Two portfolios are being compared, one built from speculative small-company stocks and one from a broad mix of large companies. What does a risk-adjusted return figure state about each that its raw return does not?

How much of its return survived inflation and taxes, so that portfolios held by investors in quite different tax brackets can be set side by side.

How much return it earned for each unit of risk it took on, so that portfolios with very different risk profiles can be set side by side.

How much of its return came from dividends and interest rather than price gains, so that portfolios with very different payout policies can be set side by side.

How much its return is likely to vary over the coming year, so that portfolios with very different lengths of track record can be set side by side.

Over the same year Fund P and Fund Q each returned nine percent while the risk-free rate was three percent. Fund P's returns had a standard deviation of six percent and Fund Q's eighteen percent. How should the two results be compared?

Fund Q's is the more attractive result, because bearing three times the risk positions it to earn a proportionately higher return in the years ahead.

The two are equally attractive, because what matters to the investor is the return actually delivered, and here the two returns are identical.

Fund P's is the more attractive result, because Fund Q bore three times the risk and would have needed a proportionately higher return to justify it.

They cannot be compared until a benchmark index is chosen for each, because a return means nothing until it is set against the index a fund tracks.

The Sharpe ratio, the Treynor ratio, Jensen's alpha and the information ratio each measure return relative to risk. In what chiefly do they differ from one another?

In the measure of risk each uses to scale the return and in the benchmark against which each one judges that return.

In the length of track record each one needs and in how often the returns feeding each one must be sampled to be reliable.

In whether each works from returns forecast in advance by analysts or from returns realised over a period that is already past.

In whether each counts the portfolio's income together with its price changes or counts the changes in its price alone.

Manager C returned eighteen percent with a standard deviation of twenty-four percent; Manager D returned twelve percent with a standard deviation of six percent. On raw return alone Manager C performed better. What do the figures show once risk is brought in?

Manager C took on four times Manager D's risk for six more points of return, so once risk is counted Manager C's result is likely still the stronger.

Manager C took on four times Manager D's risk for half as much again in return, so once risk is counted Manager D's result is likely the stronger.

Manager C took on one third more risk than Manager D for half as much again in return, so once risk is counted Manager C's result is likely the stronger.

Manager C took on four times Manager D's risk and earned a return in proportion to it, so once risk is counted the two results are likely equally strong.

A portfolio's Sharpe ratio for a year is negative. What does a negative Sharpe ratio indicate about the portfolio's result?

That the portfolio fell short of its benchmark index, a negative figure recording a return below the index it is measured against.

That the portfolio lost money over the year, its value at the end of the period standing below what investors held at the start.

That its investors would have done better in a risk-free investment, the portfolio having borne risk to earn less than the risk-free rate.

That the portfolio's volatility outweighed its return, the standard deviation of its returns exceeding the return it earned.

Fund R and Fund S each show a Sharpe ratio of zero point three computed from monthly excess returns, Fund R's from twenty-five months of returns and Fund S's from one hundred. How do the two figures compare as evidence that each fund's mean excess return is genuinely above zero?

Fund S's is about four times as strong: zero point three times one hundred months gives a t-statistic four times the one that zero point three times twenty-five months gives Fund R.

Fund R's is the stronger: a ratio reached over twenty-five months reflects the fund's current conditions and manager, while Fund S's is diluted by returns that are now years old.

They are equally strong: a Sharpe ratio already divides the mean excess return by the variability of the returns, so the length of the record adds nothing further to the evidence.

Fund S's is about twice as strong: zero point three times the square root of one hundred gives a t-statistic of three, against one and a half for Fund R.

An investor who already holds a large portfolio of stocks and bonds plans to add one fund, and chooses the fund with the highest Sharpe ratio among those on her list. Why might that fund not be the best addition to what she already owns?

Its ratio takes no account of how its returns move with those of her existing holdings, which decides how much the fund will add to them.

Its ratio is recomputed once the fund is held, its returns then being measured against her whole portfolio's standard deviation rather than its own.

Its ratio depends on the benchmark index each fund is measured against, and the funds on her list were each judged against a different index, so the ratios cannot be compared.

Its ratio is likely to have been flattered by a short record, since a ratio drawn from few returns is always higher than one drawn from many.

Fund X earned six percentage points above the risk-free rate with a standard deviation of twelve percent. Fund Y borrows at the risk-free rate to hold twice as much of Fund X as its own capital would buy. What is Fund Y's Sharpe ratio?

About zero point seven one: the excess return doubles to twelve points while the standard deviation rises by the square root of two.

Zero point five, the same as Fund X's: twelve points of excess return divided by twenty-four points of standard deviation.

Zero point two five: borrowing doubles the standard deviation to twenty-four points while the excess return stays at six points.

One: borrowing doubles the excess return to twelve points while the standard deviation stays at Fund X's twelve points.

A portfolio returned sixteen percent with a standard deviation of thirty percent over a year in which the risk-free rate was four percent and the market portfolio's standard deviation was twenty percent. What is the portfolio's M-squared, the Modigliani measure of risk-adjusted performance?

About ten point seven percent: the whole sixteen percent return scaled by twenty over thirty, the four percent risk-free rate playing no part in it.

Twenty-two percent: twelve points of excess return scaled by thirty over twenty, giving eighteen points, which are added to the four percent risk-free rate.

Twelve percent: twelve points of excess return scaled by twenty over thirty, giving eight points, which are added to the four percent risk-free rate.

Twenty-four percent: the whole sixteen percent return scaled by thirty over twenty, the four percent risk-free rate playing no part in it.

Over one period an analyst ranks ten funds first by Sharpe ratio and then by M-squared, the Modigliani measure that states risk-adjusted performance in percentage points. How do the two rankings compare?

In a different order wherever the funds' standard deviations differ from the market's, since each fund's excess return is rescaled by a factor of its own.

In the same order only among funds riskier than the market, the funds less risky than the market rising in the ranking as their returns are scaled up.

In exactly the order of the Treynor ratio rather than the Sharpe ratio, since each fund's figure is rescaled to the risk of the market portfolio.

In exactly the same order: over one period every fund shares the risk-free rate and the market's standard deviation, so M-squared changes the unit but not the order.

Over one period a portfolio's M-squared, the Modigliani measure that restates its risk-adjusted performance as a return in percentage points, was seven percent, while the market portfolio returned nine percent. What does that comparison show?

It underperformed the market once risk was equalised, because at the market's own level of risk it would have returned seven percent against the market's nine.

It outperformed the market, because a lower M-squared marks a portfolio that earned its return with less risk than the market took to earn its own.

It underperformed the market in raw return, because M-squared is the return actually earned over the period, so the shortfall is one that risk adjustment leaves as it was.

It took more risk than the market, because an M-squared below the market's return shows that the portfolio's standard deviation was the higher of the two.

A portfolio returned fourteen percent over a year in which the risk-free rate was five percent, and its beta is one point two. What is its Treynor ratio?

Ten point eight: nine points of return above the risk-free rate multiplied by the portfolio's beta of one point two.

Seven point five: nine points of return above the risk-free rate divided by the portfolio's beta of one point two.

Eleven point seven: fourteen points of return divided by the portfolio's beta of one point two, the risk-free rate set aside.

Zero point one three: the portfolio's beta of one point two divided by nine points of return above the risk-free rate.

Over the same year, against the same risk-free rate, Fund J holding four hundred stocks and Fund K holding twelve earned the same return, and each has a beta of one point one. Fund K's standard deviation is far higher because of company-specific swings in its few holdings. How do the Treynor and Sharpe ratios rank the two funds?

The Treynor ratio ranks Fund J ahead for its far broader diversification, while the Sharpe ratio ranks them level on their equal returns.

Both ratios rank Fund J ahead, since each denominator rises with the company-specific swings that Fund K's few holdings produce.

The Treynor ratio ranks them level, both having the same beta, while the Sharpe ratio ranks Fund J ahead for its lower total risk.

Both ratios rank them level, since the same return and the same beta leave neither of the two ratios anything to tell them apart by.

Over a year a portfolio with a beta of one half returned seven percent. The market returned eleven percent and the risk-free rate was three percent. What is the portfolio's Jensen's alpha?

Plus four points: seven percent less the three percent risk-free rate, the beta of one half entering only when funds are ranked.

Plus one and a half points: one half times the eleven percent market return gives an expected five and a half percent for the portfolio.

Minus one and a half points: three percent plus one half times the whole eleven percent market return gives an expected eight and a half percent.

Zero: three percent plus one half times the eight-point market premium gives an expected seven percent, which is exactly what it earned.

A portfolio has a beta of zero point six. In practical terms, what does the return the Capital Asset Pricing Model predicts for it represent?

What a passive investor would earn by holding sixty percent in the market portfolio and forty percent in the risk-free asset.

What a passive investor would earn by holding forty percent in the market portfolio and sixty percent in the risk-free asset.

What a passive investor would earn by holding the whole market portfolio plus sixty percent more bought with borrowed money.

What the portfolio's own holdings are forecast to return, averaged across the estimates of the analysts who follow them.

A trustee oversees a single active manager and has no peer group of comparable funds to set the manager's figures against. Why can Jensen's alpha tell her whether the manager added value when a Sharpe or Treynor ratio on its own cannot?

It is expressed as a percentile rank within the manager's style category, which supplies the peer comparison she otherwise lacks.

It is an absolute figure in percentage points of outperformance, and its sign shows on its own whether value was added or destroyed.

It is drawn from a full market cycle of returns, so a single unusually good or bad year cannot distort the reading that it gives.

It stays reliable over a short record, since it needs fewer periods of returns than a ratio built on a measure of variability does.

Why can a fund's Jensen's alpha be negative for a year in which the fund, net of all its fees, beat its market index?

Because alpha is measured only on the fund's holdings within the index, leaving out any gains it earned on stocks outside the index.

Because alpha is measured after the fund's fees and trading costs, which can turn a margin over the index into a shortfall below it once charged.

Because alpha is measured against what a passive holding at the fund's own beta would have earned, which a high-beta fund can fall short of.

Because alpha is measured against the average fund in the same category, which may have beaten the index by more than this fund did.

Over a year a manager returned ten and a half percent against a benchmark return of nine percent, and the tracking error of the portfolio against that benchmark was two percent. What is the manager's information ratio?

Three: one and a half points of return above the benchmark multiplied by two points of tracking error.

One point three three: two points of tracking error divided by one and a half points of return above the benchmark.

Five point two five: ten and a half points of return divided by two points of tracking error, the benchmark set aside.

Zero point seven five: one and a half points of return above the benchmark divided by two points of tracking error.

A manager's information ratio uses tracking error as its denominator. What does tracking error measure?

The standard deviation over time of the difference between the portfolio's return and the benchmark's return.

The standard deviation over time of the portfolio's own returns, whatever the benchmark happened to return.

The sensitivity of the portfolio's return to the benchmark's movements, read as the slope of one against the other.

The largest single-period shortfall of the portfolio's return below the benchmark's across the whole record.

Managers L and M each beat the same benchmark by an average of three percentage points a year over ten years. Manager L's yearly margin stayed between two and four points; Manager M's swung between fifteen points above and nine points below. How do their information ratios compare, and what does the comparison suggest?

Manager M's is higher, since margins that large show the conviction and willingness to depart from the index that active management is paid for.

Manager L's is far higher, and Manager M's erratic margin suggests concentrated positions or inconsistent bets rather than a reliably effective process.

They are about equal, since the two managers produced the same average margin over the same benchmark across the same ten years of returns.

Manager M's is higher, since its best years added more above the benchmark than its worst years took away below it across the ten-year record.

A retiree's entire invested wealth sits in a single fund of about twenty stocks, and she owns nothing else. Which measure suits judging how well that fund has rewarded her for the risk she bears, and why?

The Treynor ratio, because her tolerance for risk was set against market downturns, and beta is the measure of exposure to those.

The Treynor ratio, because company-specific swings in a fund's holdings tend to cancel out over time, leaving only market risk to reward.

The Sharpe ratio, because with nothing else held, every source of the fund's volatility, company-specific included, reaches her in full.

The information ratio, because a retiree drawing on her savings needs above all to know how steadily the fund has beaten its index.

Over a year a fund returned eight percent with a standard deviation of ten percent, and its maximum drawdown during the year was twenty percent. What is its Calmar ratio?

Minus twelve points: the eight percent return less the twenty-point maximum drawdown.

Two point five: the twenty-point maximum drawdown divided by the eight percent return.

Zero point eight: the eight percent return divided by the ten-point standard deviation.

Zero point four: the eight percent return divided by the twenty-point maximum drawdown.

Fund T and Fund U recorded the same twelve monthly returns over a year in a different order: Fund T's four losing months came one after another, while Fund U's were scattered among its gains. Of their mean monthly returns, standard deviations, Sharpe ratios and maximum drawdowns for the year, which differ?

Their maximum drawdowns, Fund T's being deeper, while their mean returns, standard deviations and Sharpe ratios are the same.

Their standard deviations, Fund T's being higher, and so their Sharpe ratios, while their maximum drawdowns are the same.

Their mean returns, Fund T's being lower as consecutive losses compound, and so every figure that is built on the mean.

None of them, since each figure for the year is computed from the same twelve returns, whatever order they arrived in.

Why is a Calmar ratio measured over one fund's ten-year record not directly comparable with a Calmar ratio measured over another fund's three-year record?

Because drawdowns shrink as a record lengthens, the later recoveries offsetting the earlier falls, so the longer record's ratio is flattered by nothing more than its length.

Because a longer record gives a fund more opportunity to suffer a deep decline, so the maximum drawdown to be expected grows with the length of the record.

Because the ten-year ratio averages ten separate yearly drawdowns, which smooths out the effect of the worst year in a way that a three-year ratio averaging three cannot.

Because a ten-year calculation sets aside every drawdown older than the latest three years, so the longer record's ratio leaves out the falls of its earlier seven years.

A fund's downside capture ratio is eighty. On that record, about how far would the fund be expected to fall in a month in which its benchmark fell ten percent?

About ten percent: the ratio records how often the fund falls along with its benchmark, not how far it falls.

About two percent: the fund has avoided eighty percent of its benchmark's falls, losing a fifth of what the benchmark did.

About eight percent: the fund has captured eighty percent of its benchmark's falls, losing less than the benchmark did.

About eighteen percent: the fund has fallen by its benchmark's ten points plus a further eighty percent of that fall.

A fund overweight in defensive stocks shows an attractive downside capture ratio across three volatile years in which the market made little progress overall. Why should an adviser not recommend the fund on its downside capture ratio alone?

Because a downside ratio is published only over a fifteen-year period, and a figure drawn from three years has not yet been confirmed across that full period.

Because a downside ratio is largely a measure of total risk, already counted in the fund's standard deviation, so it adds nothing beyond what the Sharpe ratio shows.

Because a downside ratio is set against the fund's category average rather than a single index, so it cannot be compared with the size of the market's own fall.

Because most funds trade one ratio against the other, and a fund that held up in a flat market may lag once the market turns into a sustained rise.

A fund holding mainly small-company value stocks is measured against a large-company stock index and shows upside and downside capture ratios that depart sharply from those of a fund that simply tracks the index. What is the sound reading of those figures?

They partly reflect the gap between the fund's style and the index chosen, since capture ratios depend on the benchmark and a very different style produces striking figures.

They show a manager who departed sharply from the index, for better or worse, since capture ratios are computed from the fund's own months and not from any index.

They are unreliable for every fund, since capture ratios can be computed only for funds whose holdings match the index month for month.

They show that the manager has changed style during the period, since a fund whose ratios depart far from an index-tracker's has by definition abandoned its stated approach.

What does Morningstar Risk-Adjusted Return, the measure on which the Morningstar star rating rests, measure?

The return the fund would have earned had its standard deviation been brought into line with that of the average fund in its category.

The guaranteed riskless return that would give a model investor the same utility as the fund's actual, variable excess returns.

The fund's return above its category's average return, divided by the standard deviation of the fund's margin over its category's average.

The fund's average excess return less a charge for every point of maximum drawdown it suffered over the period rated.

An investment-grade bond fund carries five Morningstar stars and an equity fund carries three. What does the difference in stars show about the two funds?

That the bond fund's manager is the more skilful of the two, the star rating being designed to reflect manager skill rather than market conditions.

That the bond fund earned the higher risk-adjusted return of the two, stars being awarded on a single scale to every fund that Morningstar rates.

Nothing directly about which is better managed, since each fund's stars come from ranking it against the other funds in its own Morningstar Category.

That the equity fund took more risk, the star rating deducting one star for each band of risk above the average of the fund's category.

A fund with twelve years of history has an overall Morningstar rating that weights its ten-year rating at fifty percent, its five-year rating at thirty and its three-year rating at twenty. Which years of its record have the greatest effect on the overall rating, and why?

The two years before the last ten, because a fund with more than ten years of history is rated over its whole record, its first years weighing the most.

The earliest five of the last ten years, because they count toward the ten-year rating alone, and the ten-year rating carries half the weight in the formula.

The fourth and fifth years back, because they fall within both the five-year and the ten-year periods, which together carry eighty percent of the weight.

The most recent three years, because those years fall within all three rated periods and so enter the ten-year and five-year ratings as well as the three-year.

Risk-Adjusted Return: Sharpe, Treynor, Alpha and Info Ratio

What Is Risk-Adjusted Return?

Risk-adjusted return is a measure of investment performance that accounts for the amount of risk taken to achieve a given level of return — expressing not merely what an investment earned but how much was earned per unit of risk assumed, enabling meaningful comparison between portfolios or securities with fundamentally different risk profiles that would be incomparable on the basis of raw return alone. The foundational insight underlying risk-adjusted return measurement is that two investments earning identical nominal returns are not equally attractive if one achieved that return with substantially more risk than the other — the higher-risk investment must earn proportionately higher returns to justify the additional uncertainty imposed on the investor. Four primary risk-adjusted performance measures — the Sharpe ratio, the Treynor ratio, Jensen’s alpha, and the information ratio — each approach the measurement of return per unit of risk from a different analytical perspective, using different measures of risk in the denominator and different benchmark frameworks, and all four are tested on the Series 65 examination.

Why Raw Return Is an Insufficient Performance Measure

An investor evaluating two portfolio managers on the basis of absolute returns alone — Manager A earned fifteen percent and Manager B earned twelve percent last year — would conclude that Manager A performed better. But if Manager A achieved fifteen percent by concentrating the portfolio in a handful of speculative small-cap stocks with a standard deviation of thirty percent, while Manager B achieved twelve percent through a diversified strategy with a standard deviation of eight percent, the raw return comparison is deeply misleading. Manager A took nearly four times as much risk as Manager B to generate only twenty-five percent more return — on a risk-adjusted basis, Manager B may have delivered substantially superior results.

This limitation of raw return comparison is the reason risk-adjusted performance measures were developed — to provide a framework that rewards managers who generate the same return with less risk, or higher return with the same risk, and that penalises managers who take excessive risk that is not justified by commensurate return.

Under the fiduciary duty of the Investment Advisers Act of 1940, which binds investment advisers, and the care obligation of Regulation Best Interest at 17 CFR 240.15l-1, which binds broker-dealers and their associated persons when they recommend a securities transaction or investment strategy to a retail customer, an investment adviser evaluating portfolio management decisions and a broker-dealer recommending an investment strategy to a retail customer must each weigh the risk taken to earn a return rather than raw return alone — the adviser’s duty of care calls for a reasonable belief that its advice is in the client’s best interest, considering whether the client can and is willing to tolerate the risks of the investments recommended, and recommending a high-return strategy without assessing the risk required to achieve that return may fail the broker-dealer care obligation’s requirement to exercise reasonable diligence in understanding the potential risks of any recommended strategy.

The Sharpe Ratio — Excess Return Per Unit of Total Risk

The Sharpe ratio — developed by Nobel laureate William F. Sharpe in 1966 and published in the Journal of Business as a measure of mutual fund performance — is the most widely used risk-adjusted performance metric in investment management. It measures the excess return earned above the risk-free rate per unit of total portfolio risk, expressed as standard deviation of returns.

Sharpe ratio equals the portfolio return minus the risk-free rate, divided by the standard deviation of portfolio returns.

The numerator — portfolio return minus the risk-free rate — is the risk premium. It measures how much above the risk-free rate the portfolio earned — the compensation the investor received for taking investment risk rather than holding risk-free assets. Subtracting the risk-free rate is essential because no rational investor should be rewarded for returns that could have been earned without any risk.

The denominator — standard deviation of portfolio returns — is the measure of total risk, capturing both systematic risk arising from broad market movements and unsystematic risk from security-specific factors. Standard deviation is the appropriate risk measure when the portfolio being evaluated represents the investor’s entire invested wealth — in that context all volatility matters, whether it comes from market-wide or company-specific sources.

A higher Sharpe ratio indicates better risk-adjusted performance — the portfolio earned more return per unit of total risk. A Sharpe ratio of one means the portfolio earned one unit of excess return for each unit of standard deviation. A Sharpe ratio of two indicates twice as much excess return per unit of risk — superior risk-adjusted performance. A negative Sharpe ratio — arising when the portfolio return falls below the risk-free rate — indicates the portfolio would have been better off in a risk-free investment.

The Sharpe ratio is most useful when comparing portfolios that are not fully diversified — where unsystematic risk is a meaningful component of total portfolio risk — because the standard deviation denominator captures both systematic and unsystematic risk in the total volatility measure. It is appropriate for evaluating standalone portfolios held in isolation.

What a Historical Sharpe Ratio Can and Cannot Show

William Sharpe’s own restatement of the measure, published in the Journal of Portfolio Management in 1994, draws out several properties that matter whenever a historical Sharpe ratio is used to choose between funds. The first is that the ratio does not depend on the scale of a position financed at the riskless rate. A fund that borrows at the riskless rate to hold twice as much of another fund doubles that fund’s excess return and doubles its standard deviation, and because the two change in the same proportion the leveraged fund’s Sharpe ratio is the same as the original’s — leverage of that kind changes the size of the position, not the excess return earned for each unit of risk.

The second is the ratio’s link to statistical significance. A Sharpe ratio computed from a series of periodic excess returns, multiplied by the square root of the number of returns used, equals the t-statistic that measures the statistical significance of the mean excess return. The same ratio is therefore much stronger evidence when it rests on a long record than on a short one: a monthly ratio of zero point three amounts to a t-statistic of about one over twelve months, but more than two over sixty months.

The third is what the ratio leaves out. Neither the forecast nor the historical Sharpe ratio takes any account of how a fund’s returns are correlated with the investor’s other assets or liabilities, so the fund with the highest ratio taken on its own is not necessarily the best addition to a portfolio the investor already holds, and for that decision the ratio needs to be supplemented. The ratio can also rank funds in a way investors find perverse when excess returns are negative. If two funds have the same negative average excess return, the fund whose returns were more volatile receives the higher, less negative Sharpe ratio, because it lost less per unit of risk — a result consistent with portfolio theory that many retail investors find counterintuitive.

Modigliani Risk-Adjusted Performance — Expressing the Sharpe Ratio in Percentage Points

A Sharpe ratio is a ratio of excess return to standard deviation, not a return, so it cannot be set directly beside the return an investor could have earned in the market. The risk-adjusted performance measure published by Franco Modigliani and Leah Modigliani in the Journal of Portfolio Management in 1997, commonly called M-squared, addresses that difficulty. It adjusts the risk of the portfolio being evaluated so that its standard deviation matches that of the market portfolio, and then states the return the portfolio would have earned at that level of risk. Because the result is a return in percentage points — the ordinary unit in which performance is quoted — it can be compared directly with the market’s own return: a portfolio whose M-squared exceeds the market’s return outperformed the market once risk is equalised, and one whose M-squared falls short of it underperformed.

The measure equals the risk-free rate plus the portfolio’s excess return multiplied by the ratio of the market’s standard deviation to the portfolio’s standard deviation. A portfolio with more risk than the market has its excess return scaled down in proportion, and a portfolio with less risk than the market has its excess return scaled up. In the Modiglianis’ own comparison of mutual funds, a fund whose raw return was higher than the market index’s had an M-squared below the index’s return because it had taken more risk than the index, while a fund with a lower raw return than the index but less risk had an M-squared above it.

Because the portfolio’s excess return divided by its standard deviation is its Sharpe ratio, M-squared is simply the risk-free rate plus the portfolio’s Sharpe ratio multiplied by the market’s standard deviation. For a set of portfolios measured over the same period, the risk-free rate and the market’s standard deviation are the same for every portfolio, so M-squared ranks the portfolios in exactly the order the Sharpe ratio ranks them. What it changes is the unit in which the result is expressed, not the ranking it produces.

The Treynor Ratio — Excess Return Per Unit of Systematic Risk

The Treynor ratio — developed by Jack Treynor in 1965, predating the Sharpe ratio — measures the excess return earned above the risk-free rate per unit of systematic risk, using beta rather than standard deviation as the risk measure.

Treynor ratio equals the portfolio return minus the risk-free rate, divided by the portfolio’s beta.

Beta measures the portfolio’s systematic risk — its sensitivity to movements in the market portfolio — and excludes the unsystematic risk that can be eliminated through diversification. The Treynor ratio is appropriate when the portfolio being evaluated is one component of a larger, diversified investment programme — in that context, only systematic risk matters because unsystematic risk is diversified away at the total portfolio level, and rewarding managers for bearing unsystematic risk would distort the performance assessment.

A higher Treynor ratio indicates better risk-adjusted performance per unit of market risk assumed. Like the Sharpe ratio, the Treynor ratio is a relative ranking tool — it identifies which portfolio earned more excess return per unit of systematic risk but does not indicate whether any specific Treynor ratio level represents adequate compensation in absolute terms.

The key distinction between Sharpe and Treynor is the measure of risk in the denominator — total risk including unsystematic risk for Sharpe, systematic risk only for Treynor. When a portfolio is well-diversified and its unsystematic risk is negligible, Sharpe and Treynor rankings produce similar results. When a portfolio is concentrated and carries significant unsystematic risk, the two measures may produce very different rankings — the Sharpe ratio will penalise the concentrated portfolio’s extra total risk while the Treynor ratio will not.

Jensen’s Alpha — Excess Return Above the CAPM Expected Return

Jensen’s alpha — developed by Michael Jensen in 1968 and published in the Journal of Finance — measures the excess return earned by a portfolio above and beyond what the Capital Asset Pricing Model would predict given the portfolio’s level of systematic risk.

Jensen’s alpha equals the actual portfolio return minus the expected return implied by the CAPM.

The CAPM expected return equals the risk-free rate plus the portfolio’s beta multiplied by the equity risk premium — the market return minus the risk-free rate. The CAPM expected return is the benchmark return that any passive investment strategy with the same beta would have earned — it is the return available to any investor simply by holding the market portfolio with the appropriate leverage or deleverage to match the portfolio’s beta.

Alpha measures the manager’s skill — the additional return earned above the passive benchmark return that can be attributed to active investment decisions rather than to market exposure. A positive alpha means the manager earned more than the market would have delivered for the risk taken — genuine outperformance attributable to skill. A zero alpha means performance exactly matched what a passive strategy would have produced — no value added or destroyed by active management. A negative alpha means the manager underperformed the passive benchmark even on a risk-adjusted basis — value was destroyed.

Alpha has two important advantages over the Sharpe and Treynor ratios. It is an absolute measure — its value can be directly interpreted as the dollars or percentage points of outperformance per year rather than requiring comparison to another portfolio’s ratio. And its sign is immediately interpretable — positive alpha is good, negative alpha is bad, regardless of the magnitude of other measures. Alpha is therefore used both to rank managers and to assess whether any specific manager’s active management is adding value in absolute terms.

The Information Ratio — Consistency of Active Management

The information ratio measures the consistency of active management by comparing the active return — the portfolio return minus the benchmark return — to the tracking error — the standard deviation of the active return over time.

Information ratio equals the portfolio return minus the benchmark return, divided by the tracking error.

The numerator — active return or alpha relative to the benchmark — measures how much the portfolio earned above its designated benchmark. The denominator — tracking error — measures how consistently the portfolio delivered that active return, capturing the volatility of performance differences between the portfolio and its benchmark over rolling periods.

A high information ratio indicates a manager who delivers consistent active returns with low variability of outperformance — the hallmark of a skilled, disciplined active manager with a repeatable process. A low information ratio despite a high average active return indicates a manager whose outperformance is highly variable — sometimes significantly above benchmark, other times significantly below — suggesting that the outperformance may be the result of taking concentrated positions or making inconsistent bets rather than applying a reliably effective investment process.

The information ratio is the performance measure most directly aligned with the evaluation of an active manager’s ability to consistently add value relative to a specific benchmark — making it more useful for institutional manager selection than the Sharpe or Treynor ratios when the investor’s goal is to identify managers who will reliably outperform a designated benchmark rather than to rank absolute risk-adjusted returns.

The Appropriate Measure for Each Context

A critical analytical skill tested on the Series 65 examination is selecting the appropriate risk-adjusted measure for the specific analytical context.

The Sharpe ratio is appropriate when the portfolio represents the investor’s entire wealth — all risk matters, and total risk including unsystematic risk should be measured. It is appropriate for comparing standalone funds where diversification within the fund is the relevant consideration.

The Treynor ratio is appropriate when the portfolio is one component of a larger diversified programme — only systematic risk matters because unsystematic risk is diversified away at the total portfolio level. It is appropriate for evaluating individual fund managers within a multi-manager institutional portfolio.

Jensen’s alpha is appropriate for measuring the absolute skill contribution of active management — the excess return above what passive market exposure with equivalent beta would have delivered — and for ranking managers on the basis of value added.

The information ratio is appropriate for evaluating the consistency and reliability of active management relative to a specific benchmark — the metric of choice for institutional investors conducting manager due diligence and ongoing manager monitoring.

Maximum Drawdown and the Calmar Ratio

Standard deviation treats every departure from the average return alike and takes no account of the order in which returns arrive, so a measure built on it cannot describe an investor’s aversion to a sustained run of losses. Maximum drawdown — the largest fall in value from a peak to a later trough during the period measured — is one of the most widely quoted measures of risk among hedge funds and commodity trading advisors, because long declines can drive investors out of a fund and cost its manager clients.

The Calmar ratio is a risk-adjusted performance measure built on maximum drawdown rather than on standard deviation. It divides the return the portfolio earned over a period by the maximum drawdown it suffered over the same period, so a fund that returned twelve percent over a year in which its largest peak-to-trough fall was twenty percent has a Calmar ratio of zero point six. A higher Calmar ratio indicates more return for each unit of drawdown suffered.

Two properties of maximum drawdown limit what a Calmar ratio can show. First, drawdown depends on the sequence of returns and not only on their average and dispersion. The same set of monthly returns arranged in a different order has the same mean and the same standard deviation, and therefore the same Sharpe ratio, but losses that arrive in an unbroken run produce a much deeper drawdown than the same losses scattered among gains. Second, the expected maximum drawdown grows with the length of the period observed, since a longer record gives a fund more opportunity to suffer a deep decline. A Calmar ratio measured over a ten-year record is therefore not directly comparable with one measured over three years, and Malik Magdon-Ismail and Amir Atiya have proposed scaling laws for drawdown-based ratios, analogous to the square-root-of-time rule for the Sharpe ratio, so that funds with track records of different lengths can be compared.

Upside and Downside Capture Ratios

Capture ratios measure a fund’s performance against a benchmark separately in rising and falling markets. The upside capture ratio takes the fund’s return in the months in which the benchmark’s return was positive and divides it by the benchmark’s return in those same months; the downside capture ratio does the same for the months in which the benchmark’s return was negative. Which months count as up or down is decided by the benchmark, not by the fund. Morningstar, which publishes both ratios, computes them over one-, three-, five-, ten- and fifteen-year periods from the geometric average of the fund’s and the index’s returns in the up months and in the down months respectively.

The two ratios read in opposite directions. An upside capture ratio above one hundred means the fund has generally gained more than its benchmark when the benchmark rose. A downside capture ratio below one hundred means the fund has lost less than its benchmark when the benchmark fell — a fund with a downside capture ratio of eighty would have fallen about eight percent when the market fell ten percent. On the upside a higher figure is better and on the downside a lower one, and a fund whose ratios are both one hundred has moved in step with its benchmark in rising and falling markets alike.

Neither ratio should be read alone. Most funds show a trade-off between the two: a fund with a sizeable advantage over its index in rising markets is likely to have weaker results when the market falls, and the reverse. A fund overweight in defensive stocks may show an attractive downside capture ratio across a volatile period in which the market makes little progress, and then lag when the market turns into a sustained rise. The ratios also depend on the benchmark chosen, since every fund in an asset class is measured against a single index, and a manager whose style differs sharply from that index can be expected to show striking capture ratios.

The Morningstar Rating — Risk-Adjusted Return Within a Peer Group

The star rating that Morningstar assigns to funds rests on a risk-adjusted measure, but on a different idea of risk adjustment from the Sharpe ratio’s. Rather than equalising the risk of two funds before comparing them, Morningstar Risk-Adjusted Return draws on expected utility theory: it is the guaranteed riskless return that would give a model investor the same level of utility as the fund’s actual, variable excess returns — a certainty equivalent. The measure always rewards return and always penalises risk, gives more weight to downside variation than to upside variation, and assumes no particular distribution of returns. Because it is an annualised return, it can be split into a return component and a risk component, Morningstar Risk, which is never less than zero.

The rating then compares each fund only with the other funds in its Morningstar Category, a peer group of funds similar enough in their exposures to risk factors that the differences in their returns reflect mainly security selection and the timing of those exposures. The aim is that a fund’s rating should be affected more by its manager’s skill than by market conditions beyond the manager’s control: in a period when falling interest rates let investment-grade bond funds outperform equity funds, a comparison across asset classes would reward bond managers for circumstances rather than skill. Within each category the funds with the highest risk-adjusted returns receive the most stars, the top ten percent receiving five stars and the bottom ten percent one.

A fund needs at least thirty-six months of returns to be rated. Ratings are calculated for three, five and ten years, and the overall rating is a weighted average of those available, which for a fund with ten or more years of history gives fifty percent to the ten-year rating, thirty percent to the five-year and twenty percent to the three-year. Although the ten-year period carries the largest weight in that formula, the most recent three years have the greatest effect on the overall rating, because they fall within all three periods.

Examination Relevance and Key Takeaways

Risk-adjusted return is tested on the Series 65 examination in the context of portfolio performance evaluation, the four primary risk-adjusted measures, their formulas, their appropriate applications, and the distinction between total risk and systematic risk measures.

The key points to retain are these.

Risk-adjusted return measures how much return was earned per unit of risk assumed — addressing the fundamental limitation of raw return comparisons that ignore risk differences between portfolios. The four primary measures are the Sharpe ratio, the Treynor ratio, Jensen’s alpha, and the information ratio.

The Sharpe ratio equals portfolio return minus the risk-free rate divided by standard deviation of returns — measuring excess return per unit of total risk. It is appropriate when the portfolio represents the investor’s entire wealth or when unsystematic risk is material. A higher Sharpe ratio indicates superior risk-adjusted performance. The Treynor ratio equals portfolio return minus the risk-free rate divided by beta — measuring excess return per unit of systematic risk only. It is appropriate when the portfolio is one component of a larger diversified programme where unsystematic risk is diversified away. Jensen’s alpha equals actual portfolio return minus the CAPM-predicted return for the portfolio’s beta level — measuring the absolute excess return attributable to active management skill. Positive alpha indicates outperformance above passive market exposure. Negative alpha indicates underperformance. Alpha is both a ranking tool and an absolute performance measure. The information ratio equals active return above the benchmark divided by tracking error — measuring the consistency of active outperformance. A high information ratio indicates reliable, repeatable active management skill rather than volatile concentration bets.

When the portfolio is not fully diversified — use Sharpe, because only its standard deviation captures the unsystematic risk that alpha and the Treynor ratio, both measured against beta, leave out. When the portfolio is a component of a larger diversified programme — use Treynor or alpha. When evaluating consistency of benchmark outperformance — use the information ratio. Jensen’s alpha rests on the CAPM, and the Treynor ratio on beta, the CAPM’s measure of systematic risk; the Sharpe ratio, built on total risk, and the information ratio, built on return relative to a chosen benchmark, do not depend on the CAPM. All four should be applied within the context of the specific portfolio’s investment mandate and benchmark.