modern portfolio Theory modern portfolio theories purpose is to maximize expected return by selecting the weights of the various assets.

MPT was pioneered by Harry Markowitz in 1952, but is still very relevant today despite new theories. It is one of the most important and influential economic theories dealing with investment.

It only uses two factors return and risk building portfolios is about balancing these factors and optimal portfolio will achieve the best possible risk, return trade-off different types of investors will position themselves along the risk return trade-off depending primarily on their risk appetite MPT takes into account many assumptions that are not always realistic or correct in the real world, but it still stands as a basis for portfolio asset allocation while the theory has been challenged mainly due to the various assumptions. It remains one of the best known in widely used the key MPT assumptions are one returns are distributed along a normal bell curve.

2 correlations do not change over time.

Three investors will make rational investment decisions and will avoid unnecessary risk 4 asset volatility is known and constant and five. There are no taxes or transaction costs.

As we've discussed risk versus return forms a key feature of the theory investors should seek the highest return for an acceptable level of risk mean variance optimization is an analysis process in which investors allocate assets based on the trade-off between risk and reward.

Efficient Frontiers can be used to evaluate portfolio efficiency by plotting them on a graph like the one shown where the top Edge is known as the efficient Frontier and identifies the most optimal combination of assets. Let's explore these in more detail.

First all investors are risk-averse. It's one of the strongest assumptions in MPT, even though it's not present in the real world an investor will take on increased risk only if compensated by higher expected returns conversely an investor who wants higher expected returns must accept more risk different investors will evaluate the trade-off differently based on individual risk aversion characteristics or risk tolerance. For example retirees. Typically choose lower risk Roots compared to younger professionals.

In this example, we see three portfolios with differing risk return values. MPT assumes all investors will choose portfolio B over portfolio a why because it can obtain a higher level of return for the same level of risk and given npt's reasonable risk of versus assumption. All investors will avoid the unnecessary risk, but what about portfolio C. It's not as straightforward differed investors will evaluate the trade-off between portfolio sees higher risk and higher return against portfolio of bees lower risk and lower return depending on their risk tolerance regardless of their risk tolerance investors will never choose portfolio a in this scenario.

Mvo provides us with a framework for determining how much weight to allocate to each asset in a portfolio either to maximize the expected Return of the portfolio for an expected level of risk or minimize the expected risk given an expected return.

In this sense mvo is a risk budgeting tool that helps investors to spend their risk budget wisely. We emphasize the word expected because the inputs to mean variance optimization are forward-looking estimates and the resulting portfolios reflect the quality of the inputs.

In this example, we have a target for portfolio risk of 21 percent and a universe of three assets three combinations with each security weighed differently are shown to achieve 21% risk, the mvo process helps us identify the optimal weights for each asset in a portfolio optimal ratings are driven by the individual risk in Return of each asset in the correlation with other assets here. We see weight option two offers the best return of 12.2% for the Target risk.

In this example, we look at it the other way around with the target portfolio return of 7% instead of a Target risk, three combinations of security weights achieve the target return again, the mvo process helps us identify the optimal weights for each asset and option two is the best as it offers the lowest risk. Keep in mind that in both examples, we can articulate the portfolio's expected return with just the given data, however, portfolio risk requires additional data on how the assets are correlated, which we didn't include here for Simplicity.