Sharpe ratio after combining portfolio with T-Bill

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In Reading 17, Example 10, the solution to part 1 says :
[content removed by moderator]
Combining portfolio with T-Bill will not lower Sharpe ratio, is it because the E(Rp) and σp is lowered at the same time?
 
This all goes to an earlier discussion in the candidate reading #17, look back to page 220 titled Cash Equivalents and Capital Market Theory”…here it states in the second paragraph
“When we assume a nominally risk-free asset and take a single-period perspective, mean–variance theory points to choosing the asset allocation represented by the perceived tangency portfolio if the investor can borrow or lend at the risk-free rate. (Borrowing in this context means using margin to buy risky assets, resulting in a negative weight on the risk-free asset.) The tangency portfolio is the perceived highest-Sharpe-ratio efficient portfolio. The investor would then use margin to leverage the position in the tangency portfolio to achieve a higher expected return than the tangency portfolio, or split money between the tangency portfolio and the risk-free asset to achieve a lower risk position than the tangency portfolio. The investor’s portfolio would fall on the capital allocation line, which describes the combinations of expected return and standard deviation of return available to an investor from combining his or her optimal portfolio of risky assets with the risk-free asset. Many investors, however, face restrictions against buying risky assets on margin. Even without a formal constraint against using margin, a negative position in cash equivalents may be inconsistent with an investor’s liquidity needs. Leveraging the tangency portfolio may be practically irrelevant for many investors.”
Combining the Tangency portfolio (CP4) with the T-bill will not lower the Sharp ratio, this approach only minimizes the Sharpe ratio while the required return of 6.5% (which is below the Tangency portfolio return of 7.24%) does not change.
I hope that helps, this is a very important subtly in this material and especially within the example 10.
Marc A. LeFebvre, CFA
Founder & President - LevelUp, LLC and LevelUp Bootcamps
www.levelupbootcamps.com
 
Thank you. Now I understand the slope of CAL line is the Sharpe ratio.
> Combining the Tangency portfolio (CP4) with the T-bill will not lower the Sharp ratio, this approach only minimizes the Sharpe ratio while the required return of 6.5% (which is below the Tangency portfolio return of 7.24%) does not change.
I guess since Sharpe ratio does not change, you mean this approach minimizes the standard deviation?
 
Think about the expected return calculation, it is a weighted average between the Tangency portfolio and the risk free rate. Ok now do the same weighted average calculation for the standard deviation. Recall, the risk free rate has zero variance (not technically but here we assume it does) and again we calculate the weighted average (linear approximation) between the Tangency portfolio risk (Sharpe ratio = portfolio with highest return and lowest risk) and the risk free rate (zero variance). BTW we DO NOT include the correlation factor as we normally would in a two asset class st dev its assumed to be zero in this special case. So to answer your question, when we average the risk of the Tangency and the risk free portfolios yes the risk is reduced. Voila, you have a free lunch.
See the last paragraph in BlueBox 10
“If the trustee’s suggestion were accepted, the portfolio 15 percent invested in T-bills and 85 percent invested in Corner Portfolio 4 (discussed in Part 1) would be optimal for CEFA; that asset allocation meets CEFA’s return objective with minimum standard deviation of return.”
Now here is the real tricky part…what is the difference between BlueBox 9 and BlueBox 10 that we have had this discussion about?
-Marc
 
> Now here is the real tricky part…what is the difference between BlueBox 9 and BlueBox 10 that we have had this discussion about?
in Bluebox 9, the tangent portfolio’s expected return cannot meet the investor’s required return. So it is combined with another portfolio to increase the expected return.
in BlueBox 10, the tangent portfolio’s expected return exceed the investor’s required return. So it is combined with a risk free asset to lower the expected return.
 
>> In Reading 17, Example 10, the solution to part 1 says :
>> [content removed by moderator]
I am curious, why did the moderator remove the content? Did i write something offensive?
 
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