Week 41 Day 1: The Sharpe Ratio: How Much Are You Getting Paid to Take Risk?

Fund A returns 15% with 30% volatility. Fund B returns 10% with 15% volatility. Which is better? Fund A earned more, but it also bounced around twice as much. The Sharpe ratio settles it: Fund A: (15% - 4%) / 30% = 0.37. Fund B: (10% - 4%) / 15% = 0.40. Fund B wins. It delivers more return per unit of risk.

The Sharpe ratio formula: Sharpe = (Return - Risk-free rate) / Standard deviation. 'Return' is the average annual return of the investment. 'Risk-free rate' is the return on Treasury bills (currently approximately 4-5%). 'Standard deviation' is the annual volatility. The numerator (Return - Risk-free rate) is the 'excess return' -- the extra return you earn above the guaranteed rate, which compensates you for taking risk. The denominator (standard deviation) is the risk you took. The ratio tells you: how much excess return per unit of risk? Sharpe ratios for common investments (approximate, long-term): Cash/T-bills: 0.0 (by definition -- it IS the risk-free rate). U.S. bonds (BND): 0.2-0.4. U.S. stocks (VTI): 0.4-0.5. 60/40 balanced portfolio: 0.5-0.6. SCHD (dividend stocks): 0.4-0.5. International stocks (VXUS): 0.3-0.4. Individual stock picking: typically 0.1-0.3 (lower because of idiosyncratic risk). Hedge funds (aggregate): 0.3-0.4. Bitcoin: 0.2-0.5 (highly variable). Key insight: a 60/40 portfolio has a HIGHER Sharpe ratio than a 100% stock portfolio. Adding bonds improves risk-adjusted returns even though it slightly reduces total return. This is because the reduction in volatility from adding bonds more than compensates for the reduction in return. The Sharpe ratio explains WHY diversification works: it improves the bang per buck of risk.

The Sharpe ratio was introduced by William Sharpe (1966) and is formally defined as S = (E[R_p] - R_f) / sigma_p, where E[R_p] is the expected portfolio return, R_f is the risk-free rate, and sigma_p is the portfolio standard deviation. Under mean-variance preferences, the Sharpe ratio is a sufficient statistic for portfolio ranking: of all portfolios on the mean-variance efficient frontier, the tangency portfolio (the one that maximizes the Sharpe ratio) is the portfolio every investor should hold (in combination with the risk-free asset) regardless of their risk preference. Investors who are more risk-averse hold more of the risk-free asset and less of the tangency portfolio; risk-seeking investors hold less of the risk-free asset (or borrow to leverage the tangency portfolio). This is the separation theorem (Tobin, 1958): the optimal risky portfolio is the same for all investors; risk preferences only determine the allocation between that portfolio and cash. Empirically, the maximum achievable Sharpe ratio for a diversified portfolio is approximately 0.5-0.7 per year. Strategies that report Sharpe ratios >1.0 over extended periods should be treated with skepticism -- they typically involve either (a) data mining (selecting the strategy post-hoc from many tested strategies), (b) survivor bias (only successful strategies are reported), (c) illiquidity premium (returns are smoothed because of infrequent trading, understating true volatility), or (d) tail risk (the strategy earns steady returns most of the time but occasionally suffers catastrophic losses that are not captured in historical volatility). The Sortino ratio (Sortino and Price, 1994) addresses one Sharpe ratio limitation by using downside deviation instead of total standard deviation, penalizing only harmful volatility (losses) rather than total volatility (which includes upside moves).