Risk vs Return: the “why” behind the numbers

Investors like to say “higher return requires higher risk.” That’s often true in broad strokes — but it’s also incomplete. The real question is: how much return are you getting for the risk you’re taking? This is why risk‑adjusted metrics exist.

Expected return is your best estimate of the average annual gain (or loss). In practice, people often use long‑term historical averages, a forward-looking model, or a personal assumption. If you expect 8% per year, that’s your μ.

Volatility measures how wildly returns swing around the average. Two investments can have the same expected return, but one may be much more turbulent. In many financial models, volatility is used as a convenient “risk” stand‑in.

Sharpe ratio compares your expected return to a risk‑free benchmark, then divides by volatility:
Sharpe = (μ − Rf) ÷ σ
Intuition: it answers, “How much extra return am I getting per 1% of risk?” A higher Sharpe ratio usually means a better risk‑return tradeoff.

VaR is a popular “worst‑case” style estimate. At 95% confidence, VaR tries to describe a loss level you might expect to exceed only about 5% of the time (under the model assumptions). This calculator uses a simple normal‑distribution approximation: the horizon return is modeled as normal with mean and volatility scaled by time.

Another way to think about risk: what’s the chance the ending value is below the starting value over your time horizon? Under a normal model, the probability is the cumulative probability that the horizon return is negative.

Important: these are standard textbook-style calculations, but real markets can be non‑normal and path‑dependent. Use them as a first-pass comparison tool, not a final decision engine.

What this calculator actually computes

We estimate a future value using your expected return and optional periodic contributions (monthly/quarterly/yearly). Contributions are treated as end‑of‑period deposits (ordinary annuity).

• Sharpe ratio using your risk‑free rate.
• 1‑year VaR (based on your initial investment) at your chosen confidence.
• Horizon VaR (scaled by time horizon) on the projected ending value.
• Probability of loss over the horizon.
• Risk‑Adjusted Score (0–100) for quick comparison & sharing.

The Risk‑Adjusted Score is a friendly summary that rewards higher Sharpe ratios and penalizes extreme volatility. It is not an industry standard — it’s a shareable shortcut that helps you compare scenarios quickly.

• Comparing two portfolios with different risk levels.
• Stress‑testing assumptions (“What if my volatility is higher?”).
• Explaining risk to friends or clients with a single screenshot.

Realistic examples (with intuition)

Suppose you expect 6% return with 10% volatility, risk‑free rate 4%, and invest $10,000 for 10 years with $200 monthly contributions. The Sharpe ratio is (6−4)/10 = 0.20, which is modest — the portfolio isn’t taking a lot of risk, but it also isn’t generating huge “excess return” above the risk‑free rate.

If you expect 8% return at 16% volatility and keep risk‑free at 4%, Sharpe becomes (8−4)/16 = 0.25. Slightly better risk‑adjusted payoff, but swings are bigger. VaR will typically increase because volatility rises.

With 15% return and 60% volatility, Sharpe becomes (15−4)/60 ≈ 0.18. Even though return looks huge, you’re “paying” for it with wild volatility. This is the risk-return trap many people fall into: big expected return doesn’t automatically mean better risk-adjusted performance.

Use the calculator like a comparison engine: change one assumption at a time and watch how Sharpe and VaR react. If you can improve Sharpe while keeping volatility reasonable, you’re usually improving the overall risk-return tradeoff.

Frequently Asked Questions

• Is volatility the same as “risk”?

  Not exactly. Volatility is a common proxy because it’s easy to measure and plug into models, but true risk can include permanent loss, leverage, liquidity, concentration, and behavior. Still, volatility is a useful baseline for comparisons.

• What is a “good” Sharpe ratio?

  There’s no universal cutoff, but a rough rule of thumb: 0–0.5 is low/modest, 0.5–1.0 is decent, 1.0+ is strong, and 2.0+ is exceptional. Context matters: time period, asset class, and estimation method all change the meaning.

• How does VaR differ from “worst-case loss”?

  VaR is not the absolute worst loss — it’s a quantile. A 95% VaR means “under this model, losses worse than this happen about 5% of the time.” In extreme crashes, losses can exceed VaR dramatically.

• Why does horizon VaR grow with time?

  In this simplified model, volatility scales with the square root of time (σ√t), while mean return scales linearly (μt). Over longer horizons, uncertainty grows, which increases the spread of possible outcomes.

• Should I use nominal or real return?

  If you care about purchasing power, use real return (inflation-adjusted). If you’re comparing to nominal debt rates or account statements, nominal return is fine. For inflation-aware investing, try the Inflation Adjusted Return Calculator.

• Does this account for diversification between assets?

  Not directly. You input a single volatility number for the overall portfolio. To model diversification, estimate volatility for the combined portfolio (or compute it using weights and correlations).