NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Discuss all general (i.e. non-personal) investing questions and issues, investing news, and theory.
User avatar
nisiprius
Advisory Board
Posts: 36216
Joined: Thu Jul 26, 2007 9:33 am
Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by nisiprius » Sun Apr 29, 2018 6:00 am

What do you call this kind of diagram?

Image

The Y axis is return, X is standard deviation, red and green dots are two financial assets, the curve is an hyperbola--sometimes called the "Markowitz bullet"--which shows the return and standard deviation of every portfolio consisting of a long-only mixture of the two assets, the blue dot on the Y axis is the riskless asset, the yellow line is the "capital markets line," and the yellow dot is the "tangent portfolio."

But what do you call a diagram of this kind as a whole?

Also, historically, where it did first appear and who published the first ones? I don't think it was Markowitz, because the early papers that defined what's now called "modern portfolio theory" have charts of curves of mean and standard deviation, but the capital markets line and the tangent portfolio seem to be associated with CAPM, which came later.
Last edited by nisiprius on Sun Apr 29, 2018 9:51 am, edited 1 time in total.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

grok87
Posts: 8228
Joined: Tue Feb 27, 2007 9:00 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by grok87 » Sun Apr 29, 2018 6:08 am

Keep calm and Boglehead on. KCBO.

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Sun Apr 29, 2018 9:23 am

It's the graphic display of the Tobin separation theorem. James Tobin was a great macroeconomist, but he got his Nobel prize basically for this paper. Tobin was a very pleasant and humble person. When asked by a newsman after receiving his Nobel when the theorem said, Tobin replied, "It says you shouldn't put all your eggs in one basket". It says considerably more than that.
In his 1958 article Tobin also led the way in showing how to deal with utility maximization under uncertainty with an infinite number of possible (future states of the world). ...

Under Tobin’s assumptions we can reformulate the person’s decision problem as being one of trading off risk and expected return. Risk, or more precisely the variance of your investment portfolio creates spread in the returns you expect. People are willing to assume more risk only if compensated by a higher level of expected return. One can thus think of a tradeoff people are willing to make between risk and expected return. They invest in risky assets to the point at which their willingness to trade off risk and return is equal to the rate at which they able to trade them off. It is difficult to exaggerate how brilliant is the simplification of the investment problem that flows from these assumptions. Instead of worrying about the investor’s optimization problem in potentially millions of possible states of the world, one need only worry about how the investor can trade off risk and return in the stock market."
Link - https://en.wikipedia.org/wiki/James_Tobin

Here is an interesting side note about the man. Jim Tobin was such an excellent Navy office trainee that Herman Wouk decided not to become a naval officer but instead a novelist. In The Caine Mutiny the character 'Tobit' performs his duties so well that the narrator of the story decides not to become an officer. Later when developing a well known econometric estimator, Tobin named his estimator the 'Tobit' model. :)
After the United States entered the war, I joined the Naval Reserve and spent ninety days in a Columbia University dormitory learning to be a naval officer. Among my friends were, for alphabetical reasons, Cyrus Vance and Herman Wouk. Wouk's thinly disguised reference to me in The Caine Mutiny was until recently my main source of notoriety. I spent nearly four years as a line officer on the destroyer U.S.S. Kearny, serving eventually as gunnery officer and then navigator and executive officer (second in command). Mostly, our ship engaged in convoy escort and other anti-submarine duty in the Atlantic and Mediterranean, but we also participated in the invasions of North Africa and Southern France and in the Italian campaign.
Above is from Tobin's autobiographical sketch presented to the Nobel committee.
Link - https://www.nobelprize.org/nobel_prizes ... n-bio.html

The Separation Theorem
Harry Markowitz was primarily concerned with the diversification of risky assets. James Tobin added the concept of combining risk-free assets, such as cash or bonds, with risky assets, such as stocks. His paper, "Liquidity Preference as Behavior Toward Risk" appeared in The Review of Economic Studies in February 1958. The concept he described is known as the Separation Theorem, because it separates Markowitz's approach from the completely different decision of dividing up the whole portfolio between risky and risk-free assets.
Liquidity Preference, Separation and Asset Pricing Tobin received the 1981 Nobel Memorial Prize “for his analysis of financial
markets and their relations to expenditure decisions, employment, production and prices ”. With Harry Markowitz [1952, 1959, 1970]
he developed what became the foundations of modern portfolio theory. The key ‘separation theorem’ proven by Tobin [1958b], is
that in a world with one safe asset and a large number of risky assets, portfolio choice by any risk-averse portfolio holder can
be described as a choice between the safe asset and the same portfolio of risky assets. The ratio of the shares in the total portfolio accounted for by any pair of risky assets is the same for all risk-averse portfolio holders. The degree of risk aversion only determines the shares
in the total portfolio accounted for by the safe asset and by the common portfolio of risky assets.
This is an important and beautiful
result, which is not done justice by Tobin’s own summary: “Don’t put all your eggs in one basket”. Indeed, Tobin’s remarkable result
is better summarised as ‘regardless of your degree of risk aversion and caution, you will only need two baskets for all your eggs’.
Link - http://eprints.lse.ac.uk/847/1/tobin.pdf

BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

User avatar
David Jay
Posts: 5278
Joined: Mon Mar 30, 2015 5:54 am
Location: Michigan

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by David Jay » Sun Apr 29, 2018 9:35 am

Many writers (for simplicity) just call it “the efficient frontier” graph.
Prediction is very difficult, especially about the future - Niels Bohr | To get the "risk premium", you really do have to take the risk - nisiprius

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Sun Apr 29, 2018 9:41 am

It is not the efficient frontier graph. There is no tangent line in the efficient frontier graph. That's the point. The efficient frontier graph is only concerned with the combination of risky assets. Markowitz's efficient frontier says nothing about how to combine risky assets with lowest risk assets.

BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

User avatar
nisiprius
Advisory Board
Posts: 36216
Joined: Thu Jul 26, 2007 9:33 am
Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by nisiprius » Sun Apr 29, 2018 9:43 am

Grok, Bob, thank you so much for pointing this out to me. Yes, it looks as if Tobin deserves credit for putting the tangent line on the diagram. However, I'm still left with two questions.

First, do working financial economists have a name for diagrams like the one I presented, above?

Second, I'd be tempted to call it a "Tobin diagram" if there isn't any other name for it... but I don't think I should coin names, and I also don't actually see it in Tobin's 1958 paper.

Grok pointed me to a helpful tutorial page by Glyn Holton. It presents what I'll call the "canonical diagram,"

Image

and credits it--or the concepts behind it--to Tobin. But nothing like it appears in the 1958 paper, Liquidity Preference as Behavior Toward Risk. The closest thing is probably this:

Image

I haven't yet tried to figure out how that diagram relates to the familiar ones; that's the only place where the word "tangent" appears in the paper... and he keeps talking about the curves as "ellipses," not hyperbolas... so this is not "the diagram as we know it."

P.S. Ummm... on staring at Glyn Holton's diagram again, I have a strong nagging feeling that the curve in her diagram is not an hyperbola. I'm sure that's artistic license, drawing packages typically having drawing tools for ellipses but not hyperbolas.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Sun Apr 29, 2018 9:58 am

Separation Theorem - Tobin
Portfolio choice is separated into two
stages:
• Find the efficient portfolio of risky assets;
• Find the optimum fraction to invest in the efficient portfolio of risky assets and the risk-free asset.

The role of risk aversion is confined to the second stage and plays no role in the first stage.

See - https://www.albany.edu/~bd445/Economics ... heorem.pdf

Given the rate of return of the lowest risk asset everyone will hold the same risky assets in the same proportion. The efficient frontier is simple a frontier of trade-offs of risk and return. It is silent on how you should allocate your holdings among the risky assets.

BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

dbr
Posts: 27207
Joined: Sun Mar 04, 2007 9:50 am

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by dbr » Sun Apr 29, 2018 10:03 am

I doubt any of the curves involved on these charts are either hyperbolas or ellipses. Is there some reason the curves should be described by those exact mathematical figures?

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Sun Apr 29, 2018 10:23 am

What Tobin's separation theorem strongly suggests, as a practical matter, is first decide on the allocation among your risky assets without considering your low risk assets. A simple example would be say 70% US stock index and 30% international stock index for your risky asset allocation. Then decide on the proportion of risky assets vs safe assets. For a person near or in retirement with assets mainly in a tax advantaged account TIPS bonds or funds would be the low risk asset. The 70/30 proportion between US stocks and foreign stocks will be the same regardless of the proportion between TIPS and stocks.

BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

User avatar
nisiprius
Advisory Board
Posts: 36216
Joined: Thu Jul 26, 2007 9:33 am
Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by nisiprius » Sun Apr 29, 2018 10:38 am

dbr wrote:
Sun Apr 29, 2018 10:03 am
I doubt any of the curves involved on these charts are either hyperbolas or ellipses. Is there some reason the curves should be described by those exact mathematical figures?
Yes. They are hyberbolas. At least in my personal implementation. If you have five numbers:

Return of asset A
Standard deviation of asset A
Return of asset B
Standard deviation of asset B
Correlation between A and B

then you can calculate the return and the standard deviation of any portfolio consisting of a mix of A and B. It's math, not finance. And it doesn't depend on anything else (it doesn't assume a normal distribution or an efficient market or anything). You just crunch six numbers, the five parameters above and the percentage of A, and you come out with a point.

And all the points form an hyperbola.

Perhaps an easy way to see this is if you imagine a very lopsided situation in which A has humongous return and standard deviation and B has tiny return and standard deviation. Since B has such a tiny effect, the curve will be nearly a straight line with a little hook at the end. Well, hyperbolas straighten out with distance--they're just slicing through a cone, and the further out you get the less the offset from the apex matters--so it's plausible that the curve would be an hyperbola.

Here's a "almost a straight line with a little hook at the end," Vanguard REIT Index plus Vanguard Treasury Money Market.

Image

A lot of the mischief of MPT comes from showing the (somewhat rare) examples where you have a nice bulgy efficient frontier curve, as in my initial post, that pushes the tangent line and pivots it up way above the two asset dots... whereas there are a lot of real-world examples where that doesn't happen at all.

For example, in the case above, you will see that the optimum allocation is about 50/50 between REITs and the money market fund. However, it's nonsense because the curve is so straight that the difference in Sharpe ratio between 100% REIT and 51.73% REIT, 48.27% money market is truly negligible. Nevertheless, I can imagine an unscrupulous person insisting that the money market fund, due to the fact that it indeed has low correlation with the REIT fund, acts as a diversifier and improves the portfolio as a whole.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Sun Apr 29, 2018 11:03 am

nisiprius wrote:
Sun Apr 29, 2018 10:38 am
A lot of the mischief of MPT comes from showing the (somewhat rare) examples where you have a nice bulgy efficient frontier curve, as in my initial post, that pushes the tangent line and pivots it up way above the two asset dots... whereas there are a lot of real-world examples where that doesn't happen at all.

For example, in the case above, you will see that the optimum allocation is about 50/50 between REITs and the money market fund. However, it's nonsense because the curve is so straight that the difference in Sharpe ratio between 100% REIT and 51.73% REIT, 48.27% money market is truly negligible. Nevertheless, I can imagine an unscrupulous person insisting that the money market fund, due to the fact that it indeed has low correlation with the REIT fund, acts as a diversifier and improves the portfolio as a whole.
Nisi,

A money market fund is a low risk asset. It does not belong in the efficient frontier of risky assets. The portfolio of assets on the efficient frontier consist solely of risky assets. There is no efficient frontier that looks almost straight with a hook on the end. The low risk asset is the asset that is duration matched to the liability to have essentially zero standard deviation. If the investment horizon is 3 months or less then a money market fund would be the surrogate for the risk free asset. For a retiree with an investment horizon of 15-30 years the low risk asset is duration matched real bonds such as TIPS.

The ratio of the shares in the total portfolio accounted for by any set of risky assets on the efficient frontier is the same for all risk-averse portfolio holders. The degree of risk aversion only determines the shares in the total portfolio accounted for by the safe asset and by the common portfolio of risky assets on the efficient frontier.

Safe assets are not on the efficient frontier.

BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Sun Apr 29, 2018 12:07 pm

Steps in applying the separation theorem.

1) Decide what risky assets to include in your portfolio. All possible combinations of the risky assets compose the minimum variance frontier of risky assets. Those combinations of risky assets that lie to the right and above the minimum variance combination and where the frontier has positive slope are said to be on the efficient frontier of risky assets.
2) Decide on your surrogate for the risk-free asset.
3) The tangency point between the straight line with vertical intercept at the risk-free asset return rate and the efficient frontier determines the optimal mix of risky assets. (This is regardless of the mix between the low risk asset and the portfolio of risky assets.)
4) Decide how much of your portfolio should be in the low risk asset and how much should be in the mix of risky assets.

Steps (3) and (4) are separate decisions and hence the name - separation theorem.

Example
Risky assets are US stock fund and international stock fund.
Pick you surrogate for the risk-free asset.
Find the tangency point between the efficient frontier of these two risky assets and the straight line with vertical intercept at the risk-free rate of return.
The tangency point in this case turns out to be 60% US stock and 40% international stock.
You decide that 30% of portfolio should be in the low risk (risk-free surrogate) asset.

Portfolio asset allocation becomes.
30% low risk asset
42% US stock fund (70% of 60%)
28% international stock fund (70% of 40%)

BobK

PS - The tangency point between the straight line and the efficient frontier is where the reward/risk ratio is highest for the portfolio of risky assets, which makes that mix of the risky assets the optimal combination. And that's true regardless of how you mix that combination of risky assets with your best surrogate for the risk-free asset.
Last edited by bobcat2 on Tue May 01, 2018 5:13 pm, edited 1 time in total.
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

User avatar
Oicuryy
Posts: 1235
Joined: Thu Feb 22, 2007 10:29 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by Oicuryy » Sun Apr 29, 2018 12:26 pm

dbr wrote:
Sun Apr 29, 2018 10:03 am
I doubt any of the curves involved on these charts are either hyperbolas or ellipses. Is there some reason the curves should be described by those exact mathematical figures?
This guy calls it a parabola and gives an equation for it.
http://www.calculatinginvestor.com/2011 ... rontier-1/

Ron
Money is fungible | Abbreviations and Acronyms

User avatar
nisiprius
Advisory Board
Posts: 36216
Joined: Thu Jul 26, 2007 9:33 am
Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by nisiprius » Sun Apr 29, 2018 12:48 pm

Oicuryy wrote:
Sun Apr 29, 2018 12:26 pm
dbr wrote:
Sun Apr 29, 2018 10:03 am
I doubt any of the curves involved on these charts are either hyperbolas or ellipses. Is there some reason the curves should be described by those exact mathematical figures?
This guy calls it a parabola and gives an equation for it.
http://www.calculatinginvestor.com/2011 ... rontier-1/

Ron
Image

That's no parabola, Ron. I'm too lazy to check through his equations, but that's no parabola.

In a parabola, the two sides gradually straighten out and become parallel to each other. His picture is showing a shape that, if it is a conic section, could very well be an hyperbola--it looks like an hyperbola, if you slice through a cone exactly at its apex you get a V shape with a sharp vertex, if you miss the apex by a bit you get that V shape with a blunt end. It can't possibly be a parabola, an ellipse, or a circle.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

User avatar
nisiprius
Advisory Board
Posts: 36216
Joined: Thu Jul 26, 2007 9:33 am
Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by nisiprius » Sun Apr 29, 2018 1:00 pm

bobcat2 wrote:
Sun Apr 29, 2018 11:03 am
...A money market fund is a low risk asset. It does not belong in the efficient frontier of risky assets. The portfolio of assets on the efficient frontier consist solely of risky assets. There is no efficient frontier that looks almost straight with a hook on the end...
I was trying to find the most extreme example for which I had data. OK, I'll try not to use almost-riskless assets in these charts.

Would you accept the Vanguard Short-Term Investment Grade bond fund as legitimate?

In my opinion, in this chart, the efficient frontier, assuming long-only portfolios, is reasonably described as "An almost straight line with a hook at the end." In my opinion. In this case, though, the tangent portfolio is a large improvement over either asset in isolation.

Image

And here's one where, under all the usual assumptions, the shape of the efficient frontier curve and the return of the riskless asset leads to a recommendation to "diversify" a 100% small-cap value holding, Fama-French "small high," by adding 32.44% to SBBI Large Stocks (S&P 500)--yet it is obvious that the improvement obtained is negligible.

Image

And then, of course, there are many pairs of assets that yield charts like this:

Image
Last edited by nisiprius on Sun Apr 29, 2018 1:21 pm, edited 2 times in total.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

dbr
Posts: 27207
Joined: Sun Mar 04, 2007 9:50 am

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by dbr » Sun Apr 29, 2018 1:02 pm

Oicuryy wrote:
Sun Apr 29, 2018 12:26 pm
dbr wrote:
Sun Apr 29, 2018 10:03 am
I doubt any of the curves involved on these charts are either hyperbolas or ellipses. Is there some reason the curves should be described by those exact mathematical figures?
This guy calls it a parabola and gives an equation for it.
http://www.calculatinginvestor.com/2011 ... rontier-1/

Ron
Thanks. In that analysis variance is a quadratic in the return, hence a parabola.

User avatar
nisiprius
Advisory Board
Posts: 36216
Joined: Thu Jul 26, 2007 9:33 am
Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by nisiprius » Sun Apr 29, 2018 1:06 pm

dbr wrote:
Sun Apr 29, 2018 1:02 pm
Oicuryy wrote:
Sun Apr 29, 2018 12:26 pm
dbr wrote:
Sun Apr 29, 2018 10:03 am
I doubt any of the curves involved on these charts are either hyperbolas or ellipses. Is there some reason the curves should be described by those exact mathematical figures?
This guy calls it a parabola and gives an equation for it.
http://www.calculatinginvestor.com/2011 ... rontier-1/

Ron
Thanks. In that analysis variance is a quadratic in the return, hence a parabola.
Picture, I worked with parabolas. I knew parabolas. Parabolas were friends of mine. Picture, you're no parabola.

Seriously, I think what he is saying there is that the variance, if plotted, would be a parabola. But he immediately goes on to say "using the standard deviation rather than the variance," plots the standard deviation (as one does), and never puts a name to the resulting shape.

Image
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Sun Apr 29, 2018 1:26 pm

nisiprius wrote:
Sun Apr 29, 2018 1:00 pm
bobcat2 wrote:
Sun Apr 29, 2018 11:03 am
...A money market fund is a low risk asset. It does not belong in the efficient frontier of risky assets. The portfolio of assets on the efficient frontier consist solely of risky assets. There is no efficient frontier that looks almost straight with a hook on the end...
I was trying to find the most extreme example for which I had data. Would you accept the Vanguard Short-Term Investment Grade bond fund as legitimate?
No.

The efficient frontier consists solely of a portfolio of risky assets. A short-term bond fund is not risky if the investment horizon is short. In that case it might be the surrogate for the risk-free asset. If the investment horizon is long, there is not much justification for holding a short-term bond fund. For in that case it has risk but a lower expected return than riskier assets such as stocks.

The idea here is to decide on your portfolio of risky assets. Those risky assets are what constitutes the efficient frontier. Then decide how to mix that portfolio of risky assets with the low risk asset. The beauty of the separation theorem is that it determines the AA among the risky assets, regardless of the mix of low risk asset to risky assets.

Who thinks their collection of risky assets should include a ST US bond fund? An emerging market bond fund is a risky asset. A high yield (junk) bond fund is a risky asset. If the investment horizon is not long a LT US bond fund is a risky asset. Most other bond funds are not very risky. The separation theorem leads to the adage - don't take risk on the bond side.

Think of the separation theorem as telling you how to pick the AA of a three fund portfolio. You pick your two risky assets. You pick your low risk asset and use the separation theorem to decide what the proportion is among the two risky assets, regardless of the AA between the low risk asset and the portfolio of the two risky assets. Then in the second step of implementing the separation theorem you decide the proportion between the low risk asset and the two risky assets.

BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

User avatar
Oicuryy
Posts: 1235
Joined: Thu Feb 22, 2007 10:29 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by Oicuryy » Sun Apr 29, 2018 1:49 pm

dbr wrote:
Sun Apr 29, 2018 1:02 pm
Thanks. In that analysis variance is a quadratic in the return, hence a parabola.
You're welcome. I'm sure nisiprius is right about the name of the curve.

Ron
Money is fungible | Abbreviations and Acronyms

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Sun Apr 29, 2018 1:53 pm

From Wikipedia.
A combination of assets, i.e. a portfolio, is referred to as "efficient" if it has the best possible expected level of return for its level of risk (which is represented by the standard deviation of the portfolio's return). Here, every possible combination of risky assets can be plotted in risk–expected return space, and the collection of all such possible portfolios defines a region in this space. In the absence of the opportunity to hold a risk-free asset, this region is the opportunity set (the feasible set). The positively sloped (upward-sloped) top boundary of this region is a portion of a hyperbola and is called the "efficient frontier."

If a risk-free asset is also available, the opportunity set is larger, and its upper boundary, the efficient frontier, is a straight line segment emanating from the vertical axis at the value of the risk-free asset's return and tangent to the risky-assets-only opportunity set. All portfolios between the risk-free asset and the tangency portfolio are portfolios composed of risk-free assets and the tangency portfolio, while all portfolios on the linear frontier above and to the right of the tangency portfolio are generated by borrowing at the risk-free rate and investing the proceeds into the tangency portfolio.
And that tangency point determines the optimal mix of risky assets, regardless of how one mixes the low risk asset with that optimal mix of risky assets. So in a sense the straight line segment is an efficient frontier once we mix the low risk asset with the portfolio of risky assets. It is uncommon though to refer to the straight line segment as the new efficient frontier. Usually when we refer to the efficient frontier we are only discussing the possible mix of risky assets curved line. Sometimes the straight line is called the optimal capital asset line (CAL).

Link- https://en.wikipedia.org/wiki/Efficient_frontier

BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

dbr
Posts: 27207
Joined: Sun Mar 04, 2007 9:50 am

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by dbr » Sun Apr 29, 2018 1:56 pm

nisiprius wrote:
Sun Apr 29, 2018 1:06 pm
dbr wrote:
Sun Apr 29, 2018 1:02 pm
Oicuryy wrote:
Sun Apr 29, 2018 12:26 pm
dbr wrote:
Sun Apr 29, 2018 10:03 am
I doubt any of the curves involved on these charts are either hyperbolas or ellipses. Is there some reason the curves should be described by those exact mathematical figures?
This guy calls it a parabola and gives an equation for it.
http://www.calculatinginvestor.com/2011 ... rontier-1/

Ron
Thanks. In that analysis variance is a quadratic in the return, hence a parabola.
Picture, I worked with parabolas. I knew parabolas. Parabolas were friends of mine. Picture, you're no parabola.

Seriously, I think what he is saying there is that the variance, if plotted, would be a parabola. But he immediately goes on to say "using the standard deviation rather than the variance," plots the standard deviation (as one does), and never puts a name to the resulting shape.

Image
I thought of that just a couple of minutes ago as it slowly dawned that people plot SD not SD^2. That curve can be a hyperbola, I think, if it can be put in the form (x-a)^2/b - (y-c)^2/d = e.
Last edited by dbr on Sun Apr 29, 2018 2:06 pm, edited 1 time in total.

User avatar
grayfox
Posts: 4928
Joined: Sat Sep 15, 2007 4:30 am

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by grayfox » Sun Apr 29, 2018 1:57 pm

nisiprius wrote:
Sun Apr 29, 2018 6:00 am

But what do you call a diagram of this kind as a whole?
Google efficient frontier diagram and it shows a bunch of graphs like that. That's what Google thinks it is.

User avatar
nisiprius
Advisory Board
Posts: 36216
Joined: Thu Jul 26, 2007 9:33 am
Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by nisiprius » Sun Apr 29, 2018 2:40 pm

bobcat2 wrote:
Sun Apr 29, 2018 1:26 pm
...Who thinks their collection of risky assets should include a ST US bond fund? An emerging market bond fund is a risky asset...
Well, we're chopping logic and setting boundaries here. We're looking at a standard deviation of 4, compared to something like 0.75 for Treasury bills and of course 0 for an ideal riskless asset. A short-term US bond fund, which sometimes goes down by a small amount, is qualitatively different from a bank account which never does. I'd have said short-term bonds are a risky asset with very low risk. I guess I'm not sure what the generally accepted definition of "risky asset" would be. Surely you would agree that the Vanguard Total Bond Market Index Fund is an appropriate asset for inclusion in one of these charts? Or not?
Last edited by nisiprius on Sun Apr 29, 2018 2:47 pm, edited 1 time in total.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

User avatar
nisiprius
Advisory Board
Posts: 36216
Joined: Thu Jul 26, 2007 9:33 am
Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by nisiprius » Sun Apr 29, 2018 2:44 pm

grayfox wrote:
Sun Apr 29, 2018 1:57 pm
nisiprius wrote:
Sun Apr 29, 2018 6:00 am

But what do you call a diagram of this kind as a whole?
Google efficient frontier diagram and it shows a bunch of graphs like that. That's what Google thinks it is.
Good idea, but most of those graphs are not showing the tangent line or the riskless asset. The efficient frontier is the curved line. But it is only part of "my" chart. I want a name for the kind of chart that includes a tangent line as an essential part. But I'm beginning to think the answer is that there isn't any name for it.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Sun Apr 29, 2018 3:32 pm

Nisiprius
We're looking at a standard deviation of 4, compared to something like 0.75 for Treasury bills and of course 0 for an ideal riskless asset. A short-term US bond fund, which sometimes goes down by a small amount, is qualitatively different from a bank account which never does. I'd have said short-term bonds are a risky asset with very low risk.
Nisi, you are missing the point.The risk-free asset is duration matched. It has essentially zero standard deviation and essentially zero correlation with the risky assets. For example, if the investment horizon is ten years, then a ten year zero coupon bond has zero standard deviation when it matures in ten years and zero correlation with the risky assets. In that case the ten year zero is an excellent surrogate for the hypothetical risk-free asset. You combine such a surrogate for the risk-free asset with your portfolio of risky assets. It seems to me that the Vanguard Total Bond Index fund would only be a good idea to include in your portfolio of risky assets when your investment horizon is long and you care about nominal rather than real outcomes.

Whether ST nominal bonds are risky assets depends on the length of your investment horizon and whether you are concerned with nominal or real outcomes. In the ST they are low risk. In the LT they aren't low risk, particularly in real terms.

What is the efficient frontier.
Every possible combination of risky assets can be plotted in risk–expected return space, and the collection of all such possible portfolios defines a region in this space. In the absence of the opportunity to hold a risk-free asset, this region is the opportunity set (the feasible set). The positively sloped (upward-sloped) top boundary of this region is a portion of a hyperbola and is called the "efficient frontier".
Wikipedia

BobK
Last edited by bobcat2 on Tue May 01, 2018 5:28 pm, edited 1 time in total.
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

User avatar
Pajamas
Posts: 6015
Joined: Sun Jun 03, 2012 6:32 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by Pajamas » Sun Apr 29, 2018 3:44 pm

Optimal portfolio chart.

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Sun Apr 29, 2018 3:46 pm

The efficient portfolio frontier is defined as the set of portfolios of risky assets offering the highest possible expected rate of return for any given standard deviation.
from Bodie and Merton, Financial Economics.

The separation theorem combines the above portfolio of risky assets with the low risk asset and determines the optimal AA of the risky assets, regardless of the mix between the low risk asset and the risky assets.

The beauty of the separation theorem is that when we find the tangency point between the straight line drawn from the vertical intercept of the risk free return to the efficient frontier of risky assets that tangency point determines the optimal mix of risky assets, regardless of how we mix the lowest risk asset with the risky assets. That outcome is both eloquent and non-intuitive.

The efficient frontier simply gives the trade-offs between risk and return for all the combinations of the risky assets. It is silent about which combination to choose. The separation theorem adds the non-risky asset and determines the optimal combination of risky assets. What is left for the investor to determine is simply the allocation between the non-risky asset and the portfolio of risky assets.

BobK
Last edited by bobcat2 on Tue May 01, 2018 4:40 pm, edited 1 time in total.
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Sun Apr 29, 2018 8:55 pm

nisiprius wrote:
Sun Apr 29, 2018 6:00 am
... because the early papers that defined what's now called "modern portfolio theory" have charts of curves of mean and standard deviation, but the capital markets line and the tangent portfolio seem to be associated with CAPM, which came later.
This has nothing to do with CAPM. If it were associated with CAPM the measurement of risk along the horizontal axis would be measured in units of beta (β), not in terms of standard deviation. This is a refinement of MPT by mixing the low risk asset with the portfolio of risky assets on the efficient frontier.

BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

User avatar
nisiprius
Advisory Board
Posts: 36216
Joined: Thu Jul 26, 2007 9:33 am
Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by nisiprius » Sun Apr 29, 2018 8:59 pm

bobcat2 wrote:
Sun Apr 29, 2018 8:55 pm
nisiprius wrote:
Sun Apr 29, 2018 6:00 am
... because the early papers that defined what's now called "modern portfolio theory" have charts of curves of mean and standard deviation, but the capital markets line and the tangent portfolio seem to be associated with CAPM, which came later.
This has nothing to do with CAPM. If it were associated with CAPM the measurement of risk along the horizontal axis would be measured in units of beta (β), not in terms of standard deviation. This is a refinement of MPT by mixing the low risk asset with the portfolio of risky assets on the efficient frontier.

BobK
Thank you.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Tue May 01, 2018 5:53 pm

The risk-free rate of return on the chart (the vertical intercept of the straight line) is not a hypothetical value. It is the rate of return for the best surrogate asset you can find for the theoretical risk-free asset. What asset to use as the best risk-free surrogate depends on the situation.

Some examples - If the investment planning horizon is 4 months or less, a money market fund or T-bills would be good surrogates. If the planning horizon is between 4 months and two years an ultra-short bond fund or T-bills or Treasury Notes would be good surrogates. If the planning horizon is long, Treasury bonds or TIPS would be good surrogates depending on whether you are concerned with nominal or real outcomes.

After deciding on the best surrogate for the risk free asset and what risky assets to include in your portfolio, you use that information to find the best mix of risky assets (the tangency point between the capital market line and the efficient frontier). Lastly, you determine how much of your portfolio consists of the very low risk asset (the surrogate for the risk-free asset) and the above optimal fixed proportions of risky assets. It is only in this last step that you take risk capacity and risk tolerance into account.

BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

staythecourse
Posts: 5813
Joined: Mon Jan 03, 2011 9:40 am

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by staythecourse » Tue May 01, 2018 6:01 pm

Nothing to add, but this is why bogleheads have to be the most intelligent group of amateur finance guys around. :D

Good luck.
"The stock market [fluctuation], therefore, is noise. A giant distraction from the business of investing.” | -Jack Bogle

User avatar
dratkinson
Posts: 4268
Joined: Thu Jul 26, 2007 6:23 pm
Location: Centennial CO

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by dratkinson » Wed May 02, 2018 3:52 pm

+1 I don't usually stay long in theory topics as they quickly get beyond my level of understanding. But did for this one.


I found grok's link especially helpful as Glyn Holton dumbed-down the related theories enough that I could understand the high-level concept.
grok87 wrote:
Sun Apr 29, 2018 6:08 am
i think tobin did it?
https://www.glynholton.com/notes/capital_market_line/
I started with grok's link, found it informative, so followed Holton's internal links to overviews of the other theorems. Then reread the clarifying discussions here.

I found this link helpful in providing a simplified high-level overview of the relationships among Markowitz (1952), Tobin (1958), and Sharpe (1964). Follow internal links to more information on each.
See: https://www.glynholton.com/notes/portfolio_theory/


Who knew the 3-fund portfolio was so sophisticated?!


Can't wait for my next encounter with a door-to-door EJ rep.

"What am I doing for myself? I'm using the theories of multiple Nobel Laureate to construct a super-efficient portfolio that I then modify for my personal risk tolerance." :)


Thanks. Interesting topic.
d.r.a, not dr.a. | I'm a novice investor, you are forewarned.

User avatar
patrick013
Posts: 2297
Joined: Mon Jul 13, 2015 7:49 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by patrick013 » Wed May 02, 2018 4:16 pm

Well you get what you pay for.

How about a X - Y Scatter Plot
age in bonds, buy-and-hold, 10 year business cycle

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Wed May 02, 2018 8:49 pm

James Tobin on how to change the amount of risk in your portfolio.
You would choose the same portfolio of nonsafe assets regardless of how risk-averse you were. Even if you wanted to change the amount of risk in the (overall) portfolio, you’d do it by changing the amount of the safe assets relative to the nonsafe assets, but not by changing the different proportions in which you held the nonsafe assets relative to each other.

Here is investment advisor Frank Armstrong explaining Tobin's separation property.
Tobin said investors should first determine their appetite for risk. With that level of risk tolerance in mind, investors can choose the equity portfolio from a Markowitz optimization. In other words, they’d choose the portfolio on the efficient frontier–or the line on a risk-return graph that includes all portfolios with the greatest expected return for a given level of risk–that has the highest return per unit of risk (the risk portfolio at the tangency point). Investors should then satisfy their liquidity and safety needs with another portfolio, called the zero-risk portfolio. In essence, investors have two buckets–an equity bucket for growth and a liquidity or safety bucket of lower-risk investments ...and simply divide their assets between them.
When using Tobin's separation property the risk-free asset is not a hypothetical asset and the risk-free rate of return is not assumed. Instead you determine the best safe asset or assets to use as the surrogate for the hypothetical risk-free asset. The risk-free rate of return used is the current rate of return of the surrogate safe asset.

BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

Lastrun
Posts: 109
Joined: Wed May 03, 2017 6:46 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by Lastrun » Wed May 02, 2018 9:44 pm

Question for BobK.

Does the risky asset have to be equity?

Stated differently, could my portfolio choice be risky = total bond, and risk free = TBills?

I am trying also to reconcile this with the concept that the risky asset is the market portfolio per Sharpe. Market portfolio of what—just stocks or stock and bonds?

Thanks so much for this great discussion.

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Wed May 02, 2018 11:48 pm

Lastrun wrote:
Wed May 02, 2018 9:44 pm
Question for BobK.
Does the risky asset have to be equity?
No, but typically the mix of risky assets is mainly equity.
Stated differently, could my portfolio choice be risky = total bond, and risk free = TBills?
It could be, but that would be unusual. That's true both on the risky asset side and the safe asset side. Keep in mind that the risk-free rate of return is the return of the best surrogate safe asset you can find for the theoretical risk-free asset. What asset to use as the best risk-free surrogate depends on the situation.

Some examples - If the investment horizon is 4 months or less, a money market fund or T-bills would be good surrogates. If the horizon is between 4 months and two years an ultra-short bond fund or T-bills or Treasury Notes would be good surrogates. If the horizon is long, Treasury bonds or TIPS would be good surrogates depending on whether you are concerned with nominal or real outcomes.

My portfolio of safe assets are a money market fund and ultra short bond fund for the first two years of my investment horizon and TIPS for the years of my horizon beyond the first two years.
I am trying also to reconcile this with the concept that the risky asset is the market portfolio per Sharpe. Market portfolio of what—just stocks or stock and bonds?
I don't get worked up with trying to figure out what the market portfolio is. If my equities are well-diversified that is good enough. I hold the total market stock portfolio in the US and for foreign stocks I hold developed market large cap, developed market mid-cap, developed market small cap, and emerging market funds. I'm probably holding over 7,000 stocks globally with about 60% in the US. That's well diversified. I don't really care if it's the market portfolio a la Sharpe. As long as you are holding a well diversified portfolio of equities and (possibly) some bonds as your risk portfolio, I don't think you should care either. Now if your risk portfolio is 50% small cap value or 50% emerging market that's something else again.

Pick a reasonable combination of risky stock and (perhaps risky) bond funds and combine them in a reasonable way without extreme weightings far from the market weights. Pick your safe assets with care. There is a different safe asset that corresponds to each possible maturity and to each unit of account. A 10 year zero coupon Treasury is the safe asset if held for 10 years and you are interested in the nominal, not real value. If you interested in the real value you need to hold TIPS bonds with a weighted duration of 10 years or I-bonds. If you want the money in two years or less short-term high quality bonds and at the short end T-bills are the safe assets.

If the number of risky funds is about 5-7 or less you can actually find the optimal portfolio on the efficient frontier without too much work. That is, you can find the tangency point of the CML with the efficient frontier of risky assets using a spreadsheet if you are a true geek at heart. I personally don't think that's worthwhile as a practical matter, but if your truly geeky with extra time on your hands - give it a shot.

Some people wonder why the tangency point is the optimal portfolio of risky assets. It's optimal because it is the point on the efficient frontier where the reward to risk ratio is highest. Capital allocation lines above the efficient frontier are impossible. Capital allocation lines below the tangency point are inferior - the reward to risk ratio is lower. The CML, capital market line, is the capital allocation line tangent to the efficient frontier. It is the optimal capital allocation line.

The important practical points I draw out of this are the following. Decide what the safest assets are for your situation. Then pick your risky or growth assets that you want to include in your risky portfolio. Decide on the proportions in which you want to hold these risky assets relative to each other. If you want to change the amount of risk in your total portfolio of safe and risky assets change the amount of the safe assets relative to the risky assets, don't do it by changing the different proportions in which you hold the risky assets relative to each other.

BobK

Edited to add the words (possibly) and (perhaps) in front of the word bonds in two places.
Last edited by bobcat2 on Thu May 03, 2018 11:31 am, edited 1 time in total.
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

User avatar
alec
Posts: 2908
Joined: Fri Mar 02, 2007 2:15 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by alec » Thu May 03, 2018 7:37 am

bobcat2 wrote:
Wed May 02, 2018 11:48 pm
Lastrun wrote:
Wed May 02, 2018 9:44 pm
Question for BobK.
Does the risky asset have to be equity?
No, but typically the mix of risky assets is mainly equity.
Stated differently, could my portfolio choice be risky = total bond, and risk free = TBills?
It could be, but that would be unusual. That's true both on the risky asset side and the safe asset side. Keep in mind that the risk-free rate of return is the return of the best surrogate safe asset you can find for the theoretical risk-free asset. What asset to use as the best risk-free surrogate depends on the situation.

Some examples - If the investment horizon is 4 months or less, a money market fund or T-bills would be good surrogates. If the horizon is between 4 months and two years an ultra-short bond fund or T-bills or Treasury Notes would be good surrogates. If the horizon is long, Treasury bonds or TIPS would be good surrogates depending on whether you are concerned with nominal or real outcomes.

My portfolio of safe assets are a money market fund and ultra short bond fund for the first two years of my investment horizon and TIPS for the years of my horizon beyond the first two years.
I am trying also to reconcile this with the concept that the risky asset is the market portfolio per Sharpe. Market portfolio of what—just stocks or stock and bonds?
I don't get worked up with trying to figure out what the market portfolio is. If my equities are well-diversified that is good enough. I hold the total market stock portfolio in the US and for foreign stocks I hold developed market large cap, developed market mid-cap, developed market small cap, and emerging market funds. I'm probably holding over 7,000 stocks globally with about 60% in the US. That's well diversified. I don't really care if it's the market portfolio a la Sharpe. As long as you are holding a well diversified portfolio of equities and some bonds as your risk portfolio, I don't think you should care either. Now if your risk portfolio is 50% small cap value or 50% emerging market that's something else again.

Pick a reasonable combination of risky stock and bond funds or etfs and combine them in a reasonable way without extreme weightings far from the market weights. Pick your safe assets with care. There is a different safe asset that corresponds to each possible maturity and to each unit of account. A 10 year zero coupon Treasury is the safe asset if held for 10 years and you are interested in the nominal, not real value. If you interested in the real value you need to hold TIPS bonds with a weighted duration of 10 years or I-bonds. If you want the money in two years or less short-term high quality bonds and at the short end T-bills are the safe assets.

If the number of risky funds is about 5-7 or less you can actually find the optimal portfolio on the efficient frontier without too much work. That is, you can find the tangency point of the CML with the efficient frontier of risky assets using a spreadsheet if you are a true geek at heart. I personally don't think that's worthwhile as a practical matter, but if your truly geeky with extra time on your hands - give it a shot.

Some people wonder why the tangency point is the optimal portfolio of risky assets. It's optimal because it is the point on the efficient frontier where the reward to risk ratio is highest. Capital allocation lines above the efficient frontier are impossible. Capital allocation lines below the tangency point are inferior - the reward to risk ratio is lower. The CML, capital market line, is the capital allocation line tangent to the efficient frontier. It is the optimal capital allocation line.

The important practical points I draw out of this are the following. Decide what the safest assets are for your situation. Then pick your risky or growth assets that you want to include in your risky portfolio. Decide on the proportions in which you want to hold these risky assets relative to each other. If you want to change the amount of risk in your total portfolio of safe and risky assets change the amount of the safe assets relative to the risky assets, don't do it by changing the different proportions in which you hold the risky assets relative to each other.

BobK
:thumbsup :thumbsup
"It is difficult to get a man to understand something, when his salary depends upon his not understanding it!" - Upton Sinclair

Lastrun
Posts: 109
Joined: Wed May 03, 2017 6:46 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by Lastrun » Thu May 03, 2018 10:18 am

Thanks BobK for the answer and your patience.

User avatar
nisiprius
Advisory Board
Posts: 36216
Joined: Thu Jul 26, 2007 9:33 am
Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by nisiprius » Thu May 03, 2018 10:32 am

Just to reiterate: this thread has been valuable to me, but I think the answer to my question is:

Image
is an "efficient frontier diagram."

Image
is an "efficient frontier diagram."

The availability of a riskless asset is of great importance. It determines a single optimal portfolio from a range of efficient portfolios. But despite the fact that it took a second Nobel prize to get there, there is no special name for the chart that includes the riskless asset and the tangent line.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

pascalwager
Posts: 1141
Joined: Mon Oct 31, 2011 8:36 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by pascalwager » Thu May 03, 2018 11:05 am

Holding Sharpe's practical version of the (risky) market portfolio* isn't difficult at all. For the equity part, just use the VG Total World Stock Fund; or for lower fees, use the two VG Total U.S. and int'l stock funds, (currently) set them at 52/48, and readjust next year.

Sharpe describes the entire market portfolio in his RISMAT paper, Section 7. The whole portfolio uses only four stock and bond funds and VG could easily provide this as a fund-of-funds. Then we could copy it to fit individual account/age requirements.

The actual market portfolio is probably most suitable for retirees. A younger investor would usually want/need to increase the stock/bond ratio (take more risk), or even go 100% stocks. But the US/int'l stock ratio would still be 52/48 (currently). Now, if you want to beat the market, Sharpe can't help you there.

No, Sharpe doesn't have a name for the green diagram shown in some of the above posts although he employs the same diagram in his RISMAT paper.

*The market portfolio should be on the efficient frontier curve, but Markowitz proved that it's really not unless leveraging is employed.

Walkure
Posts: 31
Joined: Tue Apr 11, 2017 9:59 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by Walkure » Thu May 03, 2018 11:45 am

nisiprius wrote:
Thu May 03, 2018 10:32 am
Image
is an "efficient frontier diagram."

The availability of a riskless asset is of great importance. It determines a single optimal portfolio from a range of efficient portfolios. But despite the fact that it took a second Nobel prize to get there, there is no special name for the chart that includes the riskless asset and the tangent line.
This is a very impressive diagram, and I like the idea of the separation theory between allocation of the efficient risky portfolio and the risk-aversion balance between risky and "risk-free" assets or their nearest approximation for the duration. However, there is one thing that seems counter-intuitive to me. If the expected return of risky assets was unchanged (or even declined, say, due to "high" stock valuations) while the risk-free asset increased (i.e. higher real rates on I-bonds), then the capital markets line flattens out, pushing the tangent rightward on the efficient frontier. This implies that one should increase the volatility of the risky asset as the spread between risky and risk-free return narrows?

Lastrun
Posts: 109
Joined: Wed May 03, 2017 6:46 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by Lastrun » Thu May 03, 2018 12:25 pm

pascalwager wrote:
Thu May 03, 2018 11:05 am
The market portfolio should be on the efficient frontier curve, but Markowitz proved that it's really not unless leveraging is employed.
Help my small brain reconcile the above statement with this:

"The market portfolio is always efficient. For most people, the market portfolio is the most sensible decision." Eugene Fama

Thanks in advance.

Also, I do understand the leverage concept.

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Thu May 03, 2018 12:30 pm

Walkure wrote:
Thu May 03, 2018 11:45 am
nisiprius wrote:
Thu May 03, 2018 10:32 am
Image
is an "efficient frontier diagram."

The availability of a riskless asset is of great importance. It determines a single optimal portfolio from a range of efficient portfolios. But despite the fact that it took a second Nobel prize to get there, there is no special name for the chart that includes the riskless asset and the tangent line.
This is a very impressive diagram, and I like the idea of the separation theory between allocation of the efficient risky portfolio and the risk-aversion balance between risky and "risk-free" assets or their nearest approximation for the duration. However, there is one thing that seems counter-intuitive to me. If the expected return of risky assets was unchanged (or even declined, say, due to "high" stock valuations) while the risk-free asset increased (i.e. higher real rates on I-bonds), then the capital markets line flattens out, pushing the tangent rightward on the efficient frontier. This implies that one should increase the volatility of the risky asset as the spread between risky and risk-free return narrows?
An important implication of the separation property is that when the real return of the safe asset changes in a material way over time it affects the optimal combination of risky assets even if there are no bonds in the risky asset portfolio and even if you are 100% stocks and hold none of the safe asset. If the return on the safe asset rises, the optimal risky portfolio becomes more risky but the risk/reward ratio becomes smaller. The converse is true if the return on the safe asset declines.

I was wondering if and when someone would notice this. :) I didn't mention it because the main static points are hard enough to get across when people are not familiar with the separation property and adding dynamics like changing interest rates complicates the picture. Tobin supposedly said after writing the paper that this was the first word on the subject - not the last word. I believe he was aware of this implication when he wrote the paper and felt that it was one of the aspects of this issue beyond the scope of his first word. Bodie and Merton in their undergrad textbook Finance discuss briefly how dynamics and other aspects of the problem have been addressed in later 'words'. Notice also that Tobin doesn't address duration matching but only short-term assets when considering the safe asset. This is also not surprising. The idea of duration matching for hedging risk was first suggested by a British actuary in the 1950s. Expecting Tobin to include that in a paper he was writing in New Haven from 1957-58 is unrealistic.

BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

Walkure
Posts: 31
Joined: Tue Apr 11, 2017 9:59 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by Walkure » Thu May 03, 2018 12:44 pm

Thanks for confirming that, BobK.

pascalwager
Posts: 1141
Joined: Mon Oct 31, 2011 8:36 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by pascalwager » Thu May 03, 2018 1:53 pm

Lastrun wrote:
Thu May 03, 2018 12:25 pm
pascalwager wrote:
Thu May 03, 2018 11:05 am
The market portfolio should be on the efficient frontier curve, but Markowitz proved that it's really not unless leveraging is employed.
Help my small brain reconcile the above statement with this:

"The market portfolio is always efficient. For most people, the market portfolio is the most sensible decision." Eugene Fama

Thanks in advance.

Also, I do understand the leverage concept.
I'm familiar with Fama's statement, but I suspect he was just referring to the US stock market because he has also said that one's US/int'l allocation ratio is a matter of personal taste. To me, at this point, it's just a sound bite, like Warren Buffet's various investing statements.

pascalwager
Posts: 1141
Joined: Mon Oct 31, 2011 8:36 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by pascalwager » Thu May 03, 2018 3:02 pm

pascalwager wrote:
Thu May 03, 2018 1:53 pm
Lastrun wrote:
Thu May 03, 2018 12:25 pm
pascalwager wrote:
Thu May 03, 2018 11:05 am
The market portfolio should be on the efficient frontier curve, but Markowitz proved that it's really not unless leveraging is employed.


Help my small brain reconcile the above statement with this:

"The market portfolio is always efficient. For most people, the market portfolio is the most sensible decision." Eugene Fama

Thanks in advance.

Also, I do understand the leverage concept.
I'm familiar with Fama's statement, but I suspect he was just referring to the US stock market because he has also said that one's US/int'l allocation ratio is a matter of personal taste. To me, at this point, it's just a sound bite, like Warren Buffet's various investing statements.
John Rekentheler, M*, has an article on leveraging and the market portfolio--several months back if you're interested.

pascalwager
Posts: 1141
Joined: Mon Oct 31, 2011 8:36 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by pascalwager » Thu May 03, 2018 3:23 pm

pascalwager wrote:
Thu May 03, 2018 1:53 pm
Lastrun wrote:
Thu May 03, 2018 12:25 pm
pascalwager wrote:
Thu May 03, 2018 11:05 am
The market portfolio should be on the efficient frontier curve, but Markowitz proved that it's really not unless leveraging is employed.
Help my small brain reconcile the above statement with this:

"The market portfolio is always efficient. For most people, the market portfolio is the most sensible decision." Eugene Fama

Thanks in advance.

Also, I do understand the leverage concept.
I'm familiar with Fama's statement, but I suspect he was just referring to the US stock market because he has also said that one's US/int'l allocation ratio is a matter of personal taste. To me, at this point, it's just a sound bite, like Warren Buffet's various investing statements.
I think Fama would approve of a US Total Stock Market fund because it can be managed efficiently--not a lot of companies entering and leaving the index. But he has also said that indexing is stupid, probably thinking of the Russell 2000 and other predictable indexes.

User avatar
bobcat2
Posts: 5215
Joined: Tue Feb 20, 2007 3:27 pm
Location: just barely Outside the Beltway

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by bobcat2 » Thu May 03, 2018 3:52 pm

pascalwager wrote:
Thu May 03, 2018 3:23 pm
I think Fama would approve of a US Total Stock Market fund because it can be managed efficiently--not a lot of companies entering and leaving the index. But he has also said that indexing is stupid, probably thinking of the Russell 2000 and other predictable indexes.
This thread is about Tobin's separation theorem. This post is off-topic. If you want to discuss Fama's opinion on different stock indexes then start a thread on that topic.

Thank you,
BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.

User avatar
Epsilon Delta
Posts: 7430
Joined: Thu Apr 28, 2011 7:00 pm

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by Epsilon Delta » Thu May 03, 2018 4:04 pm

bobcat2 wrote:
Sun Apr 29, 2018 9:41 am
It is not the efficient frontier graph. There is no tangent line in the efficient frontier graph. That's the point. The efficient frontier graph is only concerned with the combination of risky assets. Markowitz's efficient frontier says nothing about how to combine risky assets with lowest risk assets.

BobK
This is quite wrong. Even if it's what many economists and financiers say it's still wrong.

Pretty much every discussion of the efficient frontier will begin by defining it as the Pareto optimal set:
https://en.wikipedia.org/wiki/Efficient_frontier wrote:In modern portfolio theory, the efficient frontier (or portfolio frontier) is an investment portfolio which occupies the 'efficient' parts of the risk-return spectrum. Formally, it is the set of portfolios which satisfy the condition that no other portfolio exists with a higher expected return but with the same standard deviation of return.
In CAPM with a risk free asset the upper limb of the hyperbola does not satisfy the definition and so cannot be the efficient frontier.

In CAPM without a risk free asset the efficient frontier is the upper limb of the hyperbola.
In CAPM with a risk free asset the efficient frontier is the tangent line.
In other pricing models that consider other things the efficient frontier is something else. Efficient frontier is a very general concept and not tied to CAPM, or even portfolio optimization.

User avatar
nisiprius
Advisory Board
Posts: 36216
Joined: Thu Jul 26, 2007 9:33 am
Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

Re: NAME for the MPT or CAPM-based diagram with the hyperbola and tangent line?

Post by nisiprius » Fri May 04, 2018 7:10 am

Epsilon Delta, I think you meant "the lower limb of the hyperbola does not satisfy..."

I am trying (and not succeeding) to visualize what happens in three-space if you have a surface representing the efficient frontier for three risky assets, and you move one of the assets so that it becomes less and less risky and finally becomes riskless. I think the tangent-line diagram is just a way of expressing this special case without requiring three dimensions, but I can't see it in my mind yet.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

Post Reply