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.

References

[1] 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.

[2] 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.

[3] 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.

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

[5] 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.

Version History

Introduced in R2006b