## Active, no benchmark

Optimize the trade given various utilities and some simple constraints for the case where there are expected returns and no benchmark.

## Preparation

• vector of asset prices
• vector of expected returns
• variance matrix for the assets
• current portfolio (if it exists)
• Portfolio Probe

You need the prices at which assets trade, and a variance matrix of the asset returns.  You also need an expected return for each asset in the universe.

The holdings of the current portfolio need to be in a vector with names that are the asset identifiers.

You also need to have the Portfolio Probe package loaded into your R session:

`require(PortfolioProbe)`

If you don’t have Portfolio Probe, see “Demo or Buy”.

### Doing the example

`require(pprobeData)`

## Doing it

We’ll do a few optimizations:

• Maximize the information ratio
• Maximize a mean-variance utility
• Maximize expected return given a maximum volatility

### Preliminaries

The inputs we need in order to get our optimal portfolio are:

• vector of prices at which the assets may be traded
• variance matrix of the asset returns
• vector of expected returns
• desired value of the new portfolio
• appropriate constraints
• current portfolio (optional)

#### prices

We start by naming the vector of prices that we want to use:

`priceVector <- xassetPrices[251,]`

These are the prices at the close of the last trading day of 2006. The first few values are:

```> head(priceVector)
XA101  XA103  XA105  XA107  XA108  XA111
33.56  72.25  74.39 192.06   5.91  15.98```

The requirement for the prices is that it be a vector of positive numbers with names (that are the asset identifiers).

#### expected returns

We can get the MACD signal value for the same time point as the prices to use as the expected returns:

```> expRet <- xaMACD[251,] / 100
XA101        XA103        XA105
0.009178291  0.009114743  0.011844481
XA107        XA108        XA111
0.008825741  0.014268342 -0.003350204```

#### current portfolio

We create an object to serve as the current portfolio:

```curPortfol <- (1:10) * 1000
names(curPortfol) <- colnames(xassetPrices)[1:10]```

What is expected is a numeric vector of the number of units of each asset in the portfolio. The names of the vector are the identifiers of the assets that are used in the price vector and the variance matrix.

```> curPortfol
XA101 XA103 XA105 XA107 XA108 XA111 XA113 XA115
1000  2000  3000  4000  5000  6000  7000  8000
XA120 XA126
9000 10000```

#### portfolio value

The value of the portfolio that we should specify is the current value of the existing portfolio adjusted by whatever cash flow is desired. Here we assume we want to add \$20,000 to the portfolio:

```cashFlow <- 20000
grossVal <- as.numeric(valuation(curPortfol,
priceVector, collapse=TRUE)) + cashFlow```

We get the value of the current portfolio assuming the prices we are using and then add the cash flow. (The `as.numeric` is merely for cosmetic reasons to make the result simpler.) We end up with:

```> grossVal
 3033430```

The gross value of the portfolio that we want is slightly more than \$3 million.

### Optimization: information ratio

We’re now ready to do an optimization. The only constraint that we impose besides the gross value and being long-only is that no more than 10 assets may be in the portfolio.

```opMaxInfo <- trade.optimizer(priceVector,
variance=xaLWvar06, expected.return=expRet,
existing=curPortfol, gross=grossVal, long.only=TRUE,
port.size=10)```

We are not specifying what utility to use, so it will use the default which is to maximize the information ratio.

### Optimization: mean-variance utility

If we are using a mean-variance utility, we need to decide what risk aversion to use.

```opMeanVar <- trade.optimizer(priceVector,
variance=xaLWvar06, expected.return=expRet,
existing=curPortfol, gross=grossVal, long.only=TRUE,
port.size=10, utility="mean-variance",
risk.aversion=10)```

We’ve added two arguments to the previous optimization.  We use the `utility` argument to state the form of utility to use, and we use the `risk.aversion` argument to state the risk aversion that we want to use.

Note that the form of the utility is: expected return minus risk aversion times variance.  Some have a one-half in the last term.

### Optimization: maximum return with volatility constraint

An often convenient form of optimization is to constrain the volatility to some maximum value and then maximize the expected return.  Here we constrain volatility to 8%.

```opMaxExpVC <- trade.optimizer(priceVector,
variance=xaLWvar06, expected.return=expRet,
existing=curPortfol, gross=grossVal, long.only=TRUE,
port.size=10, utility="maximum return",
var.constraint=.08^2/252)```

We need to translate our 8% volatility into the scale of the variance, which is daily.

### Print results

The resulting object is printed like:

```> opMaxExpVC
\$new.portfolio
XA199 XA280 XA298 XA351 XA420 XA481 XA802 XA891
35808 12235  3939   345  5388  2847  1366  6415
XA893 XA980
9164  4593

XA101  XA103  XA105  XA107  XA108  XA111  XA113
-1000  -2000  -3000  -4000  -5000  -6000  -7000
XA115  XA120  XA126  XA199  XA280  XA298  XA351
-8000  -9000 -10000  35808  12235   3939    345
XA420  XA481  XA802  XA891  XA893  XA980
5388   2847   1366   6415   9164   4593

\$results
objective     negutil        cost     penalty
-0.02012791 -0.02012791  0.00000000  0.00000000

\$converged
 TRUE

\$objective.utility
 "maximum return"

\$alpha.values
A0
0.02012791

\$var.values
V0
2.539682e-05

\$utility.values
 -0.02012791

\$existing
XA101 XA103 XA105 XA107 XA108 XA111 XA113 XA115
1000  2000  3000  4000  5000  6000  7000  8000
XA120 XA126
9000 10000

\$violated
NULL

\$timestamp
 "Tue Sep 25 09:52:46 2012"
 "Tue Sep 25 09:52:55 2012"

\$call
trade.optimizer(prices = priceVector, variance = xaLWvar06, expected.return = expRet,
existing = curPortfol, gross = grossVal, long.only = TRUE,
port.size = 10, utility = "maximum return", var.constraint = 0.08^2/252)```

The first two components are the new (optimal) portfolio and the trade to achieve that. There are some additional components to the object that are not shown.

## Explanation

### Optimization strategy

The optimization with the volatility constraint is the easiest to do in practice.  This is because we don’t need the variance and the expected returns to be on the same scale.  We merely need to decide what (expected) volatility we are willing to tolerate.

The mean-variance formulation assumes either that the variance and expected returns are matched in scale, or that the risk aversion takes the mismatch into account.  Maximizing the information ratio assumes that the scales are matched.

### Technical details

#### portfolio value

It is mandatory that the value of the resulting portfolio be specified. For long-only portfolios it is sufficient to state the desired gross value. The actual value of the portfolio will (usually) be slightly less than the specification:

```> format(grossVal, nsmall=2, big.mark=",")
 "3,033,430.00"
> grossVal - as.numeric(valuation(opMaxExpVC, collapse=TRUE))
 18.99```

#### utility

If both expected returns and variance are given, then the default utility is to maximize the information ratio.  If expected returns are not given, then the default is to minimize variance.

#### other output components

One component of the output to pay special attention to is ‘`violated`‘ — this states which constraints, if any, are violated. You want this to be `NULL`.

It is probably not important whether ‘`converged`‘ is `TRUE` or `FALSE`. The optimization is likely to be good enough with or without convergence.

## Further Details

You can see more about the optimization with the summary of the object:

```> summary(opMaxExpVC)
\$results
objective     negutil        cost     penalty
-0.02012791 -0.02012791  0.00000000  0.00000000

\$objective.utility
 "maximum return"

\$alpha.values
A0
0.02012791

\$var.values
V0
2.539682e-05

\$number.of.assets
10                20
new              open
10                10
close    universe.total
10               350
350               350
0

\$opening.positions
 "XA199" "XA280" "XA298" "XA351" "XA420"
 "XA481" "XA802" "XA891" "XA893" "XA980"

\$closing.positions
 "XA101" "XA103" "XA105" "XA107" "XA108"
 "XA111" "XA113" "XA115" "XA120" "XA126"

\$value.limits
lower   upper
gross 3033127 3033430
net   3033127 3033430
long  3033127 3033430
short       0       0

\$valuation.new.portfolio
gross     net    long   short
3033411 3033411 3033411       0

gross        net       long      short
6046841.01   19981.01 3033411.01 3013430.00

gross         net        long       short
1.993413023 0.006586977 1.000000000 0.993413023```

This has some pieces that are also in the `print` method, but new information as well. We see that all of the current portfolio was sold off — a trade to make the broker happy.

## Troubleshooting

• The variance matrix needs to contain all of the assets that are in the price vector. It can have additional assets — except for benchmarks, these will be ignored. The order of the assets in the variance does not matter.
• All of the prices need to be in the same currency. You have to check that — the code has no way of knowing.
• It will still work if the object given as the prices is a one-column or one-row matrix. But it will complain about other matrices.  The same is true for expected returns.