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Using the Capital Market Line to Replicate a Portfolio Return

Article Quant Q&A · Author: standarddev
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Summary

The document asks how to assess three portfolios against the Capital Market Line (CML) and how to allocate capital to achieve a target return. With a 2% risk-free rate and a market portfolio expected to return 10% with 20% standard deviation, the CML slope is 0.4. A portfolio with a 12% expected return matches that slope at 25% standard deviation, so it lies on the CML under the stated assumptions.

To target 12%, combine the market portfolio and the risk-free asset using a 125% market allocation and a -25% risk-free allocation. For an investment amount X, that means investing 1.25X in the market portfolio and borrowing 0.25X at the risk-free rate. This assumes borrowing and lending at the stated rate and that the market portfolio is the relevant efficient risky portfolio. The example does not provide asset-level weights within the market portfolio. A portfolio above the CML would imply a higher Sharpe ratio than the market and would be inconsistent with the model's assumptions, rather than simply being infeasible by definition.

Key ideas

  • The CML relates expected return to total portfolio standard deviation when combining the risk-free asset with the market portfolio.
  • The market portfolio's stated expected return is 10%, since its excess return is 8% above the 2% risk-free rate.
  • A 12% target return requires a 125% allocation to the market portfolio and borrowing equal to 25% of capital at the risk-free rate.
  • The corresponding standard deviation is 25% if the market portfolio's standard deviation is 20% and the CML assumptions hold.
  • The method assumes borrowing and lending are available at the risk-free rate and does not specify holdings inside the market portfolio.

Tags

Portfolio constructionRisk managementPosition sizingStatistics
Full text
# Understanding CAPM, CML, and efficient portfolios


# Understanding CAPM, CML, and efficient portfolios












I'm trying to understand the CAPM model and how we can use it to understand efficient portfolios. Specfically, I'm trying to use the CML line (mapping expected returns and standard deviations of portfolios) to value proposed portfolios.

In this scenario: risk free rate = 2%. Expected excess return on market portfolio is 8% (so, I'm assuming, the expected return on the the market portfolio is 10%). The last given value is that the standard deviation of the market portfolio is 20.

I have to analyze 3 portfolios:

A: E(r) = 8%, SD = 10% B: E(r) = 12% SD = 25% C: E(r) = 13% SD = 30%

Based on the Sharpe Ratio (ie: the slope of the CML), I deduced that portfolio A is unfeasible and C is inefficient, whereas B falls on the CML and must therefore be efficient for the level of risk.

The next question I'm posed with is "How can the expected return of the wining portfolio be achieved? Specify the amount invested in each asset/portfolio of assets?"

It is given that I have some number X to invest, but I'm not quite sure how to approach this problem. The question does not seem too clear to me.

## Answer by phdstudent (score 1)

https://quant.stackexchange.com/a/21875

It is really simple and probably not a question for this forum.

You just need: $\alpha * .08 + (1-\alpha) * .02 = .12$. Solve for alpha and then check the standard deviation that should be .25.

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.