Risk–Reward Trade-Offs in Quantitative Portfolio Optimization

Summary

The document answers whether quantitative finance considers the risk–reward trade-off. It connects the idea to mean–variance portfolio optimization, where portfolio weights are chosen to minimize variance subject to a target expected return. It also identifies the Sharpe ratio as a common performance measure that relates excess return to return volatility.

This frames risk and reward at the portfolio level rather than prescribing a minimum payoff ratio for an individual trade. The answer offers no specific threshold, empirical comparison, or detailed treatment of alternative risk measures; it describes the basic framework and notes that the Sharpe ratio is a crude summary. Its central takeaway is that quantitative approaches express the trade-off through optimization objectives and performance metrics rather than a universal fixed risk-to-reward rule.

Key ideas

• Mean–variance optimization formalizes the balance between portfolio risk and expected return.
• Portfolio variance can be minimized subject to a constraint on expected return.
• The Sharpe ratio compares excess return with return variability as a performance measure.
• The document does not establish a universal minimum risk–reward ratio for individual trades.

Tags

# Is the risk-reward ratio considered in Quantitative Finance?


# Is the risk-reward ratio considered in Quantitative Finance?












Many discretionary traders swear by risk-reward ratio, as in "The minimum risk-reward ratio for a Forex trade is 1:2."

Do quantative traders use risk-to-reward ratio as well? If so, how do you calculate the minimum risk-reward ratio?

## Answer by Shane (score 3, accepted)

https://quant.stackexchange.com/a/9384

Maximizing expected return while minimizing risk is at the heart of the quantitative revolution in finance in modern portfolio theory.

Starting with Harry Markowitz (1952) "Portfolio Selection", a huge portion of quantitative finance is dedicated to refining the ideas around mean-variance portfolio optimization. The objective is to find a weight vector $w$ that will minimize:

$$w^T \Sigma w$$

subject to:

$$R^T w = \mu$$

When evaluating performance, the Sharpe ratio is the most widely used performance measure, and it directly (if a little crude) addresses the trade-off between risk and reward.

$$S = \frac{E[R-R_f]}{\sqrt{\mathrm{var}[R]}}$$

I recommend reading Peter Bernstein's "Capital Ideas" as a gentle introduction to this subject.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.