What you will learn
- Understand what a risk-free asset is
- See why mixing it with a risky portfolio traces a straight line
- Identify the tangency portfolio and the capital market line
- Grasp two-fund separation: everyone holds the same risky portfolio
The efficient frontier from the last lesson was built only from risky assets, and it was a curve. Now we add one more ingredient, a risk-free asset, and something neat happens: the curve straightens into a line, and a single best risky portfolio emerges that, in theory, everyone should hold. It is one of the most elegant results in finance.
The capital market line (Edspira)
Introduces the risk-free asset and shows how combining it with a risky portfolio creates the capital market line. Watch how the line beats the curved frontier.
The risk-free asset
A risk-free asset is an investment with essentially no risk: its return is known in advance, its volatility is zero, and it does not move with anything else. The usual real-world stand-in is short-term government debt, like Treasury bills, which are as close to certain repayment as finance offers. We call its return the risk-free rate. Because its volatility is zero, it behaves very differently from any risky asset when you mix it into a portfolio.
Mixing risk-free and risky gives a straight line
Suppose you split your money between the risk-free asset and a single risky portfolio. Put a fraction x in the risky portfolio and the rest in the risk-free asset. Then two clean things happen. Your expected return is the risk-free rate plus x times the risky portfolio's excess return: return = rf + x·(rp - rf). And because the risk-free asset has zero volatility, your portfolio's risk is simply x times the risky portfolio's volatility: risk = x·σp. Both return and risk scale in a straight line as you change x, so on the risk-return chart your options trace a straight line starting at the risk-free rate and passing through the risky portfolio's point.
Key terms
- Risk-free asset
- An investment with a known return and zero volatility, such as a short-term Treasury bill.
- Tangency portfolio
- The one risky portfolio that gives the steepest, best line when combined with the risk-free asset. Often called the market portfolio.
- Capital market line
- The straight line from the risk-free rate through the tangency portfolio, showing the best possible risk-return combinations.
- Two-fund separation
- The result that every investor holds a mix of just two things: the risk-free asset and the same tangency portfolio.
The tangency portfolio and the capital market line
You could draw such a line through any risky portfolio, but you want the steepest one, because a steeper line means more return for every unit of risk. The steepest line you can draw from the risk-free rate just touches the efficient frontier at exactly one point. That point is the tangency portfolio, and the line is the capital market line. The slope of that line is the tangency portfolio's Sharpe ratio, its excess return divided by its volatility, so the tangency portfolio is the risky portfolio with the highest possible Sharpe ratio. Every point on the capital market line beats the old curved frontier, offering more return for the same risk.
Moving along the capital market line
The risk-free rate is 2 percent. The tangency portfolio has an expected return of 8 percent and a volatility of 15 percent. If you put 50 percent of your money in the tangency portfolio and 50 percent in the risk-free asset, what are your expected return and risk?
- Expected return. rf + x times (rp - rf) is 2 percent plus 0.5 times (8 percent minus 2 percent), which is 2 plus 0.5 times 6, or 2 plus 3, which is 5 percent.
- Risk. x times σp is 0.5 times 15 percent, which is 7.5 percent.
- Read the slope. The capital market line's slope is the tangency Sharpe ratio, (8 minus 2) over 15, which is 0.4.
Why it matters: Blending in the risk-free asset lets you dial risk up or down along a straight line, always earning 0.4 units of extra return per unit of risk. That constant slope is the tangency portfolio's Sharpe ratio.
Compute a point on the line
Same setup: risk-free rate 2 percent, tangency portfolio return 8 percent, volatility 15 percent. If you put 60 percent in the tangency portfolio and 40 percent in the risk-free asset, what is your expected return, in percent?
Add a truly safe asset to the mix and the best portfolios fall on a single straight line. Your only real choice becomes how far along it to stand.
The capital market line, CML (Bionic Turtle)
A more formal look at the CML and the tangency portfolio. Reinforces that its slope is the market Sharpe ratio.
Two-fund separation
Here is the conclusion. If everyone faces the same risk-free asset and the same efficient frontier, then everyone should hold the very same risky portfolio, the tangency portfolio, no matter their risk tolerance. Investors differ only in how they split their money between that one risky portfolio and the risk-free asset. A cautious investor holds mostly the risk-free asset with a slice of the tangency portfolio, sitting low on the line. An aggressive investor holds all tangency portfolio, or even borrows at the risk-free rate to hold more than 100 percent of it, sitting high on the line. This is called two-fund separation, and it is the theoretical justification for holding a broad market index fund plus bonds or cash, tuned to your risk tolerance.
How should two investors differ?
Under two-fund separation, a cautious retiree and an aggressive young investor should differ in what way?
Both investors hold the same tangency portfolio. They differ only in the split between it and the risk-free asset: the retiree keeps more in the safe asset, the young investor more in the risky tangency portfolio.Why a line beats a curve
In your own words, explain why the capital market line offers better risk-return tradeoffs than the curved efficient frontier of risky assets alone.
Write an answer before comparing it with the model response.
Model answer
The curved frontier is the best you can do with risky assets only. Once you add a risk-free asset, you can combine it with the single best risky portfolio, the tangency portfolio, and because the safe asset has zero volatility, that combination traces a straight line from the risk-free rate through the tangency point. That line sits above the curved frontier everywhere except where they touch, so for any level of risk the line offers a higher expected return. In other words, blending the tangency portfolio with the risk-free asset dominates holding risky assets alone, which is why the line is the better menu.
The tangency portfolio is the best portfolio of risky assets, and the capital market line prices whole portfolios. The next lesson zooms in from portfolios to individual assets, asking what return a single asset should earn for its risk. That is the Capital Asset Pricing Model.