# CAL, Indifference Curve, MVF, Efficient Frontier relationship

Hey guys, I’ve been reviewing PM this weekend and am still getting lost on the relationship between the CAL/CML, Indifference Curve, MVF, and Efficient Frontier. I understand why each one exists by itself, but can’t really figure out what the relationships between them are. Basically I know enough to get most questions right (optimal portfolio is point of tangency, portfolios above MVF are impossible, etc.), but still feel a little shaky on the fundamentals. Does anyone have a laymans explanation of these various charts and their relationships with each other? Thanks!

Ok here’s my understanding. 1) Minimum variance frontier – this curve represents the portfolios with the lowest level of risk for a certain level of return. Ie if you considered all of the risky assets in the world, and formed them into portfolios, the MVF would show you the “best” you can get in terms of the lowest risk (at each level of return). 2) The global minimum variance portfolio is on the MVF, at the lowest level of risk (the MVF looks like a C, and the global minimum variance portfolio is at the most left-hand point of the C, if that makes sense). It is the lowest risk you can have, amongst all risky assets (note this DOESN’T include adding a risk-free asset, we’ll get to that later.) 3) Now think a bit more about what the ‘C’ shaped curve (the MVF) actually shows us – returns are on the Y axis and risk is on the X axis. Because of the ‘C’ shape, there are two parts on the curve where risk is the same, but returns are different. Why would you want to invest in the portfolio on the ‘bottom half’ of the C, which essentially has the same risk but worse returns? That’s how we get to the Markowitz efficient frontier – if you cut the C in half from left to right through its middle, the efficient frontier is the top half. 4) Now if you look at a graph that just shows the efficient frontier (the top half of the ‘C’). The risk free rate is on the Y axis somewhere. What you want is the risky asset portfolio that results in the steepest slope between the RFR and the efficient frontier (because a steep slope suggests that returns are growing faster than risk). You’re going to get that at the point of tangency between the efficient frontier and a straight line drawn from the RFR. (less steep and you’re not maximising returns, more steep is unattainable). That point of tangency defines the market portfolio, ie the best possible risky portfolio to invest in. 5) The CAL and the CML show the same thing except the CAL graphs the combo of the risk free rate and ANY risky portfolio whereas the CML shows the risk free rate and the market portfolio. 6) To work out where on the CML each investor should be, you use the point of tangency with indifference curves . hope that helps…

good explanations…further note: CML is a special case of CAL where with CML all investors agree on returns, variances and covariances and therefore the market portfolio exists and is the same for all investors… versus the CAL where every investor could have a difference CAL depending on their expectations. Once the CAL or CML is determined, the indifference curve plots where the investor lies which is a function of their risk aversion these have the highest reward to risk ratio (think Sharpe for slope) with the risk free rate as the intercept. hope this makes sense…probably didn’t add much, but I enjoy this topic

Thanks guys, this helps a lot - especially kia, you obviously have it down solid. Just had a quick question - when you talk about the the Markowitz Frontier and how you’re looking for the straight line thats tangent between the RF rate and the frontier, is this the same as the CAL line? I think the one thing I have trouble understanding is which lines you plot against which charts to find the “optimal portfolio”. In particular, I’ve seen the term optimal portfolio come up twice: 1) The line tangent to the Markowitz frontier connecting to the RF rate 2) The point tangent to the investors highest indifference curve on the CAL/CML. Whats the difference between these two “optimal portfolios”? I know I’m missing something very basic here…

the first “optimal portfolio” you mention (line tangent to efficient frontier) is known as the market portfolio, and it’s the optimal risky asset portfolio for ANY investor. (ie given all the different assets in the market, it’s the best possible return for the risk.) that line is the CML. the second “optimal portfolio” (tangency between CML and indifference curve) is the optimal portfolio for an INDIVIDUAL investor – ie, their optimal combination of the market portfolio and the risk free asset. this is dependent on how risk averse they are (which is why you use indifference curves) – someone who is more risk averse will have more invested in the risk free asset and less in the market portfolio, etc. as far as I understand it, any line that tracks the risk/return of the risk free asset combined with a risky asset portfolio, is a CAL. the CML is a special case of the CAL, where the risky asset portfolio is the best possible one (ie, the market portfolio).

@Kia Is Global minimum variance portfolio efficient? and is it on CML? I got a question I don’t remember if it was about efficiency or about being on CML. From this it looks like, it should be efficient but should not be on CML. right?

And optimal portfolio for an investor occurs at the point of tangency between investor’s highest attainable indifference curve and Markowitz efficient frontier. What happens when there is no point of contact between these two curves? Portfolio not possible or just not efficient?

Above the frontier is impossible, below is inefficient.

Thanks kia!

Matori Wrote: ------------------------------------------------------- > Above the frontier is impossible, below is > inefficient. Is there any degree of how above and how below it is or it’s just a general rule?

Its just a general rule. If it doesn’t touch the frontier, it is note achievable. If it is below the frontier, it is not efficient. I believe there is a graph in the text that has 3 indifference curve (one above, one tangent, one below going through two points)