Assessing a New Asset by Comparing Portfolio Sharpe Ratios
Summary
The document presents a decision rule for adding a prospective asset to an existing portfolio: compare the Sharpe ratio of the proposed combined portfolio with that of the current portfolio, and prefer the addition when the new ratio is at least as high. It expresses the combined portfolio return as a weighted average of the existing portfolio and the candidate asset, then rearranges the comparison into a minimum expected return criterion for the candidate.
The derivation assumes a zero-return cash benchmark and uses expected returns and portfolio standard deviations. It cites a risk management text as the source, but provides no empirical test or worked numerical example. The criterion is sensitive to the chosen allocation weight and to the covariance of the new asset with the existing portfolio, which affects the combined portfolio’s risk; it is not a standalone assessment of the asset’s attractiveness.
Key ideas
- The proposed investment is accepted when it raises or preserves the portfolio’s Sharpe ratio.
- The combined portfolio return depends on the allocation to the candidate asset and the existing portfolio.
- A minimum expected return for the candidate can be derived by rearranging the Sharpe ratio comparison.
- The candidate’s effect on portfolio volatility depends on its relationship with existing holdings.
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# Answer by user86422 (score 1)
# How to prove that the return criteria for adding an investment A to an existing portfolio can be represented using Sharpe Ratio Approach
How can I prove that the return criteria for adding an investment A to an existing portfolio can be represented as the below inequality using the Sharpe Ratio Approach for risk adjusted returns as applied to portfolio decision making?
## Answer by user86422 (score 1)
https://quant.stackexchange.com/a/78833
Denote the old Sharpe ratio by $SR_p^{old}$, and the new one by $SR_p^{new}$. Buy the new asset (i.e., go from old to new) if and only if $SR_p^{new} \ge SR_p^{old}$. let $R_p^{old}$ be the return to the old portfolio and note that the return to the cash benchmark is zero. The differential return, $d^{old}$, is therefore simply $R^{old}$ and the standard deviation is $\sigma_{d^{old}}$. The ratio of $d^{old}$ to $\sigma_{d^{old}}$ then gives us our existing (old) Sharpe ratio.
Suppose that the prospective new portfolio consists of the new asset $A$ and the existing portfolio, in relative proportions $a$ and $1-a$. The expected return on the new portfolio is $R_p^{new} = aR_A + (1 - a)R_p^{old}$, which is also the expected differential for the new portfolio, $d^{new}$.
We should make the investment if $$ SR^{new}=d^{new}/\sigma_{R_p^{new}} \ge d^{old}/\sigma_{R_p^{old}}$$
Which when substituted and re-arranged, becomes $$ R_A \ge R_p^{old} + [\sigma_{R_p^{new}} / \sigma_{R_p^{old}} - 1]R_p^{old}/a $$
This answer is found on page 146, Chapter 7, of Dowd, K. (1998). Beyond value at risk: The New Science of Risk Management. Hoboken, NJ: John Wiley & Sons IncShown 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.