Modern Portfolio Theory (MPT) Risk & Diversification Calculator
The formula
Harry Markowitz and "The Only Free Lunch in Finance"
In finance, higher returns almost universally require taking on higher risk. The single mathematical exception discovered by Harry Markowitz in 1952 is diversification. When two assets have a correlation below +1.0, combining them in a portfolio eliminates unsystematic risk without sacrificing expected return.
Frequently asked questions
What is a good Sharpe Ratio?
Source & methodology
Markowitz H. Portfolio Selection (J Finance, 1952); Sharpe WF. Mutual Fund Performance (J Business, 1966).
Modern Portfolio Theory (MPT) Risk & Diversification: what this tool answers
This page was written for one job: turning your own inputs into a defensible figure, with the working shown. There are plenty of pages about Modern Portfolio Theory (MPT) Risk & Diversification; almost none of them show the working. This one does: the formula, every input, a fully computed example, and a sensitivity analysis calculated live from this engine’s own defaults.
It helps to separate the mechanics from the interpretation. The mechanics are fixed: σ_p = √[ w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ ] | Sharpe = (E(R_p) − R_f) / σ_p. Anyone, anywhere, entering the same inputs will get the same 10.40%. The interpretation is where experience counts: whether 10.40% is good, normal or alarming depends on comparisons covered below, and that is the part most articles about Modern Portfolio Theory (MPT) Risk & Diversification never explain.
In plain terms, Modern Portfolio Theory (MPT) Risk & Diversification answers the question "Total Portfolio Volatility (Risk σ)?". The measurement matters because it converts a situation that feels complicated — several inputs pulling in different directions — into a single number you can act on. Within Investing & Markets, Portfolio Management is the context where this figure is quoted most often, and professionals treat it as a first-pass screen rather than a verdict.
What makes a calculation page trustworthy is that its copy is derived from its own arithmetic: here total Portfolio Volatility (Risk σ) = 10.40%, recomputed every time a field changes, and the sensitivity section ranks the inputs that move the result — Asset 1 annual volatility / standard deviation first, at ≈8.9 % of the headline for a ten-percent error either way in that input. Read the sections in order and you will be able to reproduce every figure by hand.
Every input in Modern Portfolio Theory (MPT) Risk & Diversification, and what it does to the answer
One formula, published in full: σ_p = √[ w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ ] | Sharpe = (E(R_p) − R_f) / σ_p. Anything you can read, you can reproduce in a spreadsheet, and that reproducibility is the whole point of printing it rather than hiding it behind a button.
Asset 1 allocation weight (%) (example value 60) is a measured number in the unit named by the label; the formula does not convert for you. Enter it exactly as your document states it — quoted annual, monthly and effective rates are three different numbers.
Asset 1 expected annual return (%) (example value 10) is a quantity rather than a category — precision here is cheap and worth having. Rates are the most mis-transcribed input on any page like this; copy it from the document rather than retyping from memory.
Asset 1 annual volatility / standard deviation (%) (example value 16) is a measured number in the unit named by the label; the formula does not convert for you. Enter it exactly as your document states it — quoted annual, monthly and effective rates are three different numbers.
Asset 2 expected annual return (%) (example value 5) is a quantity rather than a category — precision here is cheap and worth having. Enter it exactly as your document states it — quoted annual, monthly and effective rates are three different numbers.
Asset 2 annual volatility / standard deviation (%) (example value 7) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. Rates are the most mis-transcribed input on any page like this; copy it from the document rather than retyping from memory.
Correlation coefficient between assets (ρ) (example value 0) is a measured number in the unit named by the label; the formula does not convert for you. Replace the example with the figure from your own records — the tool recomputes instantly and the printed steps follow your numbers.
Risk-free benchmark rate (%) (example value 4) is a measured number in the unit named by the label; the formula does not convert for you. Check the convention first: if the source says “6.5 % per year”, type 6.5 and let the formula do the rest.
Modern Portfolio Theory (MPT) Risk & Diversification with real numbers, step by step
Abstract formulas are hard to trust, so here is the complete substitution using the default inputs, written the way an examiner would want it. Starting figures: Asset 1 allocation weight (%), Asset 1 expected annual return (%), Asset 1 annual volatility / standard deviation (%), Asset 2 expected annual return (%). The computed total Portfolio Volatility (Risk σ) comes out at 10.40%, which the interpretation band labels “🟢 Powerful Diversification: Negative or low correlation shaves 2.00 percentage points off total risk!”.
- Asset Weights: w₁ = 60%, w₂ = 40%
- Expected Return = (0.60 × 10%) + (0.40 × 4.5%) = 7.80%
- Portfolio Variance σ² = (0.0960)² + (0.0280)² + 2(0.60)(0.40)(0.15)(0.1600)(0.0700) = 0.010806
- Portfolio Volatility σ = √Variance = 10.40%
- Risk Reduced via Diversification = 12.40% − 10.40% = 2.00 percentage points
Expected Portfolio Return — 7.80%. This is the number experienced users quote when the headline alone would mislead.
Portfolio Sharpe Ratio — 0.34. This is the number experienced users quote when the headline alone would mislead.
Diversification Free Lunch (Risk Saved) — 2.00%. Compare it against the main result of 10.40% before drawing conclusions.
Weighted Average Volatility (No Diversification) — 12.40%. If your inputs change, this is usually the first companion figure to move.
Asset 1 Allocation Weight — 60%. Read next to the headline, this is the figure that shows the shape of the trade-off.
Overwrite any input above and every line of this example updates with it — the substitution is live, not a screenshot of somebody else’s numbers.
The Modern Portfolio Theory (MPT) Risk & Diversification corridor: your result across a realistic band
Read this table as the practical error bar on the headline figure. It answers the question people actually have after seeing a single number: how wrong could this be if the main input is off by ten or twenty percent?
| Asset 1 annual volatility / standard deviation setting | Total Portfolio Volatility (Risk σ) |
|---|---|
| 80 % of the example value | 8.56% |
| 90 % of the example value | 9.47% |
| example inputs | 10.40% |
| 110 % of the example value | 11.32% |
| 120 % of the example value | 12.26% |
That spread — 8.56% at the low end, 12.26% at the high — is what a careful read of Modern Portfolio Theory (MPT) Risk & Diversification looks like: a band you can state out loud, with the reason for each edge. Reproduce it in the form above by editing Asset 1 annual volatility / standard deviation alone; the substitution steps will follow your figures exactly.
Sensitivity: which inputs change Modern Portfolio Theory (MPT) Risk & Diversification the most
Nobody should have to guess which input matters. The engine was re-run once per field below — ten percent up, ten percent down, everything else frozen — and the results ranked by how far they moved the total Portfolio Volatility (Risk σ). Read it as a map of where Modern Portfolio Theory (MPT) Risk & Diversification is fragile — drawn for the example inputs, which is the map you want when deciding what to measure carefully.
- Asset 1 annual volatility / standard deviation: ten percent up gives 11.32%, ten percent down gives 9.47% — ≈8.9 % of the total Portfolio Volatility (Risk σ) either way
- Asset 1 allocation weight +10 % → Modern Portfolio Theory (MPT) Risk & Diversification = 11.17% · −10 % → 9.66% (the total Portfolio Volatility (Risk σ) shifts ≈7.4 %)
- Asset 2 annual volatility / standard deviation: ten percent up gives 10.51%, ten percent down gives 10.28% — ≈1.1 % of the total Portfolio Volatility (Risk σ) either way
- Correlation coefficient between assets +10 % → Modern Portfolio Theory (MPT) Risk & Diversification = 10.43% · −10 % → 10.36% (the total Portfolio Volatility (Risk σ) shifts ≈0.4 %)
Asset 1 annual volatility / standard deviation is the dominant lever here: ten percent either way moves the result by about 8.9 %, from 9.47% to 11.32%. That is where a careless figure does the most damage, so check it against a source, and treat Asset 1 allocation weight as the second priority (≈7.4 % per ten percent). The rest of the inputs earn only a rounding-error of attention, and that asymmetry is the practical payoff of measuring rather than asserting.
Is your Modern Portfolio Theory (MPT) Risk & Diversification good — reading the result bands
Nobody acts on a bare figure — they act on a figure with a comparison attached. On the example inputs the band reads “🟢 Powerful Diversification: Negative or low correlation shaves 2.00 percentage points off total risk!”; your own figures place you on a different rung of the same ladder. How much of that band is luck? On Asset 1 annual volatility / standard deviation alone ten percent either way covers 9.47% to 11.32%. The practical reading rules: read the band as a starting hypothesis, then test it: change the two dominant inputs by ten percent and see whether the verdict survives. Verdicts that survive stress are the ones worth acting on.
In Investing & Markets, Modern Portfolio Theory (MPT) Risk & Diversification is one instrument in an orchestra: on its own it is a number, next to Expected Portfolio Return and Portfolio Sharpe Ratio it becomes a diagnosis. The tools linked above exist so the headline never has to carry that weight alone.
The errors that quietly break a Modern Portfolio Theory (MPT) Risk & Diversification figure
Copying a single headline number without checking the companion outputs. The tool reports Expected Portfolio Return, Portfolio Sharpe Ratio and Diversification Free Lunch (Risk Saved) alongside the main result precisely because Modern Portfolio Theory (MPT) Risk & Diversification in isolation misleads — the secondary columns are where outliers and trade-offs show up.
Copying a single headline number without checking the companion outputs. The tool reports Expected Portfolio Return, Portfolio Sharpe Ratio and Diversification Free Lunch (Risk Saved) alongside the main result precisely because Modern Portfolio Theory (MPT) Risk & Diversification in isolation misleads — the secondary columns are where outliers and trade-offs show up.
Attention is also misallocated between fields. Having measured this engine, we can say where it belongs: Asset 1 annual volatility / standard deviation first, everything else second.
A final trap is presentation: reporting Modern Portfolio Theory (MPT) Risk & Diversification to more decimal places than the inputs support. The tool prints what the arithmetic gives; you should quote only what your least precise field justifies.
Six things to confirm before acting on Modern Portfolio Theory (MPT) Risk & Diversification
- Compare the result with one independent method or published benchmark.
- Confirm the example values are plausible for your own situation before replacing them.
- Check Asset 1 annual volatility / standard deviation before anything else: it is the measured dominant input, ≈8.9 % of Modern Portfolio Theory (MPT) Risk & Diversification per ten percent.
- Record the date and the inputs alongside the number so future-you knows what was assumed.
- State, in one sentence, which convention each rate, unit or period in the form is using.
- Repeat the ±10 % test on the inputs that matter and see whether the decision — not just the number — still holds.
A routine for decision-grade Modern Portfolio Theory (MPT) Risk & Diversification
Using a calculator well sounds trivial, but the difference between a decision-grade figure and a throwaway one lives entirely in the routine. The routine for Modern Portfolio Theory (MPT) Risk & Diversification takes a minute to learn and applies to every input above.
Step 1 — one document, one pass. Open the statement, label or record that holds asset 1 allocation weight (%), asset 1 expected annual return (%), asset 1 annual volatility / standard deviation (%), asset 2 expected annual return (%), asset 2 annual volatility / standard deviation (%), correlation coefficient between assets (ρ) and the remaining fields listed above and transcribe straight into the form. Transcription from a single source removes the mismatch errors that appear when figures are assembled from three different places. Numbers are cheap to produce and expensive to retract; the extra minute is the cheapest insurance available.
Step 2 — start from the case you can defend. Enter pessimistic-but-honest figures first; the optimistic run is a ceiling, not a plan. A conclusion that holds on conservative inputs holds everywhere. Say the answer out loud before you save it; if it sounds wrong, the model did not change, your understanding did.
Step 3 — change one input at a time. Nudge a single field by ten percent, watch the headline and its companions move, then undo it. Two or three fields will dominate — the sensitivity table above already ranks them for this engine. If the verdict changes with a ten-percent nudge, report the nudge alongside the number.
Step 4 — run the scenario trio. Expected, pessimistic, optimistic. The gap between pessimistic and expected is the number to plan against; the optimistic result is useful only as a ceiling. A figure with a date beside it becomes evidence; without one it stays an anecdote.
Step 5 — write the one-line memo. “Total Portfolio Volatility (Risk σ) = 10.40% on this date, from these inputs.” That line is the entire deliverable; everything else on this page is how you earned the right to write it. The point of transparency is that a reader can disagree with a specific line rather than with you.
From calculation to decision: where Modern Portfolio Theory (MPT) Risk & Diversification earns its keep
Investing & Markets arguments end at arithmetic or they do not end. Modern Portfolio Theory (MPT) Risk & Diversification is the arithmetic: 10.40% on the example inputs, reproducible by anyone who types the same numbers. Comparisons are cheap: run the identical calculation for the alternative and look at the gap.
Professionals in Portfolio Management use this style of calculation as a screening filter before spending real money or effort: a quick Modern Portfolio Theory (MPT) Risk & Diversification check eliminates weak options in minutes, and the survivors earn the deeper, slower analysis. Precision is a promise you make about your data; make sure your inputs can keep it.
Re-running this page costs nothing and catches the slow drifts that never announce themselves. In Investing & Markets the expensive mistakes are rarely sudden; they are small changes in Asset 1 allocation weight (%) that nobody recalculated for a year. A calculation you cannot explain to somebody else is a calculation you cannot defend.
People also ask about Modern Portfolio Theory (MPT) Risk & Diversification
What affects Modern Portfolio Theory (MPT) Risk & Diversification the most?
The short answer: Asset 1 annual volatility / standard deviation. The longer answer is the sensitivity table on this page, where every numeric input was probed by ten percent and the total Portfolio Volatility (Risk σ) recalculated — Asset 1 annual volatility / standard deviation moved it furthest, by roughly 8.9 % (9.47% to 11.32%).
Which matters more: asset 1 allocation weight or asset 1 annual volatility / standard deviation?
By the numbers, Asset 1 annual volatility / standard deviation: a ten-percent change in it moves the result ≈8.9 %, versus ≈7.4 % for Asset 1 allocation weight. Spend your measuring effort in that order — precision on the weaker lever buys almost nothing.
What is a typical Modern Portfolio Theory (MPT) Risk & Diversification?
On the worked example above Modern Portfolio Theory (MPT) Risk & Diversification comes out at 10.40%. Treat that as one point on a distribution rather than a norm: the bands, the companion outputs and the sensitivity ranking tell you how wide the plausible range really is.
How do I calculate Modern Portfolio Theory (MPT) Risk & Diversification by hand?
σ_p = √[ w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ ] | Sharpe = (E(R_p) − R_f) / σ_p is the entire method. Work the inputs in that order — Asset Weights: w₁ = 60%, w₂ = 40% — and you will reach 10.40% on the default figures, the same number this tool prints.
What happens to Modern Portfolio Theory (MPT) Risk & Diversification if asset 1 annual volatility / standard deviation changes by 10 %?
The engine was re-run with that field moved by ten percent while everything else stayed fixed: the total Portfolio Volatility (Risk σ) goes across 9.47% to 11.32%, a swing of roughly 8.9 % around the 10.40% baseline. That ratio is the practical definition of how much this input matters.
Is the Modern Portfolio Theory (MPT) Risk & Diversification result of 10.40% good or bad?
Good and bad are comparisons, so the honest test is three questions: where does your own Modern Portfolio Theory (MPT) Risk & Diversification sit against the 10.40% reference above, how far would a plausible input error move it (≈8.9 % per ten percent on Asset 1 annual volatility / standard deviation), and what does the alternative option compute to?
How often should I check Modern Portfolio Theory (MPT) Risk & Diversification?
Whenever an input changes by more than a few percent, and otherwise on the rhythm of your decisions — before a purchase, a review, a renewal. Modern Portfolio Theory (MPT) Risk & Diversification is a snapshot, not a subscription; stale snapshots are the most common silent error in Investing & Markets planning.
Can Modern Portfolio Theory (MPT) Risk & Diversification be negative or empty?
Yes, and that is informative rather than broken: a negative Modern Portfolio Theory (MPT) Risk & Diversification usually means the inputs encode an outflow or a shortfall, and the verdict band above labels exactly that situation instead of hiding it.
What is a good Sharpe Ratio?
A Sharpe Ratio under 1.0 is standard; 1.0 to 1.99 is considered good; 2.0 to 2.99 is very good; and 3.0+ is considered elite institutional performance.
Is my data private on this page?
Yes — computation happens locally in your browser, with no account, no tracking of your inputs and nothing sent to a server. The printed formula is the whole story of what happens to your figures.
Key terms on this page
- Expected Portfolio Return
- 7.80% for the Modern Portfolio Theory (MPT) Risk & Diversification worked case; it sits next to the main result because it carries context the headline leaves out.
- Portfolio Sharpe Ratio
- 0.34 on the example inputs — a companion figure printed beside the headline of Modern Portfolio Theory (MPT) Risk & Diversification so that no result is read in isolation.
- Diversification Free Lunch (Risk Saved)
- 2.00% for the Modern Portfolio Theory (MPT) Risk & Diversification worked case; it sits next to the main result because it carries context the headline leaves out.
- Total Portfolio Volatility (Risk σ)
- The primary output of this tool: 10.40% for the default inputs, computed exactly from σ_p = √[ w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ ] | Sharpe = (E(R_p) − R_f) / σ_p.
- Worked example
- The calculation written out in full so that nothing on this page has to be taken on trust.
- Verdict band
- A named zone on the range of plausible results; the example inputs land in “🟢 Powerful Diversification: Negative or low correlation shaves 2.00 percentage points off total risk!”, and your inputs may not.
- Sensitivity (±10 %)
- How far the Modern Portfolio Theory (MPT) Risk & Diversification moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
The provenance of this Modern Portfolio Theory (MPT) Risk & Diversification calculation
Everything above runs on one stated formula, applied without hidden coefficients: σ_p = √[ w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ ] | Sharpe = (E(R_p) − R_f) / σ_p. Source of record: Markowitz H. Portfolio Selection (J Finance, 1952); Sharpe WF. Mutual Fund Performance (J Business, 1966). Any other implementation of this model should return the same figure for the same inputs. If one does not, the difference is in its assumptions — which this page prints and that one probably does not.
This page is an estimation aid, not professional advice. The design rule is simple: two people entering identical figures must get identical answers. Nothing is smoothed, weighted or adjusted behind the scenes; where judgement enters, this page says so and names the input responsible.
Next steps after the calculation
Re-run the calculation with your own figures, stress the two most sensitive inputs by ±10 %, and only then act on the result.
This page was written for people who need one trustworthy number before their next decision, and the promise is that nothing above required trust. Bookmark the tool, re-run it whenever an input moves by more than a few percent, and keep the date with the number.