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portalpha

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

Syntax

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

Arguments

`Asset`

`NUMSAMPLES x NUMSERIES` matrix with `NUMSAMPLES` observations of asset returns for `NUMSERIES` asset return series.

`Benchmark`

`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.

`Cash `

(Optional) 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.

`Choice`

(Optional) A character vector, or cell array of character vectors to indicate one or more measures to be computed from among a number of 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
`'capm'`Jensen's Alpha
`'mm'`Modigliani & Modigliani
`'gh1'`Graham-Harvey 1
`'gh2'`Graham-Harvey 2
`'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.

Description

Given `NUMSERIES` assets with `NUMSAMPLES` returns in a `NUMSAMPLES`-by-`NUMSERIES` matrix `Asset`, a `NUMSAMPLES` vector of `Benchmark` returns, and either a scalar `Cash` return or a `NUMSAMPLES` vector of `Cash` returns, compute risk-adjusted alphas and returns for one or more methods specified by `Choice`.

To summarize the outputs of `portalpha`:

• `Alpha` is a `NUMCHOICES`-by-`NUMSERIES` matrix of risk-adjusted alphas for each series in `Asset` with each row corresponding to a specified measure in `Choice`.

• `RAReturn` is a `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.

Examples

References

John Lintner. "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.

John R. Graham 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.

Franco Modigliani 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.

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

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

See Also

Introduced in R2006b

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