How to use the two asset portfolio calculator
- Enter the weight of asset A as a percent. Asset B automatically gets the rest, so the two always add up to 100%.
- Enter each asset's expected return and risk (standard deviation), both as percents a year. These are assumptions of yours, so the starting values are examples to replace.
- Enter the correlation between the two assets, from -1 (they move in opposite directions) to 1 (they move together).
- Read the expected return and risk of your mix, the risk removed by mixing, and the weights that would give the lowest risk. The table below shows the same inputs at 0%, 10% up to 100% in asset A. Use Copy, Print or CSV to keep it.
Formula
Expected return = wA x rA + wB x rB
Variance = wA^2 x sA^2 + wB^2 x sB^2 + 2 x wA x wB x correlation x sA x sB
Risk = square root of the variance
Lowest risk weight of A = (sB^2 - correlation x sA x sB) / (sA^2 + sB^2 - 2 x correlation x sA x sB)
wA and wB are the weights as fractions, with wB = 1 - wA. The lowest risk weight is limited to 0% through 100% because short positions are not modeled.
60% in A and 40% in B
Take asset A with an expected return of 8% and risk of 15%, asset B with 4% and 6%, and a correlation of 0.3. At 60% in A and 40% in B, the expected return is 0.6 x 8 + 0.4 x 4 = 6.4%.
The variance is 0.36 x 225 + 0.16 x 36 + 2 x 0.6 x 0.4 x 0.3 x 15 x 6 = 81 + 5.76 + 12.96 = 99.72, so the risk is the square root, 9.99%. The two risks averaged by weight would be 11.4%, so mixing removes 1.41 points of risk. The lowest risk mix is 4.35% in A and 95.65% in B, at 5.97% risk.
Why mixing two assets can lower risk
Risk adds up less than return does because the assets do not always move together. When one falls the other may hold steady or rise, and the ups and downs partly cancel. The lower the correlation, the larger the effect. At a correlation of 1 there is no benefit, and risk is just the weighted average of the two risks. At -1 the risks can cancel completely, as two equal-risk assets at 50% each do.
This is the idea behind diversification, and it is why the table of mixes often shows risk falling as you move away from 100% in the riskier asset before it rises again. The curve is not a forecast of what will happen, only the arithmetic of the numbers you entered.
What the inputs really mean
Expected return and risk come from your own assumptions, often based on past data. Past data changes with the period you pick, so two careful people can reach different numbers. Correlation also shifts, and it tends to rise when markets are under stress. To explore the risk number on its own, use the standard deviation calculator, and to compare growth of two investments over time use the investment return comparison calculator.
Risk here is the standard deviation of yearly returns. It describes how spread out results tend to be, not the chance of losing money or the worst case, so a low number is not a guarantee.
Assumptions and limits
- All returns, risks and the correlation are inputs you enter. They are examples at first and are not forecasts or averages of any market.
- Returns are treated as a single yearly figure and the mix is held at fixed weights. Rebalancing, contributions, fees and taxes are not modeled.
- Short selling and leverage are not modeled, so the lowest risk weight is limited to between 0% and 100%.
- The calculation is educational and is not investment advice.
Frequently asked questions
What does correlation of 0.3 mean?
It means the two assets tend to move in the same direction a little, but only loosely. A value near 0 means no consistent link, 1 means they move in lockstep and -1 means they move in opposite directions.
Why is the portfolio risk lower than the weighted average of the two risks?
Because the two assets do not rise and fall together on every occasion, some of the swings offset each other. The difference between the two numbers is shown as the risk removed by mixing.
Can the lowest risk mix be 100% in one asset?
Yes. If one asset has much lower risk and the correlation is high, mixing in the riskier asset only adds risk, and the lowest risk weight is limited to 0% or 100% in that asset.
Is a higher expected return always better?
Not on its own. A higher expected return usually comes with higher risk, and what suits you depends on your goals and how much variation you can accept. The table lets you compare the pairs side by side.