Updated for the 2026-2027 CFA® Level I curriculum.
An investor rarely holds only risky assets. Adding a risk-free asset, such as a T-bill, changes both the expected return and the risk of the total holding. This combined holding is called the complete portfolio. Level I tests whether you can calculate its expected return and standard deviation, and whether you can tell the difference between lending and borrowing at the risk-free rate.
Quick Answer
The optimal complete portfolio combines a risk-free asset with a risky portfolio using the weights an investor chooses. Expected return is:
and risk is:
where is the weight in the risk-free asset. When is positive, the investor lends at the risk-free rate. When is negative, the investor borrows at the risk-free rate to build a leveraged portfolio. Because the risk-free asset has zero standard deviation, only the weight on the risky portfolio drives total risk.
Key Takeaways About Combining a Risk-Free Asset with a Risky Portfolio
The complete portfolio combines a risk-free asset and a risky portfolio in weights chosen by the investor.
Expected return is a weighted average of the risk-free rate and the risky portfolio's expected return.
Risk depends only on the weight in the risky asset, since the risk-free asset has zero standard deviation.
A positive weight in the risk-free asset creates a lending portfolio with lower risk than the risky portfolio alone.
A negative weight in the risk-free asset creates a leveraged portfolio with higher risk than the risky portfolio alone.
Adding a risk-free asset extends the investor's opportunity set beyond what risky assets alone can offer.
What You Need to Know for CFA Level I
Set up complete portfolio weights correctly, with assigned to the risk-free asset and assigned to the risky portfolio.
Calculate expected return and standard deviation of the complete portfolio from those weights.
Recognize a lending portfolio when the risky weight is below 1.
Recognize a leveraged portfolio when the risky weight exceeds 1.
Know that combining a risk-free asset with a risky portfolio changes the investor's opportunity set from a curve to a straight line.
Keep the risk-free asset's standard deviation at zero in every calculation.
Complete Portfolio Weights
A complete portfolio is the full mix of an investor's wealth across a risk-free asset and a risky portfolio. Two weights describe it:
= weight invested in the risk-free asset
= weight invested in the risky portfolio
These weights always sum to 1, but is not limited to values between 0 and 1. An investor can set to any number, including a negative one. That flexibility is what separates lending from leverage, covered below.
Expected Return and Risk of the Complete Portfolio
Two formulas describe the complete portfolio.
Expected return:
Standard deviation:
Where:
- Expected return of the complete portfolio
- Risk-free rate
- Expected return of the risky portfolio
- Weight invested in the risk-free asset
- Weight invested in the risky portfolio
- Standard deviation of the complete portfolio
- Standard deviation of the risky portfolio
The risk-free asset's standard deviation does not appear in the formula because it equals zero. This is why total risk depends only on how much weight sits in the risky portfolio.
Lending When the Risky Weight Is Below 1
When is positive, the investor puts some wealth into the risk-free asset and less than 100% into the risky portfolio. This is a lending portfolio, because the investor is effectively lending money at the risk-free rate (buying a T-bill is lending to the government).
In a lending portfolio:
is lower than
sits between and
Leverage When the Risky Weight Exceeds 1
When w is negative, the investor borrows at the risk-free rate and invests the borrowed funds, along with existing wealth, into the risky portfolio. This is a leveraged portfolio.
In a leveraged portfolio:
is higher than
is higher than , assuming
Case | Weight in Risk-Free Asset | Weight in Risky Portfolio | Risk vs. |
|---|---|---|---|
Lending | Less than 1 | Lower | |
Fully invested | Equal to 1 | Same | |
Leveraged | Greater than 1 | Higher |
Effect on the Opportunity Set
Without a risk-free asset, an investor's opportunity set is the curved efficient frontier of risky assets. Adding a risk-free asset changes this. Every combination of the risk-free asset and a given risky portfolio plots on a straight line, since both and move in fixed proportion to . This straight-line relationship is why combining a risk-free asset with a risky portfolio expands what an investor can achieve, offering risk-return combinations that pure risky-asset portfolios cannot match.
Worked Example
Scenario: An investor is deciding how to combine risky portfolio P with a risk-free asset. Portfolio P has an expected return of 12% and a standard deviation of 20%. The risk-free rate is 4%. The investor considers two allocations.
Case 1: Lending portfolio
The investor puts 30% in the risk-free asset and 70% in portfolio P.
,
Case 2: Leveraged portfolio
The investor borrows an amount equal to 25% of the original investment at the risk-free rate and invests the total in portfolio P.
,
Lending reduces both expected return and risk below portfolio P's own values. Leverage increases both above portfolio P's own values. The investor did not change the risky portfolio itself. The investor only changed how much weight sits in it, and that weight change drives the entire result.
Common Exam Traps
Giving the risk-free asset a nonzero standard deviation
The risk-free asset has zero standard deviation by definition. If a candidate adds any risk contribution from it, every calculation will be wrong.
Using inconsistent risky-asset weights across formulas
The weight must be the same number in both the return formula and the risk formula. Recalculating it differently in each step is a common source of error.
Treating leverage as a weight between 0 and 1
Leverage means w is negative and exceeds 1. A candidate who keeps weights inside the 0-to-1 range will never model a leveraged portfolio correctly.
Confusing the complete portfolio with the optimal risky portfolio
The risky portfolio (often called P or the optimal risky portfolio) is only one piece. The complete portfolio includes the risk-free allocation too. These are different portfolios with different risk and return values.
Practice Question
An investor has $100,000 in a risky portfolio with an expected return of 15% and a standard deviation of 18%. The risk-free rate is 3%. The investor borrows an additional $20,000 at the risk-free rate and invests the full $120,000 in the risky portfolio. What are the expected return and standard deviation of the complete portfolio?
Correct Answer: A
The investor borrows $20,000 against $100,000 of own capital, so , and .
Option B: This uses the risky portfolio's own return and risk, ignoring the leverage entirely. Borrowing to invest more than 100% always changes the result.
Option C: This applies the borrowing as if it were lending, using a positive weight of 0.20 in the risk-free asset instead of a negative one. The sign of w must match the direction of the cash flow: borrowing is negative, lending is positive.
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FAQs About Combining a Risk-Free Asset with a Risky Portfolio
What is the difference between a lending portfolio and a leveraged portfolio?
A lending portfolio has a positive weight in the risk-free asset, so less than 100% of wealth sits in the risky portfolio. A leveraged portfolio has a negative weight in the risk-free asset, meaning the investor borrows to invest more than 100% of wealth in the risky portfolio.
Why does the risk-free asset have zero standard deviation?
By definition, a risk-free asset has a certain, known return with no variability, such as a short-term government T-bill held to maturity. Because there is no variability, its standard deviation is zero and it does not add risk to the complete portfolio.
Does combining a risk-free asset with a risky portfolio always reduce risk?
No. It reduces risk only when the investor lends, meaning the weight in the risky portfolio is below 1. If the investor borrows to increase the risky weight above 1, total risk increases above the risky portfolio's own standard deviation.