# Utility curve: PM

Utility curves of all the investors will essentially touch more than one point in capital market line. Why? I thought it would touch only on the market portfolio. Can someone help clarify this?

Correct me if I am wrong but I believe the CML takes into account the RF asset and a risky security so that anything that sits on the CML is a diversified portfolio. This doesn’t however mean that it is efficient, or for all intents and purposes, sits on the efficient frontier. The utility curves just represents the utility the investor is achieving. The BEST alternative is for the utility to touch the market portfolio where the efficeint frontier and the CML are tangent, but obviously with different weights of a risk free and risky assets with different profiles the utility curve would touch a different points and not necessarily always at the MP.

Oksy. Thanks I have no idea why this would be the best alternative (nor even how to define “best” in this context). Why do you think it’s best?

Every investor has an infinite number of utility curves, which generally are convex upward. If you want to visualize them, think of a parabola – y = x² – and then move it up and down; each new parabola is a new utility curve, and the higher up the utility curve, the higher the utility to the investor.

The capital market line slopes up to the right, and intersects the y-axis at the risk-free rate (above zero); picture it as y = x + 1. If you draw that line over the drawing of the parabolas, you’ll divide the parabolas – utility curves – into three groups:

• The lowest group has infinitely many parabolas each of which intersects the line in two points. These represent levels of utility that the investor can achieve, but are less than the best.
• The highest group has infinitely many parabolas none of which intersects (or even touches) the line. These represent levels of utility that the investor cannot achieve.
• The middle group has one parabola that touches the line in one point: it’s tangent to the line. This represents the best (highest) utility that the investor can achieve, and he achieves it by investing in the portfolio at the tangent point. If that tangent point is the market portfolio, that’s an amazing coincidence, but it doesn’t matter a bit.

sir, i have a doubt. What is the use of the formula:

U=E®-0.5A*S.d^2

Sorry not best alternative; it would be considered the best portfolio or choice an investor can make to invest because it provides the investor with the highest level of return for the lowest level or risk. I shouldn’t have said “alternative”.

Thank you sir. Shouldn’t it be convex to the left? Why convex upwards? convex upwards would be x^2=negative y.

Also, using the term “all investors” should make the statement “Utility curves of all the investors will essentially touch more than one point in capital market line.” incorrect, right?

Same doubt.

This is a common example of a utility curve, but it certainly isn’t the only example of a utility curve. If they ask you on the exam to compute “the” utility to an investor, this is the formula they’ll want you to you (you’ll recognize its application because they’ll have to give you a value for A), but realize that in the real world the formula (if you can even write a correct formula) will be much different.

how does every investor has an infinite no of utility curves? maybe i need to revise portfolio When finance people say “convex upward” they mean a shape like y = x². Think about the shape of the price/yield curve of a normal bond, which is said to have positive convexity. It may not be the way mathematicians would describe the curve, but it is the way finance people describe it.

Each investor will have one utility curve that is tangent to the CML, infinitely many that intersect the CML at more than one point, and infinitely many that are completely above the CML (and, so, do not intersect it).

They have a curve for utility = 1, and utility = 1.1, and utility = 1.11, and utility = 1.111 and . . . .