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# portalpha

Compute risk-adjusted alphas and returns for one or more assets

## Syntax

``portalpha(Asset,Benchmark)``
``portalpha(Asset,Benchmark,Cash)``
``[Alpha,RAReturn] = portalpha(Asset,Benchmark,Cash,Choice)``

## Description

example

````portalpha(Asset,Benchmark)` computes risk-adjusted alphas.```

example

````portalpha(Asset,Benchmark,Cash)` computes risk-adjusted alphas using the optional argument `Cash`. ```

example

````[Alpha,RAReturn] = portalpha(Asset,Benchmark,Cash,Choice)` computes risk-adjusted alphas and returns for one or more methods specified by `Choice`. ```

## Examples

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This example shows how to calculate the risk-adjusted return using `portalpha` and compare it with the fund and market's mean returns.

Use the example data with a fund, a market, and a cash series.

```load FundMarketCash Returns = tick2ret(TestData); Fund = Returns(:,1); Market = Returns(:,2); Cash = Returns(:,3); MeanFund = mean(Fund)```
```MeanFund = 0.0038 ```
`MeanMarket = mean(Market)`
```MeanMarket = 0.0030 ```
`[MM, aMM] = portalpha(Fund, Market, Cash, 'MM')`
```MM = 0.0022 ```
```aMM = 0.0052 ```
`[GH1, aGH1] = portalpha(Fund, Market, Cash, 'gh1')`
```GH1 = 0.0013 ```
```aGH1 = 0.0025 ```
`[GH2, aGH2] = portalpha(Fund, Market, Cash, 'gh2')`
```GH2 = 0.0022 ```
```aGH2 = 0.0052 ```
`[SML, aSML] = portalpha(Fund, Market, Cash, 'sml')`
```SML = 0.0013 ```
```aSML = 0.0025 ```

Since the fund's risk is much less than the market's risk, the risk-adjusted return of the fund is much higher than both the nominal fund and market returns.

## Input Arguments

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Asset returns, specified as a `NUMSAMPLES x NUMSERIES` matrix with `NUMSAMPLES` observations of asset returns for `NUMSERIES` asset return series.

Data Types: `double`

Returns for a benchmark asset, specified as a `NUMSAMPLES` vector of returns for a benchmark asset. The periodicity must be the same as the periodicity of `Asset`. For example, if `Asset` is monthly data, then `Benchmark` should be monthly returns.

Data Types: `double`

(Optional) Riskless asset, specified as a either a scalar return for a riskless asset or a vector of asset returns to be a proxy for a “riskless” asset. In either case, the periodicity must be the same as the periodicity of `Asset`. For example, if `Asset` is monthly data, then `Cash` must be monthly returns. If no value is supplied, the default value for `Cash` returns is `0`.

Data Types: `double`

(Optional) Computed measures, specified as a character vector or cell array of character vectors to indicate one or more measures to be computed from among various risk-adjusted alphas and return measures. The number of choices selected in `Choice` is `NUMCHOICES`. The list of choices is given in the following table:

CodeDescription
`'xs'`Excess Return (no risk adjustment)
`'sml'`Security Market Line — The security market line shows that the relationship between risk and return is linear for individual securities (that is, increased risk = increased return).
`'capm'`Jensen's Alpha — Risk-adjusted performance measure that represents the average return on a portfolio or investment, above or below that predicted by the capital asset pricing model (CAPM), given the portfolio's or investment's beta and the average market return.
`'mm'`Modigliani & Modigliani — Measures the returns of an investment portfolio for the amount of risk taken relative to some benchmark portfolio.
`'gh1'`Graham-Harvey 1 — Performance measure developed by John Graham and Campbell Harvey. The idea is to lever a fund's portfolio to exactly match the volatility of the S&P 500. The difference between the fund's levered return and the S&P 500 return is the performance measure.
`'gh2'`Graham-Harvey 2 — In this measure, the idea is to lever up or down the fund's recommended investment strategy (using a Treasury bill), so that the strategy has the same volatility as the S&P 500.
`'all'`Compute all measures.

`Choice` is specified by using the code from the table (for example, to select the Modigliani & Modigliani measure, `Choice` = `'mm'`). A single choice is either a character vector or a scalar cell array with a single code from the table.

Multiple choices can be selected with a cell array of character vectors for choice codes (for example, to select both Graham-Harvey measures, `Choice` = `{'gh1','gh2'}`). To select all choices, specify `Choice` = `'all'`. If no value is supplied, the default choice is to compute the excess return with `Choice` = `'xs'`. `Choice` is not case-sensitive.

Data Types: `char` | `cell`

## Output Arguments

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Risk-adjusted alphas, returned as an `NUMCHOICES`-by-`NUMSERIES` matrix of risk-adjusted alphas for each series in `Asset` with each row corresponding to a specified measure in `Choice`.

Risk-adjusted returns, returned as an `NUMCHOICES`-by-`NUMSERIES` matrix of risk-adjusted returns for each series in `Asset` with each row corresponding to a specified measure in `Choice`.

Note

`NaN` values in the data are ignored and, if `NaN`s are present, some results could be unpredictable. Although the alphas are comparable across measures, risk-adjusted returns depend on whether the `Asset` or `Benchmark` is levered or unlevered to match its risk with the alternative. If `Choice` = `'all'`, the order of rows in `Alpha` and `RAReturn` follows the order in the table. In addition, `Choice` = `'all'` overrides all other choices.

 Graham, J. R. and Campbell R. Harvey. "Market Timing Ability and Volatility Implied in Investment Newsletters' Asset Allocation Recommendations." Journal of Financial Economics. Vol. 42, 1996, pp. 397–421.

 Lintner, J. "The Valuation of Risk Assets and the Selection of Risky Investments in Stocks Portfolios and Capital Budgets." Review of Economics and Statistics. Vol. 47, No. 1, February 1965, pp. 13–37.

 Modigliani, F. and Leah Modigliani. "Risk-Adjusted Performance: How to Measure It and Why." Journal of Portfolio Management. Vol. 23, No. 2, Winter 1997, pp. 45–54.

 Mossin, J. "Equilibrium in a Capital Asset Market." Econometrica. Vol. 34, No. 4, October 1966, pp. 768–783.

 Sharpe, W.F., "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk." Journal of Finance. Vol. 19, No. 3, September 1964, pp. 425–442.

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