When you’re building a portfolio, you’re making choices about risk and return.
Our simulation tool helps you see these trade-offs clearly by using historical data from Professor Aswath Damodaran at NYU Stern (source).
Let’s walk through the building blocks — starting with the simplest case and moving toward a fully diversified portfolio.
1. Risk-Free Investments
- What they are: Assets with a guaranteed return and no uncertainty, such as U.S. Treasury bills.
- Expected return: Equal to the risk-free rate ((R_f)) — for example, 3% annually.
- Risk: Zero standard deviation of returns. You always get the same payoff.
- Role in a portfolio: Provides stability and a base from which you can add risky assets.
2. Single Risky Investment
- What it is: One asset with uncertain future returns, such as the S&P 500, a corporate bond, or a REIT.
- Expected return: The average you expect over time, based on historical patterns.
- Risk: Measured by the standard deviation of returns — how widely actual returns can vary from the average.
- Key point: If your entire portfolio is just this asset, you carry all of its ups and downs. No diversification benefit.
3. Mixing One Risky Asset with a Risk-Free Asset
This is where the Capital Allocation Line (CAL) comes in.
- Portfolio return: [ E[R_p] = w_r E[R_r] + (1 - w_r)R_f ]
- Portfolio risk: [ \sigma_p = |w_r| \sigma_r ]
- How it works:
- Put some money in the risky asset for growth potential.
- Put some in the risk-free asset for stability.
- Adjust the mix ((w_r)) to match your comfort with risk.
The CAL shows you all the combinations of risk and return available from this mix. A steeper CAL means a better risk/return trade-off.
4. Two Risky Investments
- Why it matters: Now you can take advantage of diversification.
- Expected return: [ E[R_p] = w_A E[R_A] + (1 - w_A)E[R_B] ]
- Risk: [ \sigma_p^2 = w_A^2 \sigma_A^2 + (1 - w_A)^2 \sigma_B^2 + 2w_A(1 - w_A)\sigma_{A,B} ]
- Covariance:
- Positive covariance → assets move together → less diversification benefit.
- Negative covariance → assets move in opposite directions → greater diversification benefit.
Bottom line: Even if both assets are risky, holding them together can reduce your total portfolio risk — as long as they don’t move in lockstep.
5. N Risky Investments
Real portfolios hold many assets.
The math expands, but the idea is the same:
- Expected return: [ E[R_p] = \sum_{i=1}^N w_i E[R_i] ]
- Risk: [ \sigma_p^2 = \sum_{i=1}^N w_i^2 \sigma_i^2 + \sum_{i=1}^N \sum_{j \neq i} w_i w_j \sigma_{i,j} ]
- Diversification effect:
- The more uncorrelated assets you add, the more you reduce unsystematic risk (the risk unique to each asset).
- Systematic risk (market risk) remains — no matter how diversified you are.
6. How the Simulation Tool Helps
Our simulation tool models these concepts using historical returns, volatility, and correlations for seven major asset classes.
You can:
- Adjust portfolio weights and see the impact on expected return and risk.
- Explore how diversification changes volatility.
- Test different mixes of risk-free and risky assets.
- See the role of correlation and covariance in shaping outcomes.
Better information → better decisions.
By understanding how these portfolio building blocks fit together, you can design a strategy that balances risk and reward in a way that works for you.