Model explainer — theory, significance, and formulas.
Each model's background and principle in plain language, so newcomers to finance can follow along, followed in each section by how DIVA Quantizer implements it and how to use it.
Markowitz Mean–Variance Optimization
When you hold several assets, how much of each should you mix to manage the portfolio’s risk and return together?
Markowitz’s answer: look not only at returns but at how the assets relate to each other. DIVA applies the theory to real market data so you can see the structure of your own portfolio.
Look at the whole portfolio, not one holding.
Diversification is not about the number of holdings. Mixing assets that move differently lowers the risk of the whole portfolio.
Portfolio risk is not the sum of individual risks; it also reflects how the holdings relate — their covariance.
In DIVA, you see your portfolio’s diversification as a number.
DIVA turns the risk removed by the relationships in this formula into a single number, the diversification benefit: how many percent the actual portfolio volatility sits below the weight-averaged volatility of the holdings.
It is one of the inputs to the portfolio personality (archetype) at the top of the result page.
Right below, the key metrics show where the combined portfolio’s annual volatility stands against the market.
The Easygoing Stroller평화주의 산책꾼
Spreading assets wide brings you peace of mind as you head, slowly, toward your destination.
In the example portfolio (AAPL 25% · MSFT 20% · SPY 25% · TLT 30%), the holdings’ weight-averaged volatility is 19.25%, but held together the actual volatility drops to 15.43%. That gap is a diversification benefit of 19.9%, one of the grounds for the archetype “The Easygoing Stroller”.
The archetype is one of eight, set by three criteria: risk (vs. the market), efficiency and diversification. A diversification benefit of 15% or more counts as “diversified”.
The efficient frontier
Every possible weight mix has its own expected return (μ) and risk (σ). Plot all of them and a curve emerges.
The line joining the mixes that give more return at the same risk, or less risk at the same return is the efficient frontier.
Its leftmost point is the minimum variance portfolio (MVP) — the least volatile mix possible.
DIVA compares your portfolio against that curve.
DIVA draws this curve from mixes of your own holdings only, and marks your portfolio and each holding on the same chart.
So you can see right away how far your portfolio sits from the curve.
In other words, you can see “how efficient are my current weights?” at a glance.
The farther below the curve, the more another weight mix could earn at the same risk.
The example sits about 5.83%p below the curve at the same risk. Within 0.05%p of the curve, the dot turns green.
Lowest risk and highest efficiency are not the same thing.
The MVP is simply the least volatile portfolio. Low risk does not mean high expected return.
Measured from the risk-free rate Rf, the portfolio where a line touches the frontier is the tangency portfolio — the maximum-Sharpe portfolio.
The Sharpe ratio shows how much excess return you earned for the risk you took.
In DIVA, your portfolio’s position matters more than “the right weights”.
DIVA does not present the max-Sharpe portfolio as recommended weights; it uses it as a yardstick for how close your portfolio is to the frontier.
It also shows expected return, volatility, Sharpe and diversification together, so no single number drives the judgment.
In the equalizer you can move the weights yourself and watch these metrics change.
The MVP sits at the curve’s left tip, but the tangency point, where the line from Rf touches the curve, earns the most excess return per unit of risk. The example portfolio sits below the curve, away from both.
The tangency here uses the raw historical curve (Sharpe 0.88). DIVA’s efficiency verdict uses an error-corrected curve (max Sharpe 0.69).
From the formula to real data
The Markowitz problem can be written as:
That is, fix a target return and find the lowest-risk weights that achieve it.
How DIVA fills in each term of the formula
- μ · Σ
- Estimated from monthly returns over the last 6 years, in the window all holdings share (min. 12 months).
- wi ≥ 0
- No short selling — each holding 0–100%, summing to 100%.
- R*
- 40 targets from the minimum-variance return up to the best holding’s return; solving the formula for each and joining the points draws the efficient frontier.
- Rf
- Current 13-week US T-bill rate (^IRX), used site-wide (3.99% in the example).
- Display
- Monthly values are annualized — return ×12, volatility ×√12.
Moments when Markowitz helps
- 01Check whether your current weights are efficient
The distance to the efficient frontier shows where your portfolio stands.
- 02Check diversification before adding a holding
Compare the diversification benefit and total portfolio risk before and after.
- 03See where your portfolio risk is concentrated
Risk Contribution shows what share of total risk each holding accounts for.
- 04See how the results change when you shift weights
Move weights in the equalizer and see return, volatility, Sharpe and diversification change instantly.
To set your return against the market and split it into market beta and alpha, continue with CAPM. If you hold many names, or care more about the risk structure than about return estimates, compare with HRP. To see where return comes from factor by factor, a Fama–French model may fit better.
Reading the result with care
Markowitz relies on the assumption that mean returns and covariances estimated from the past will roughly hold in the future. With more holdings, the covariance estimate can become unstable. So optimization results should not be read as the right weights for the future.
DIVA does not present the max-Sharpe portfolio as recommended weights.
Instead, it shows how your portfolio relates to the frontier, alongside the result’s reliability and how concentrated the risk is.
If the result looks unstable, run the same portfolio through HRP and compare the two approaches.
References
- Wikipedia — Modern Portfolio Theory · Overview of the efficient frontier, MVP and the tangency portfolio
- Investopedia — Efficient Frontier · Beginner-friendly explanation
- Nobel Prize — Harry Markowitz (1990) · Official Nobel citation
- H. Markowitz, “Portfolio Selection,” Journal of Finance, 1952.
Now apply the theory to your own portfolio.
Enter your holdings and amounts to see the Markowitz analysis as real numbers.