Enter Values

%
Annualized expected return
%
Annualized standard deviation
%
T-bill or other risk-free rate
Higher = more risk averse. Typical range: 2–6
Ryan O'Connell, CFA, FRM
Created by Ryan O'Connell, CFA, FRM

Optimal Allocation

Optimal Risky Allocation (y*) --

Complete Portfolio Details

Risk-Free Allocation --
Expected Return E(rC) --
Portfolio Risk σC --
CAL Slope (Sharpe) --
Portfolio Utility --
Risk-Free Utility --

Higher utility = preferred under this model

Step-by-Step Calculation

y* = [E(rP) − rf] / (A × σP²)
BKM Equation 6.7 — Optimal Allocation to Risky Asset

Interpretation Guide

Range Status Meaning
0% < y* ≤ 100% Standard Normal allocation — split between risky and risk-free assets
y* > 100% Leveraged Borrowing at risk-free rate to invest more than 100% in risky portfolio
y* ≤ 0% Short / No Risky Negative allocation: short the risky portfolio and invest more than 100% in risk-free
Model Assumptions
  • Mean-variance utility function (quadratic utility or normally distributed returns)
  • Single risky portfolio and single risk-free asset
  • Unlimited borrowing and lending at the risk-free rate (for y* > 1)
  • Risk aversion coefficient is constant and known
  • No transaction costs or taxes

For educational purposes. Not financial advice. Market conventions simplified.

Understanding Capital Allocation

Video Explanation

Video: Capital Allocation Explained

What Is Capital Allocation?

Capital allocation is the fundamental decision of how to divide your investment portfolio between risky assets (stocks, bonds, real estate) and a risk-free asset (Treasury bills). In the BKM framework (Chapter 6), this decision is formalized using a mean-variance utility function that balances expected return against risk based on the investor's personal risk aversion.

Utility Function (BKM Eq. 6.1)
U = E(r) − ½ × A × σ²
Higher utility = preferred portfolio under mean-variance framework

Optimal Risky Allocation (y*)

The optimal weight in the risky portfolio maximizes the investor's utility. Taking the derivative and solving yields:

Optimal y* (BKM Eq. 6.7)
y* = [E(rP) − rf] / (A × σP²)
Risk premium divided by (risk aversion × variance)

This formula reveals that investors allocate more to the risky portfolio when the risk premium is higher or volatility is lower, and less when they are more risk-averse.

The Capital Allocation Line (CAL)

The Capital Allocation Line plots all possible combinations of the risk-free asset and the risky portfolio in expected return vs. standard deviation space. Its slope equals the Sharpe ratio of the risky portfolio:

  • CAL Slope = [E(rP) − rf] / σP
  • A steeper CAL means a better risk-return tradeoff
  • Every investor on the same CAL has the same risky portfolio but different y* values based on their risk aversion

When to Use This Calculator

Capital Allocation Calculator

Quantitative optimizer — BKM Chapter 6 mean-variance utility maximization. Input specific return, volatility, and risk aversion to compute the mathematically optimal y*.

Asset Allocation Calculator

Qualitative questionnaire — guided risk-tolerance assessment that recommends a stock/bond/cash split based on your answers. No formulas required.

Tip: Use this calculator if you know your risk aversion coefficient and want a precise, formula-driven allocation. Use the Asset Allocation Calculator for a guided approach based on your personal risk tolerance.

Frequently Asked Questions

The optimal allocation (y*) is determined by the formula y* = [E(rP) - rf] / (A × σP²), where E(rP) is the expected return on the risky portfolio, rf is the risk-free rate, A is the investor's risk aversion coefficient, and σP² is the variance of the risky portfolio. This formula maximizes the investor's mean-variance utility function U = E(r) - ½Aσ². A higher risk premium or lower volatility increases the optimal risky allocation, while higher risk aversion decreases it.

The risk aversion coefficient (A) measures how much an investor dislikes risk relative to expected return. Typical values range from 2 to 6, where A=2 represents a relatively aggressive investor and A=6 a conservative one. It can be estimated through revealed preference (observing actual portfolio choices), questionnaires, or calibration against historical portfolio decisions. In the BKM framework, A determines how much return an investor demands per unit of variance.

When y* exceeds 100%, the model recommends a leveraged position — borrowing at the risk-free rate to invest more than your total wealth in the risky portfolio. For example, y*=150% means borrowing 50% of your wealth at rate rf and investing 150% in the risky portfolio. This is theoretically optimal under BKM assumptions (unlimited borrowing at rf) but impractical for most investors due to margin requirements, borrowing costs exceeding rf, and amplified downside risk.

The Capital Allocation Line represents all possible combinations of the risk-free asset and a single risky portfolio, plotted in expected return vs. standard deviation space. Its slope equals the Sharpe ratio of the risky portfolio: S = [E(rP) - rf] / σP. The CAL starts at the risk-free rate (σ=0) and extends through the risky portfolio point. Every point on the CAL represents a different allocation y between 0 (100% risk-free) and beyond 1 (leveraged).

The mean-variance utility function U = E(r) - ½Aσ² assigns a score to any portfolio based on its expected return and risk. Higher expected return increases utility while higher variance decreases it, scaled by the investor's risk aversion A. The optimal portfolio maximizes this utility. For a risk-free asset, utility simply equals rf (since σ=0). A higher utility score means the portfolio is preferred under this framework.

The CAL slope equals the Sharpe ratio of the risky portfolio: [E(rP) - rf] / σP. A steeper CAL means a better risk-return tradeoff — more expected return per unit of risk. The optimal allocation y* places the investor at the point on the CAL that maximizes their personal utility given their risk aversion. All investors using the same risky portfolio face the same CAL, but each chooses a different point on it based on their risk aversion coefficient A.
Disclaimer

This calculator is for educational purposes only and assumes the mean-variance utility framework from BKM Chapter 6. Actual portfolio allocation decisions involve additional factors like liquidity needs, investment horizon, tax considerations, and behavioral biases. The model assumes unlimited borrowing at the risk-free rate and normally distributed returns, which may not hold in practice. This tool should not be used as the sole basis for investment decisions.

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Course by Ryan O'Connell, CFA, FRM

Portfolio Analytics & Risk Management Course

Master portfolio theory and risk management from fundamentals to advanced analytics. Covers modern portfolio theory, risk metrics, performance evaluation, and factor models.

  • Sharpe, Sortino, Treynor & Information Ratio deep dives
  • Modern Portfolio Theory and efficient frontier construction
  • Factor models including CAPM and Fama-French
  • Hands-on exercises with real portfolio data
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