Skip to content
All library documents

Scaling a Tangency Portfolio with a Risk-Free Asset

Article Quant Q&A · Author: auhan
Read the original

Summary

The accepted answer explains how to raise a portfolio’s volatility by combining a tangency portfolio with a risk-free asset. Since the risk-free asset has zero volatility and is uncorrelated with the tangency portfolio, the combined portfolio’s volatility is the tangency portfolio’s volatility multiplied by its weight. Dividing the target volatility by the tangency portfolio’s volatility gives the required allocation.

In the example, the target volatility exceeds the tangency portfolio’s volatility, so the allocation to the tangency portfolio is above one and the risk-free asset weight is negative. This represents borrowing to invest more in the original portfolio. The answer then scales each constituent stock’s original weight by the leverage factor. The method assumes the stated risk-free asset and volatility inputs, and preserves the tangency portfolio’s internal composition; it does not address borrowing costs, changing correlations, or practical leverage constraints.

Key ideas

  • A portfolio combining a tangency portfolio and a risk-free asset can target a different volatility.
  • With a zero-volatility risk-free asset, combined volatility scales with the tangency portfolio weight.
  • A target volatility above the tangency portfolio’s volatility requires a weight above one.
  • The negative risk-free asset weight represents borrowing to increase exposure.
  • Scaling the tangency portfolio’s constituent weights preserves its internal composition.

Tags

Portfolio constructionRisk managementPosition sizing
Full text
# How to calculate weight of two stocks without knowing their correlation?


# How to calculate weight of two stocks without knowing their correlation?












I have difficulty to solve attached question. The question asks me to find new weight of stocks by just changing standard deviation of the portfolio. I really do not have any idea how to solve it. Maybe the key point is the risk-free asset but I am not sure, can somebody give me a hint?

## Answer by SRKX (score 3, accepted)

https://quant.stackexchange.com/a/22423

You are supposed to create a new portfolio using the tangency portfolio $P_t$ and the risk-free rate $r$.

You know that the volatility of the tangency portfolio $\sigma_{P_t}=0.20$.

You also know that the risk-free asset has:



- Is not correlated with anything $\rho_{P_t,r} = 0$

So you're asked to create a portfolio with a higher risk, which means you are going to need to borrow some money to buy more of the tangency portfolio (you leverage).

You know that, by definition of volatility, the volatility of a portfolio $P$ formed of the tangency portfolio with weight $w$ and the risk-free asset with weight $1-w$ is :

$$\sigma_P^2 = w^2 \sigma_{P_t}^2 + (1-w)^2 \sigma_r^2 + 2w(1-w)\sigma_{P_t}\sigma_r\rho_{P_t,r} = w^2 \sigma_{P_t}^2$$

You hence get

$$w = \frac{\sigma_P}{\sigma_{P_t}} = \frac{0.24}{0.20} = 1.2$$

So the overall weight of the tangency portfolio will be $w=1.2$, the weight of the risk-free asset will be $1-w = -0.2$ (short). Within tangency portfolio, nothing has change in terms of weighting so you just multiply the 1.2 by the original weights which yields $w_A = 0.60 \cdot 1.2 =0.72$ and $w_B = 0.40 \cdot 1.2 =0.48$.

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.