.145
## 5 1 2010-05-28 0.0245
## 6 1 2010-06-30 -0.0308
## 7 1 2010-07-30 0.000600
## 8 1 2010-08-31 0.0474
## 9 1 2010-09-30 0.222
## 10 1 2010-10-29 0.0789
## # … with 206 more rows
```

Let’s assess the output. We now have a single, “long” format data
frame of portfolio returns. It has three groups with the aggregated
portfolios blended by mapping the `weight_table`

.

These steps are the exact same as the Single Portfolio example.

First, we merge with the baseline using “date” as the key.

```
<- left_join(portfolio_returns_monthly_multi,
RaRb_multiple_portfolio
baseline_returns_monthly,by = "date")
RaRb_multiple_portfolio
```

```
## # A tibble: 216 × 4
## # Groups: portfolio [3]
## portfolio date Ra Rb
## <int> <date> <dbl> <dbl>
## 1 1 2010-01-29 -0.0489 -0.0993
## 2 1 2010-02-26 0.0482 0.0348
## 3 1 2010-03-31 0.123 0.0684
## 4 1 2010-04-30 0.145 0.0126
## 5 1 2010-05-28 0.0245 -0.0748
## 6 1 2010-06-30 -0.0308 -0.0540
## 7 1 2010-07-30 0.000600 0.0745
## 8 1 2010-08-31 0.0474 -0.0561
## 9 1 2010-09-30 0.222 0.117
## 10 1 2010-10-29 0.0789 0.0578
## # … with 206 more rows
```

Finally, we calculate the performance of each of the portfolios using
`tq_performance`

. Make sure the data frame is grouped on
“portfolio”.

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra, Rb = Rb, performance_fun = table.CAPM)
```

```
## # A tibble: 3 × 13
## # Groups: portfolio [3]
## portfolio Activ…¹ Alpha Annua…² Beta `Beta+` `Beta-` Corre…³ Corre…⁴ Infor…⁵
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 0.231 0.0193 0.258 0.908 0.741 0.312 0.472 0 0.917
## 2 2 0.219 0.0192 0.256 0.886 0.660 0.436 0.438 0.0001 0.809
## 3 3 0.319 0.0308 0.439 0.721 0.394 -0.179 0.252 0.0325 0.774
## # … with 3 more variables: `R-squared` <dbl>, TrackingError <dbl>,
## # TreynorRatio <dbl>, and abbreviated variable names ¹ActivePremium,
## # ²AnnualizedAlpha, ³Correlation, ⁴`Correlationp-value`, ⁵InformationRatio
```

Inspecting the results, we now have a multiple portfolio comparison
of the CAPM table from `PerformanceAnalytics`

. We can do the
same thing with `SharpeRatio`

as well.

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra, Rb = NULL, performance_fun = SharpeRatio)
```

```
## # A tibble: 3 × 4
## # Groups: portfolio [3]
## portfolio `ESSharpe(Rf=0%,p=95%)` `StdDevSharpe(Rf=0%,p=95%)` VaRSharpe(Rf=0…¹
## <int> <dbl> <dbl> <dbl>
## 1 1 0.172 0.355 0.263
## 2 2 0.146 0.334 0.236
## 3 3 0.150 0.317 0.238
## # … with abbreviated variable name ¹`VaRSharpe(Rf=0%,p=95%)`
```

We’ve only scratched the surface of the analysis functions available
through `PerformanceAnalytics`

. The list below includes all
of the compatible functions grouped by function type. The table
functions are the most useful to get a cross section of metrics. We’ll
touch on a few. We’ll also go over `VaR`

and
`SharpeRatio`

as these are very commonly used as performance
measures.

`tq_performance_fun_options()`

```
## $table.funs
## [1] "table.AnnualizedReturns" "table.Arbitrary"
## [3] "table.Autocorrelation" "table.CAPM"
## [5] "table.CaptureRatios" "table.Correlation"
## [7] "table.Distributions" "table.DownsideRisk"
## [9] "table.DownsideRiskRatio" "table.DrawdownsRatio"
## [11] "table.HigherMoments" "table.InformationRatio"
## [13] "table.RollingPeriods" "table.SFM"
## [15] "table.SpecificRisk" "table.Stats"
## [17] "table.TrailingPeriods" "table.UpDownRatios"
## [19] "table.Variability"
##
## $CAPM.funs
## [1] "CAPM.CML" "CAPM.CML.slope" "CAPM.RiskPremium" "CAPM.SML.slope"
## [5] "CAPM.alpha" "CAPM.beta" "CAPM.beta.bear" "CAPM.beta.bull"
## [9] "CAPM.dynamic" "CAPM.epsilon" "CAPM.jensenAlpha" "TimingRatio"
## [13] "MarketTiming"
##
## $SFM.funs
## [1] "SFM.CML" "SFM.CML.slope" "SFM.alpha" "SFM.beta"
## [5] "SFM.dynamic" "SFM.epsilon" "SFM.jensenAlpha"
##
## $descriptive.funs
## [1] "mean" "sd" "min" "max"
## [5] "cor" "mean.geometric" "mean.stderr" "mean.LCL"
## [9] "mean.UCL"
##
## $annualized.funs
## [1] "Return.annualized" "Return.annualized.excess"
## [3] "sd.annualized" "SharpeRatio.annualized"
##
## $VaR.funs
## [1] "VaR" "ES" "ETL" "CDD" "CVaR"
##
## $moment.funs
## [1] "var" "cov" "skewness" "kurtosis"
## [5] "CoVariance" "CoSkewness" "CoSkewnessMatrix" "CoKurtosis"
## [9] "CoKurtosisMatrix" "M3.MM" "M4.MM" "BetaCoVariance"
## [13] "BetaCoSkewness" "BetaCoKurtosis"
##
## $drawdown.funs
## [1] "AverageDrawdown" "AverageLength" "AverageRecovery"
## [4] "DrawdownDeviation" "DrawdownPeak" "maxDrawdown"
##
## $Bacon.risk.funs
## [1] "MeanAbsoluteDeviation" "Frequency" "SharpeRatio"
## [4] "MSquared" "MSquaredExcess" "HurstIndex"
##
## $Bacon.regression.funs
## [1] "CAPM.alpha" "CAPM.beta" "CAPM.epsilon" "CAPM.jensenAlpha"
## [5] "SystematicRisk" "SpecificRisk" "TotalRisk" "TreynorRatio"
## [9] "AppraisalRatio" "FamaBeta" "Selectivity" "NetSelectivity"
##
## $Bacon.relative.risk.funs
## [1] "ActivePremium" "ActiveReturn" "TrackingError" "InformationRatio"
##
## $Bacon.drawdown.funs
## [1] "PainIndex" "PainRatio" "CalmarRatio" "SterlingRatio"
## [5] "BurkeRatio" "MartinRatio" "UlcerIndex"
##
## $Bacon.downside.risk.funs
## [1] "DownsideDeviation" "DownsidePotential" "DownsideFrequency"
## [4] "SemiDeviation" "SemiVariance" "UpsideRisk"
## [7] "UpsidePotentialRatio" "UpsideFrequency" "BernardoLedoitRatio"
## [10] "DRatio" "Omega" "OmegaSharpeRatio"
## [13] "OmegaExcessReturn" "SortinoRatio" "M2Sortino"
## [16] "Kappa" "VolatilitySkewness" "AdjustedSharpeRatio"
## [19] "SkewnessKurtosisRatio" "ProspectRatio"
##
## $misc.funs
## [1] "KellyRatio" "Modigliani" "UpDownRatios"
```

Returns a basic set of statistics that match the period of the data passed in (e.g., monthly returns will get monthly statistics, daily will be daily stats, and so on).

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra, Rb = NULL, performance_fun = table.Stats)
```

```
## # A tibble: 3 × 17
## # Groups: portfolio [3]
## portfolio Arith…¹ Geome…² Kurto…³ LCLMe…⁴ Maximum Median Minimum NAs Obser…⁵
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 0.0293 0.0259 1.14 0.0099 0.222 0.0307 -0.266 0 72
## 2 2 0.029 0.0252 1.65 0.0086 0.227 0.037 -0.3 0 72
## 3 3 0.0388 0.0313 1.81 0.01 0.370 0.046 -0.403 0 72
## # … with 7 more variables: Quartile1 <dbl>, Quartile3 <dbl>, SEMean <dbl>,
## # Skewness <dbl>, Stdev <dbl>, `UCLMean(0.95)` <dbl>, Variance <dbl>, and
## # abbreviated variable names ¹ArithmeticMean, ²GeometricMean, ³Kurtosis,
## # ⁴`LCLMean(0.95)`, ⁵Observations
```

Takes a set of returns and relates them to a benchmark return. Provides a set of measures related to an excess return single factor model, or CAPM.

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra, Rb = Rb, performance_fun = table.CAPM)
```

```
## # A tibble: 3 × 13
## # Groups: portfolio [3]
## portfolio Activ…¹ Alpha Annua…² Beta `Beta+` `Beta-` Corre…³ Corre…⁴ Infor…⁵
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 0.231 0.0193 0.258 0.908 0.741 0.312 0.472 0 0.917
## 2 2 0.219 0.0192 0.256 0.886 0.660 0.436 0.438 0.0001 0.809
## 3 3 0.319 0.0308 0.439 0.721 0.394 -0.179 0.252 0.0325 0.774
## # … with 3 more variables: `R-squared` <dbl>, TrackingError <dbl>,
## # TreynorRatio <dbl>, and abbreviated variable names ¹ActivePremium,
## # ²AnnualizedAlpha, ³Correlation, ⁴`Correlationp-value`, ⁵InformationRatio
```

Table of Annualized Return, Annualized Std Dev, and Annualized Sharpe.

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra, Rb = NULL, performance_fun = table.AnnualizedReturns)
```

```
## # A tibble: 3 × 4
## # Groups: portfolio [3]
## portfolio AnnualizedReturn `AnnualizedSharpe(Rf=0%)` AnnualizedStdDev
## <int> <dbl> <dbl> <dbl>
## 1 1 0.360 1.26 0.286
## 2 2 0.348 1.16 0.301
## 3 3 0.448 1.06 0.424
```

This is a wrapper for calculating correlation and significance against each column of the data provided.

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra, Rb = Rb, performance_fun = table.Correlation)
```

```
## # A tibble: 3 × 5
## # Groups: portfolio [3]
## portfolio `p-value` `Lower CI` `Upper CI` to.Rb
## <int> <dbl> <dbl> <dbl> <dbl>
## 1 1 0.0000284 0.270 0.634 0.472
## 2 2 0.000122 0.229 0.608 0.438
## 3 3 0.0325 0.0220 0.457 0.252
```

Creates a table of estimates of downside risk measures for comparison across multiple instruments or funds.

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra, Rb = NULL, performance_fun = table.DownsideRisk)
```

```
## # A tibble: 3 × 12
## # Groups: portfolio [3]
## portfolio DownsideDe…¹ Downs…² Downs…³ GainD…⁴ Histo…⁵ Histo…⁶ LossD…⁷ Maxim…⁸
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 0.045 0.0488 0.045 0.0538 -0.141 -0.0811 0.0517 0.402
## 2 2 0.0501 0.0538 0.0501 0.0528 -0.162 -0.0842 0.0588 0.438
## 3 3 0.0684 0.0721 0.0684 0.0831 -0.222 -0.131 0.0813 0.595
## # … with 3 more variables: `ModifiedES(95%)` <dbl>, `ModifiedVaR(95%)` <dbl>,
## # SemiDeviation <dbl>, and abbreviated variable names
## # ¹`DownsideDeviation(0%)`, ²`DownsideDeviation(MAR=10%)`,
## # ³`DownsideDeviation(Rf=0%)`, ⁴GainDeviation, ⁵`HistoricalES(95%)`,
## # ⁶`HistoricalVaR(95%)`, ⁷LossDeviation, ⁸MaximumDrawdown
```

Table of Monthly downside risk, Annualized downside risk, Downside potential, Omega, Sortino ratio, Upside potential, Upside potential ratio and Omega-Sharpe ratio.

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra, Rb = NULL, performance_fun = table.DownsideRiskRatio)
```

```
## # A tibble: 3 × 9
## # Groups: portfolio [3]
## portfolio Annualiseddo…¹ Downs…² Omega Omega…³ Sorti…⁴ Upsid…⁵ Upsid…⁶ month…⁷
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 0.156 0.0198 2.48 1.48 0.652 0.0491 0.986 0.045
## 2 2 0.173 0.0217 2.34 1.34 0.579 0.0507 0.952 0.0501
## 3 3 0.237 0.0294 2.32 1.32 0.567 0.0681 0.937 0.0684
## # … with abbreviated variable names ¹Annualiseddownsiderisk,
## # ²Downsidepotential, ³`Omega-sharperatio`, ⁴Sortinoratio, ⁵Upsidepotential,
## # ⁶Upsidepotentialratio, ⁷monthlydownsiderisk
```

Summary of the higher moments and Co-Moments of the return distribution. Used to determine diversification potential. Also called “systematic” moments by several papers.

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra, Rb = Rb, performance_fun = table.HigherMoments)
```

```
## # A tibble: 3 × 6
## # Groups: portfolio [3]
## portfolio BetaCoKurtosis BetaCoSkewness BetaCoVariance CoKurtosis CoSkewness
## <int> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 0.756 0.196 0.908 0 0
## 2 2 0.772 1.71 0.886 0 0
## 3 3 0.455 0.369 0.721 0 0
```

Table of Tracking error, Annualized tracking error and Information ratio.

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra, Rb = Rb, performance_fun = table.InformationRatio)
```

```
## # A tibble: 3 × 4
## # Groups: portfolio [3]
## portfolio AnnualisedTrackingError InformationRatio TrackingError
## <int> <dbl> <dbl> <dbl>
## 1 1 0.252 0.917 0.0728
## 2 2 0.271 0.809 0.0782
## 3 3 0.412 0.774 0.119
```

Table of Mean absolute difference, Monthly standard deviation and annualized standard deviation.

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra, Rb = NULL, performance_fun = table.Variability)
```

```
## # A tibble: 3 × 4
## # Groups: portfolio [3]
## portfolio AnnualizedStdDev MeanAbsolutedeviation monthlyStdDev
## <int> <dbl> <dbl> <dbl>
## 1 1 0.286 0.0658 0.0825
## 2 2 0.301 0.0679 0.0868
## 3 3 0.424 0.091 0.122
```

Calculates Value-at-Risk (VaR) for univariate, component, and marginal cases using a variety of analytical methods.

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra, Rb = NULL, performance_fun = VaR)
```

```
## # A tibble: 3 × 2
## # Groups: portfolio [3]
## portfolio VaR
## <int> <dbl>
## 1 1 -0.111
## 2 2 -0.123
## 3 3 -0.163
```

One of the best features of `tq_portfolio`

and
`tq_performance`

is to be able to pass features through to
the underlying functions. After all, these are just wrappers for
`PerformanceAnalytics`

, so you probably want to be able
**to make full use of the underlying functions**. Passing
through parameters using the `...`

can be incredibly useful,
so let’s see how.

The `tq_portfolio`

function is a wrapper for
`Return.portfolio`

. This means that during the portfolio
aggregation process, we can make use of most of the
`Return.portfolio`

arguments such as
`wealth.index`

, `contribution`

,
`geometric`

, `rebalance_on`

, and
`value`

. Here’s the arguments of the underlying function:

`args(Return.portfolio)`

```
## function (R, weights = NULL, wealth.index = FALSE, contribution = FALSE,
## geometric = TRUE, rebalance_on = c(NA, "years", "quarters",
## "months", "weeks", "days"), value = 1, verbose = FALSE,
## ...)
## NULL
```

Let’s see an example of passing parameters to the `...`

.
Suppose we want to instead see how our money is grows for a $10,000
investment. We’ll use the “Single Portfolio” example, where our
portfolio mix was 50% AAPL, 0% GOOG, and 50% NFLX.

Method 3A, Aggregating Portfolio Returns, showed us two methods to aggregate for a single portfolio. Either will work for this example. For simplicity, we’ll examine the first.

Here’s the original output, without adjusting parameters.

```
<- c(0.5, 0.0, 0.5)
wts <- stock_returns_monthly %>%
portfolio_returns_monthly tq_portfolio(assets_col = symbol,
returns_col = Ra,
weights = wts,
col_rename = "Ra")
```

```
%>%
portfolio_returns_monthly ggplot(aes(x = date, y = Ra)) +
geom_bar(stat = "identity", fill = palette_light()[[1]]) +
labs(title = "Portfolio Returns",
subtitle = "50% AAPL, 0% GOOG, and 50% NFLX",
caption = "Shows an above-zero trend meaning positive returns",
x = "", y = "Monthly Returns") +
geom_smooth(method = "lm") +
theme_tq() +
scale_color_tq() +
scale_y_continuous(labels = scales::percent)
```

This is good, but we want to see how our $10,000 initial investment
is growing. This is simple with the underlying
`Return.portfolio`

argument,
`wealth.index = TRUE`

. All we need to do is add these as
additional parameters to `tq_portfolio`

!

```
<- c(0.5, 0, 0.5)
wts <- stock_returns_monthly %>%
portfolio_growth_monthly tq_portfolio(assets_col = symbol,
returns_col = Ra,
weights = wts,
col_rename = "investment.growth",
wealth.index = TRUE) %>%
mutate(investment.growth = investment.growth * 10000)
```

```
%>%
portfolio_growth_monthly ggplot(aes(x = date, y = investment.growth)) +
geom_line(size = 2, color = palette_light()[[1]]) +
labs(title = "Portfolio Growth",
subtitle = "50% AAPL, 0% GOOG, and 50% NFLX",
caption = "Now we can really visualize performance!",
x = "", y = "Portfolio Value") +
geom_smooth(method = "loess") +
theme_tq() +
scale_color_tq() +
scale_y_continuous(labels = scales::dollar)
```

Finally, taking this one step further, we apply the same process to the “Multiple Portfolio” example:

- 50% AAPL, 25% GOOG, 25% NFLX
- 25% AAPL, 50% GOOG, 25% NFLX
- 25% AAPL, 25% GOOG, 50% NFLX

```
<- stock_returns_monthly_multi %>%
portfolio_growth_monthly_multi tq_portfolio(assets_col = symbol,
returns_col = Ra,
weights = weights_table,
col_rename = "investment.growth",
wealth.index = TRUE) %>%
mutate(investment.growth = investment.growth * 10000)
```

```
%>%
portfolio_growth_monthly_multi ggplot(aes(x = date, y = investment.growth, color = factor(portfolio))) +
geom_line(size = 2) +
labs(title = "Portfolio Growth",
subtitle = "Comparing Multiple Portfolios",
caption = "Portfolio 3 is a Standout!",
x = "", y = "Portfolio Value",
color = "Portfolio") +
geom_smooth(method = "loess") +
theme_tq() +
scale_color_tq() +
scale_y_continuous(labels = scales::dollar)
```

Finally, the same concept of passing arguments works with all the
`tidyquant`

functions that are wrappers including
`tq_transmute`

, `tq_mutate`

,
`tq_performance`

, etc. Let’s use a final example with the
`SharpeRatio`

, which has the following arguments.

`args(SharpeRatio)`

```
## function (R, Rf = 0, p = 0.95, FUN = c("StdDev", "VaR", "ES"),
## weights = NULL, annualize = FALSE, SE = FALSE, SE.control = NULL,
## ...)
## NULL
```

We can see that the parameters `Rf`

allows us to apply a
risk-free rate and `p`

allows us to vary the confidence
interval. Let’s compare the Sharpe ratio with an annualized risk-free
rate of 3% and a confidence interval of 0.99.

Default:

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra,
performance_fun = SharpeRatio)
```

```
## # A tibble: 3 × 4
## # Groups: portfolio [3]
## portfolio `ESSharpe(Rf=0%,p=95%)` `StdDevSharpe(Rf=0%,p=95%)` VaRSharpe(Rf=0…¹
## <int> <dbl> <dbl> <dbl>
## 1 1 0.172 0.355 0.263
## 2 2 0.146 0.334 0.236
## 3 3 0.150 0.317 0.238
## # … with abbreviated variable name ¹`VaRSharpe(Rf=0%,p=95%)`
```

With `Rf = 0.03 / 12`

(adjusted for monthly
periodicity):

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra,
performance_fun = SharpeRatio,
Rf = 0.03 / 12)
```

```
## # A tibble: 3 × 4
## # Groups: portfolio [3]
## portfolio `ESSharpe(Rf=0.2%,p=95%)` `StdDevSharpe(Rf=0.2%,p=95%)` VaRSharpe(…¹
## <int> <dbl> <dbl> <dbl>
## 1 1 0.157 0.325 0.241
## 2 2 0.134 0.305 0.216
## 3 3 0.141 0.296 0.222
## # … with abbreviated variable name ¹`VaRSharpe(Rf=0.2%,p=95%)`
```

And, with both `Rf = 0.03 / 12`

(adjusted for monthly
periodicity) and `p = 0.99`

:

```
%>%
RaRb_multiple_portfolio tq_performance(Ra = Ra,
performance_fun = SharpeRatio,
Rf = 0.03 / 12,
p = 0.99)
```

```
## # A tibble: 3 × 4
## # Groups: portfolio [3]
## portfolio `ESSharpe(Rf=0.2%,p=99%)` `StdDevSharpe(Rf=0.2%,p=99%)` VaRSharpe(…¹
## <int> <dbl> <dbl> <dbl>
## 1 1 0.105 0.325 0.134
## 2 2 0.0952 0.305 0.115
## 3 3 0.0915 0.296 0.117
## # … with abbreviated variable name ¹`VaRSharpe(Rf=0.2%,p=99%)`
```