There is a positive relationship between risk and return. But linear relationship is a debatable topic.  

The CAPM is a model for pricing an individual security or portfolio. The expected return of an asset is equal to the risk free rate plus the excess return of the market above the risk-free rate, adjusted for the asset's overall sensitivity to market fluctuations or its beta. Mathematically, the capital asset pricing model can be written as: E(Ri) = Rf + β(E(Rm) - Rf), where R is the return, E(R) is the expected return, i denotes any asset, f is the risk-free asset, and m is the market

not yet. The risk-return trade-off is sometimes negative. There are lots of discussion about  this irregularity, such as Bowman paradox(1980). The explanation has been given on the reference effect.

In very long term for the US capital market yes. Surprisingly, in all other markets positive and linear relationship between risk and return was not confirmed.

No, it is not true that there is a linear relationship between risk and return i.e. high risk associated with the high return and low risk with the low return. In some cases, low risk has been associated with higher returns, as was witnessed in a study of infrastructure stocks in Indian stock markets, available at pages 122-130 at the link:  http://www.amity.edu/abs/pdf/abr_dec2013.pdf

Similar results for obtained for infrastructure & other defensive stocks in other markets.

Since this is a deviation from the earlier accepted axiom in finance that “share returns and risk are always inextricably linked”, this phenomenon has been called 'the low volatility anomaly'.

This is a tricky question:

There should be a positive relationship between return and risk (please note I am changing the order and I am saying should, and only pretending to say positive, nothing else...).

But it should be better if we separate between ex ante risk and return and ex post (realized) return and risk (if you use some kind of market related measure of ex post risk).

If you are talking about the ex ante (let me say expected) relationship between return and risk you must (I would recommend) starting with the premise that there is a positive relationship between both. At least, more expected return should include more risk (risks known or unknown in advance... but that is another story) 

Finally, I would also say that ortodox finance asumes that  there is always a positive relationship between (at least ex ante) return and risk. If not, you cant (should not) use neither Portfolio Optimization (Markovitz) nor CAPM or related frameworks, even more elaborated like Black-Litterman, APT, ... and so on and so forth.

Dear Prof. Thonse Hawaldar,

                                                    We most agree with Prof. Fernandez-Herraiz on this String, but, the algebraic sense of linearity found in statistical panel data (experiments) and geometric sense of linearity found in visual graphical plots using systems engineering (spreadsheet) methods have been proved by us to satisfy Dbranes String Theory Functor Algebra Calculus (Mallick, Hamburger, Mallick (2016), Mallick(2014,2015)) on www.researchgate.net/Soumitra K. Mallick and www.econometricsociety.org/[email protected] websites, which we may dare say has been discovered by us, and have been able to solve the Millenium Mathematics Prize P vs. NP and some other problems in certain econophysical fields. If you care to you can take a look.

Soumitra K. Mallick

for Soumitra K. Mallick, Nick Hamburger, Sandipan Mallick

It is not easy to give one ultimate, definite answer to this. But, unlike Yu, I would say: Generally yes. Although there are some observations of paradox relations, the general rule is that higher risk is correlated with higher returns. This has been observed for the financial markets long since, eg. Sharp (1964), Fama/McBeth (1973) etc., the whole Capital Asset Pricing Model (CAPM) and others are more or less build on this assumption. From my view the relation is logical, not only in the capital markets. If an investment opportunity is subject to high returns and low risk at the same time, capital will be allocated to this investment very quickly once uncertainty is resolved. If this is not known or uncertain, then the investment is risky by definition. So, I think that empirically and from a practical perspective, the general rule holds true.  But, that does of course not mean that exceptions with paradox relations exists. However, they should be corrected as soon as uncertainty around them is resolved.   

In finance courses we see that there exist positive relationship between risk and return.And this leads to the foundation of modern portfolio theorty. But then there are market annomalies and investor irrationality which goes some what opposite to the standard finance theories. So in my opinion we can not say that it is always the case that there will be linear relationship 

I agree with profesor Carlos Fernandez and add that: in the ex-ante, it would be poor decision making to accept low return to a high risk investment, but an other factor has to be taken in account, and that is: probability of the risk happening; if the probability that the risk occurs is high and the consequences of it are also high, it would mean "economic suicide" to invest in that economic activity. For the ex-post, every case is diferente and the number of variables participating is to high to get conclusive conclusions.

In the short-run, anomalies in the market may cause some investments to have associated low risk and high return (paradox correlation). In the long-run, the market would adjust and smoothen such anomalies, reverting risk and return to positive correlation  (high risk and high return). A few of the short-run studies have reported the paradox situation, but there is overwhelming report of positive correlation in both short-run and long-run studies. Thus, risk and return largely exhibit linear relations in the same direction.

The CAPM is not valid, in particular the relationship between return and systemic risk is not linear. 

Article Is the CAPM valid? An Empirical Analysis in USA Stock Exchange

Deleted research item The research item mentioned here has been deleted

Maybe not linear, but positive. 

Theoretically that is correct. Given any concave utility function, where u'>0 and u''

There is a positive relationship but it is not necessary a linear because the relationship between the risk and return is moderated by the investor's investment objective. Sometimes the investor increases the portfolio risk but the portfolio return increases with less proportion.

There is a positive relationship but it is not necessary a linear, for example:

There is a positive relationship between risk and return. But linear relationship is a debatable topic.  

The CAPM is a model for pricing an individual security or portfolio. The expected return of an asset is equal to the risk free rate plus the excess return of the market above the risk-free rate, adjusted for the asset's overall sensitivity to market fluctuations or its beta. Mathematically, the capital asset pricing model can be written as: E(Ri) = Rf + β(E(Rm) - Rf), where R is the return, E(R) is the expected return, i denotes any asset, f is the risk-free asset, and m is the market

Have a look at the work done by Ross & Rawls (Arbitrage Pricing Theory) and some of the stuff taught by Damodaran.  Fernandes each year asks all accounting professors for their veiw of the risk premium (but that's hardly logical to ask people who never have any money to invest).  Logic dictates that there is a positive relationship, but the issue goes back to how do you measure risk - a single Beta is hardly a good measure and probably does not account for more than about 30% of the answer.  I think you need to look at some of the work on behavioural finance to see that, unlike the assumptions behind CAPM, investors are not really rational - and without that assumption (amongst others) there can be no linear relationship.

Dear Iqbal Thonse Hawaldar

From a theoretical point of view and after some personal experiences and looking for such behavior at the international level, I can tell you the following:

• This direct relationship is questionable where the greater risk is the greater return of some investment.
• Let's look at the example of the balance between alpha factors (investor, investment manager) and beta factor (investment type). The greater profitability is achieved when you have an expert investor with moderate risk in the investment exchange. In this case the relationship is not directly proportional.
• In situations of Fluctuating Markets of Great Return for Investments, even if you have an Expert Investor, for some time and generally in the short term the profitability (Return on Investment) can be very high; but because it is a Stock Exchange or a High Risk Investment Fund (work table for example) the risk of losing all investment is also very high, so it is recommended to make differentiated investments for different financial products.

 regards

Jose Luis

Estimado Iqbal Thonse Hawaldar 

Desde el punto de vista teórico y luego de algunas experiencias personales y buscando dicho comportamiento a nivel internacional, puedo comentarle lo siguiente:

• Es cuestionable esta relación directa donde a mayor riesgo se tiene mayor retorno de alguna inversión.
• Veamos el ejemplo del equilibrio entre los factores alfa (inversor, director de inversiones) y factor beta (tipo de inversión). La mayor rentabilidad se logra cuando se tiene un experto inversor con riesgo moderado en la bolsa de inversión. En este caso la relación no es directamente proporcional.
• En situaciones de Mercados Fluctuantes de Gran Rentabilidad para las Inversiones, aun se tenga un Inversor experto, durante algún tiempo y generalmente a corto plazo la rentabilidad (retorno de la Inversión) puede ser muy alta; pero por ser Bolsa o Fondos de Inversión de Alto Riesgo (mesa de trabajo por ejemplo) el riesgo de perder toda la inversión también es muy alta, por lo cual se recomienda hacer inversiones diferenciadas para diferentes productos financieros.

Saludos

José Luis

Yes. It depend on type of the market and securities. 

Its a polynomial approximation with extremal points. Not all risk increases return, but nearly all return involves some sort of risk. Great return can actually create new risk.

So the answer is a sound no, except in special cases and small intervals.

There is a clear (if not linear) relationship between risk and returns.

The first thing we need to know about risk and reward is that under certain limited circumstances, taking more risk is associated with a higher expected return.

The second thing we need to understand about the relationship between risk and reward is that there in many cases there is no relationship.

http://www.investorsfriend.com/risk-and-reward/

https://www.tradejini.com/blog/?p=263

Positive relationship between risk and return, does not necessarily be a linear relationship....

Dear Iqbal Thonse Hawaldar

I am glad that the discussion network is current in relation to this issue of the risk vs. return relationship. I agree with some of the comments and clarifications as coordinator. I confirm with the same arguments or opinion issued about a month ago

Dear all,

When we introduce a proxy to capture investor's emotions, it is possible to show that a return-risk relationship does exist but that seems to be time-varying and nonlinear.

Since t

FORMAT TO DEFINE CONDITIONS TO INVEST IN PROJECT IN ACCORDANCE WITH THE SENSITIVITY ANALYSIS

OCURRENCE PROBABILITY OF THE VARIATIÓN

LOW

HIGH

 S

 E

 N

 S

LOW

YES

NO

 I

 T

 I

 V

MEDIUM

DEPENDING ON RETURN

ONLY WITH HIGH RETURN

 I

 T

 Y

 

HIGH

NO

(ONLY WITH VERY HIGH RETURN)

NO

                   YES  = SUGEST TO GO AHEAD WITH PROJECT

NO= SUGEST NOT TO GO AHEAD WITH PROJECT