Finding the Tangency Portfolio with a Risk-Free Asset
Summary
This discussion addresses mean–variance allocation when investors can combine risky assets with a risk-free asset. Under the standard assumptions behind the Capital Allocation Line, efficient portfolios are formed by mixing the risk-free asset with a risky tangency portfolio, which maximizes the Sharpe ratio. A target volatility can then be reached by scaling exposure to that risky portfolio and allocating the remainder to the risk-free asset, subject to the model’s borrowing and short-sale assumptions.
The response gives two routes to identifying the tangency portfolio: maximize the Sharpe ratio directly, or solve a quadratic variance minimization with a return target and then normalize the risky weights to sum to one. It does not provide the covariance matrix or compute numerical weights, and the excerpt does not answer the expected-return calculation explicitly. In a consistent setup, portfolio expected return is the weighted average of risky-asset expected returns and the risk-free return, with weights summing to one.
Key ideas
- With a risk-free asset, efficient allocations lie along a line combining it with a tangency portfolio under standard assumptions.
- The tangency portfolio maximizes expected excess return per unit of volatility.
- A target volatility is obtained by scaling the risky tangency portfolio and combining it with the risk-free asset.
- A return-targeted variance optimization can identify risky weights that are normalized to form the tangency portfolio.
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# Mean–variance portfolio optimisation with risk-free asset and fixed volatility target
# Mean–variance portfolio optimisation with risk-free asset and fixed volatility target
I am working on a mean–variance portfolio optimisation problem and would like to verify both the theoretical logic and the numerical approach.
Setup: • 3 risky assets and 1 risk-free asset • Expected returns vector: μ = (0.06, 0.02, 0.04) • Covariance matrix Σ is given • Risk-free rate r_f = 0.01
Question 1
We are asked to construct an efficient portfolio with a target volatility of 5%, allowing investment in the risk-free asset.
From theory, my understanding is:
- When a risk-free asset is available, the efficient set lies on the Capital Allocation Line.
- The optimal risky portfolio is therefore the tangency (maximum Sharpe ratio) portfolio.
- To achieve a fixed volatility of 5%, the tangency portfolio is scaled and combined with the risk-free asset.
Is this interpretation correct? In particular, is it correct that the solution is based on the tangency portfolio rather than solving a full mean–variance optimisation over all assets with an explicit volatility constraint?
Question 2
Once the portfolio weights from Question 1 are determined (including the risk-free asset), is the correct way to compute the expected portfolio return simply:
E[R_p] = wᵀ μ + w_f r_f
where w are the risky-asset weights and w_f is the weight on the risk-free asset?
Additional clarification
I am using a numerical solver (Excel Solver), and I am mainly interested in: • Confirming that the correct theoretical portfolio is being targeted • Understanding common numerical pitfalls (e.g. local optima, constraint specification errors) that might cause the solver to converge to an incorrect solution
I am looking to validate the logic and final numerical approach, not to circumvent coursework requirements.
## Answer by Kermittfrog (score 3)
https://quant.stackexchange.com/a/85449
Your intuition is correct.
When the risk-free asset is available, the optimal portfolio is on the tangency line between the risk-free rate and the so called "market portfolio".
You will still have to find this portfolio, either by modeling and maximizing the sharpe ratio, or by solving the program
$$ \min_w \frac{1}{2}w^T\Sigma w - \lambda_1(w^T\mathbf{\mu}+\left(1-\mathbf{1}^Tw\right)r_f-m) $$
given some (any) target value $m$. Then, the tangency portfolio is simply
$$ w_{\mathrm{market}} = \frac{w^*(m)}{\mathbf{1}^Tw^*(m)} $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.