# Interpretation of optimal weights in portfolio for risk-adjusted return maximization

To start, I'm not an expert in portfolio management. My research involves examining the effects that one financial asset has on another, specifically looking at the spillovers between cryptocurrency and stock market assets, both of which involve risks. Currently, there's no widely accepted model that explains why these connections exist.

Based on my own understanding, which draws from existing theoretical work, it appears that investors aim to maximize their returns while considering the associated risks. In this particular study, they've come up with a formula represented as:

$$$$w_1 = \frac{µ_1σ_{2,2} − µ_2σ_{1,2}}{ µ_1(σ_{2,2} − σ_{1,2}) + µ_2(σ_{1,1} − σ_{1,2})}$$$$ This equation suggests that the weight assigned to Asset 1 in a portfolio increases when the return on Asset 1 is higher and when the risk (variance) of Asset 2 is greater. Conversely, this weight decreases when the return on Asset 2 is higher and when there's a strong positive relationship (covariance) between the two assets.

However, I'm having difficulty fully understanding the implications of the denominator of the fraction in this formula. Could someone provide further insight?

• You should use LaTeX when writing equations so it's easier for people to read. Generally speaking, when you try to minimise the correlation in your portfolio, a lot of the correlation exists because traders are buying/selling similar stocks. So traders who are bullish in tech would be mostly long tech stocks and short other stocks, hence there becomes a correlation. Wrt crypto, the market is minuscule. If traders receive bad/good information about some S&P500 company, that does not mean they will short/long crypto. Thus, there shouldn't be much correlation to begin with. Commented Oct 23, 2023 at 23:13
• Thank you for the comment, I've modified the equation ! I'm looking for the theoretical channels through which shocks from a specific asset are transmitted to another (=spillovers). In my spirit, shocks are transmitted through the portfolio rebalancing channel (meaning that the variation of weights following a shock do transmit a shock) and after modelling this channel the paper I've mentionned find that the optimal weight of asset 1 corresponds to the equation. However, I cannot fully understand the denominator ! Commented Oct 24, 2023 at 10:03
• There definitely can be market spillovers if a whole sector goes down a lot, or confidence & trust is lost. But I don't think an individual asset within a market type could induce a spillover, unless the asset makes up a large proportion of the market i.e. Bitcoin. Commented Oct 25, 2023 at 22:25
• I totally agree with you ! I was just looking for a theoretical channel explaining spillovers (here, portfolio rebalancing : the rebalancing of weights following a shock on risk-adjusted returns generate spillovers to the other assets that are possibly included in the portfolio (their weights change as well). When I conducted a panel local projection analysis, I found puzzling results (spillovers increase when returns/volatility between the two assets converge) which I wanted to explain, and the answers did help me a lot ! Commented Oct 30, 2023 at 14:19

With regards to your question about the denominator, if we expand the denominator, it becomes:

$$$$w_1 = \frac{\mu_1\sigma_2^2 - \mu_2\rho\sigma_1\sigma_2} {\mu_1\sigma_2^2 - \mu_1\rho\sigma_1\sigma_2 + \mu_2\sigma_1^2 - \mu_2\rho\sigma_1\sigma_2}$$$$

Therefore, the weight assigned to asset 1 increases when either of the following occurs:

• Asset 1 return increase (the 1st term in the numerator, and 1st and 2nd terms in the denominator are relevant): Because the numerator increases more than the denominator (by a factor of $$\rho\sigma_1\sigma_2$$).
• Asset 2 volatility increases (both terms in the numerator, and 1st 2nd and last terms in the denominator are relevant): The denominator decreases proportionally by a factor of $$\mu_1\rho\sigma_2$$ and the other relevant terms we mentioned increase by the same amounts because they are the same terms in the numerator and denominator.
• Correlation decreases (the 2nd term in the numerator, and 2nd and last terms in the denominator are relevant): Because the numerator increases more than the denominator (by a factor of $$\mu_1\sigma_1\sigma_2$$).

Actually, you can use this expanded form and test it for yourself numerically using Excel or code for other variables.

• Edit. I might not have answered directly to your question, just gave an answer as to what happens to the weight when changes are applied to different variables. But hopefully my answer gives you some ideas. Commented Oct 24, 2023 at 12:55
• Thank you for your comment, it actually answers my question perfectly ! Commented Oct 24, 2023 at 13:10
• @krauuuus I am glad this helped you, I was worried I might have made it more confusing. You can consider accepting this as the answer (with the tick on the left to my solution). Commented Oct 24, 2023 at 13:23