College teachers showed that diversification is free lunch for getting rid of part of the risk, but does this argument based on the assumption that expected return of each asset is always in proportion to its risk? The teacher illustrate this by using a portfolio where the one with higher return always have higher risk. But, I wondered, for example, if I have a asset A with expected return of 100% and risk(measured by standard deviation) 1%, and asset B with expected return of 1% and risk 100%, would it be rational to put asset B into the portfolio ?
Diversifying your portfolio between asset A and asset B only reduces the portfolio risk if asset A and asset B are not correlated. If they have either a low correlation or a negative correlation to each other, then you benefit from combining them in a portfolio in terms of risk reduction.
The standard deviation of returns will be lower in a portfolio of low or uncorrelated assets.
If on the other hand you combine two correlated assets into a portfolio you are doubling down on the same assumption, which means you are not reducing your risk. You are also wasting capital because now you have allocated capital to 2 separate trades / investments yet they have shown a high tendency of moving together.
Here is an article that discusses this further: Why Diversify your Stock Portfolio
if I have a asset A with expected return of 100% and risk(measured by standard deviation) 1%, and asset B with expected return of 1% and risk 100%, would it be rational to put asset B into the portfolio ?
No, because Modern Portfolio Theory would say that if there is another asset (B2) with the same (or higher) return but less risk (which you already have in asset A), you should invest in that. If those are the only two assets you can choose from, you would invest completely in Asset A.
The point of diversification is that, so long as two assets aren't perfectly positively correlated (meaning that if one moves up the other always moves up), then losses in one asset will sometimes be offset by gains in another, reducing the overall risk.
If you have 100% of your money in one security that is inherently more risky than splitting your money 50/50 between two securities, regardless of the purported riskiness of the two securities.
The calculations people use to justify their particular breed of diversification may carry some assumptions related risk/reward calculations. But these particular justifications don't change the fact that spreading your money across different assets protects your money from value variances of the individual assets. Splitting your $100 between Apple and Microsoft stock is probably less valuable (less well diversified) than splitting your money between Apple and Whole Foods stock but either way you're carrying less risk than putting all $100 in to Apple stock regardless of the assumed rates of return for any of these companies stock specifically.
Edit: I'm sure the downvotes are because I didn't make a big deal about correlation and measuring correlation and standard deviations of returns and detailed portfolio theory. Measuring efficacy and justifying your particular allocations (that generally uses data from the past to project the future) is all well and good. Fact of the matter is, if you have 100% of your money in stock that's more stock risk than 25% in cash, 25% in bonds and 50% in stock would be because now you're in different asset classes. You can measure to your hearts delight the effects of splitting your money between different specific companies, or different industries, or different market capitalizations, or different countries or different fund managers or different whatever-metrics and doing any of those things will reduce your exposure to those specific allocations.
It may be worth pointing out that currently the hot recommendation is a plain vanilla market tracking S&P 500 index fund (that just buys some of each of the 500 largest US companies without any consideration given to risk correlation) over standard deviation calculating actively managed funds. If you ask me that speaks volumes of the true efficacy of hyper analyzing the purported correlations of various securities.
If you are diversifying just for diversification purposes then all you are doing is averaging down your returns.
You shouldn't just buy two securities because you think it is safer than putting all your money into one.
A better method is to use money management and position sizing to limit your risk and exposure in any one security. You should know what your maximum risk is before you buy any security and know when it is time to get out of it. There are better ways to manage your risk.
Don't put all your eggs in the one basket - yes, but don't diversify just for diversification purposes.