Quantitative Methods
coreSharpe Ratio
Builds onSample Standard Deviation · Holding Period Return (HPR) — if this page feels steep, start there.
- the Sharpe ratio — excess return earned per unit of risk taken
- the portfolio's return (the p subscript = 'portfolio')
- the risk-free rate (f = 'free'): what cash would have paid with no risk
- the excess return: only the part of the return that required taking risk
- sigma is the standard deviation — the portfolio's volatility
Reading the notation
Why it must be true
Return alone says nothing about skill — you can always buy more return by taking more risk. The Sharpe ratio asks the fairer question: how much excess return did each unit of risk buy? The numerator strips out the risk-free rate (what you'd earn doing nothing risky); the denominator is total volatility.
Comparing two funds on raw return rewards leverage. Comparing on Sharpe rewards efficiency — the fund that converts risk into reward at the better rate.
The derivation
Start from the reward for bearing risk — the excess return over the risk-free rate:
Standardize it by the amount of risk taken, measured as the standard deviation of portfolio returns:
The result is dimensionless — excess return per unit of volatility — which is exactly what makes portfolios directly comparable.
When to reach for it
Comparing funds or strategies on how efficiently they convert risk into excess return.
Listen for
Back-of-the-envelope
Estimate it in your head first — then the calculator only confirms.
- ≈
Subtract, then divide by a rounded σ: 14% − 4% = 10 points of excess on ~20% vol → 0.5. One subtraction, one easy division.
- ≈
Sanity band: real-world Sharpes mostly land between 0.2 and 1.0; above 2 is exceptional. A candidate answer of 3.4 with ordinary inputs is the inverted or variance distractor.
Traps in applying it
- ✗Forgetting to subtract the risk-free rate — gross return over σ is not a Sharpe ratio.
- ✗Dividing by variance instead of standard deviation.
- ✗Comparing ratios computed at different frequencies without consistent annualization (both numerator and √t-scaled denominator).
Limits & criticisms
Sharpe punishes upside volatility like downside, and it assumes returns are roughly normal — strategies with smooth returns and hidden tail risk (option selling, LTCM-style carry) post superb Sharpes right up until they blow up. It is also period-dependent, and negative Sharpe ratios rank perversely (less volatility makes a losing fund look worse).
Where it came from
William F. Sharpe proposed the "reward-to-variability ratio" in 1966 to answer an awkward question: mutual funds were advertising raw returns while taking wildly different risks. Sharpe's fix — excess return per unit of volatility — was so useful the profession renamed it after him; he shared the 1990 Nobel Prize for the CAPM work it grew from.
Today it is the default risk-adjusted score for funds, strategies and backtests, and the target variable of risk budgeting: hedge funds are hired and fired on this one number.
One identity, 2 questions
The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.
Reward per unit of risk
The comparison face: strips out leverage and scale so two portfolios can be ranked on efficiency alone.
Return required for a target Sharpe
The planning face: at a given volatility, the return a manager must deliver to hit a Sharpe target — risk budgeting in one line.
Where it leads
Master this and the following come almost for free: