Harry W. Margarita, the father of “Modern Portfolio theory’, developed the mean- variance analysis, which focuses on creating portfolios of assets that minimizes the variance of returns I. E. Risk, given a level of desired return, or maximizes the returns given a level of risk tolerance. This theory aids the process of portfolio construction by providing a quantitative take on it. It integrates the field of quantitative analysis with portfolio management. Mean variance analysis has found wide applications both inside and outside financial economics.

However it is based on certain assumptions which do not hold good in practice. Hence there have been certain revisions to it, so as to make it a more useful tool in portfolio management. Mean Variance Analysis. 1 Market assumptions The mean and variance analysis we do here uses many simplifying assumptions and approximations. A complete list would degenerate into philosophy, so here are Just a few. The matrix and the vector are known exactly. The investor can purchase any amount of any asset, either positive or negative.

Having a negative amount of a risky asset is short selling, or simply shorting. Having a negative amount of the risk free asset is called borrowing. There is no restriction that the number of shares owned should be an integer. 3 The investor is a price taker, which means that the investor may purchase any amount of the asset, and nothing the investor does will e etc the asset price. The price per share is independent of the amount purchased. The price is the same for long and short positions. This really is a combo- nation of previous points.

There are no transaction costs, which means that if an investor rest buys w worth of any asset then immediately sells it back, the net change of wealth of the investor is zero. If there were transaction costs, this round trip would have a non-zero net cost to the investor. This also is implicit in the above points. None of these is exactly true in actual markets. Market parameters are not known exactly. There are costs associated with borrowing beyond paying the risk free rate. There are limits to the investor’s ability to borrow, limits that have become more ever in the past year.

Even small investors are not pure price takers. For example, if the ask price for a given security is X, that means that someone has an outstanding o ere to sell shares of that security at the price X. In a typical situation, there is a relatively small number of shares available at the ask price, as few as 100 shares. If the investor wishes to purchase 200 shares, he or she will have to buy the second 100 at a slightly higher price. This is called moving the market. The total price per share for the 200 shares will be slightly more than X.

There usually is a gap between the lowest ask price and the highest bid price, the price at which there is an outstanding o ere to buy. The did Renee between these is the bid-ask spread. 4 If an investor would buy then immediately sell a share, he or she would lose the bid-ask spread, even if the market did not move (the bid and ask prices did not change) and there were no other transaction costs. An asset is liquid if it is easy to buy or sell. It is rare that an asset is totally illiquid { cannot be bought or sold at any price.

More commonly, the degree of liquidity of a arrest traded asset is measured by its bid-ask spread and how much transactions move the market. Very liquid assets are, for example, market index futures. Less liquid are individual corporate bonds. Unknown liquidity is a form of risk as important as unknown and . Published prices typically are only the best bid and ask prices. It may be hard to know in advance how many shares will be available at the bid and ask prices, or how much a given size of transaction will move the market. For example, suppose X(t) is the time varying price of a given asset.