# Theory

Keywords: portfolio management, risk, risk aversion, utility function, certainty equivalent, risk-free return, yield on stock, buy-and-hold strategy, Black-Scholes model, lognormal distribution, most probable return, mode of a probability distribution

# Application

Identification of what proportion of stock to hold based on your expectation of return and risk disposition

Keywords: S&P 500 market, market index, fixed-term deposit, AAA bond, savings book, investment period, yield on stock, buy-and-hold strategy, risk-free return, volatility, loss probability, total loss

Suppose you want to achieve a certain return in a fixed time period on a portfolio that contains two different assets: a “market” set of stocks and a “risk-free” investment. Our methods allow an estimate of what return you would receive for various stock proportions and what the loss probability would be for the portfolio as a whole (stocks + risk-free assets).

The basic assumption is that you buy the “market” (with ETFs, index certificates or a mixture of stocks that represents the market) and that you keep the percentage of stock approximately constant over the defined time period. This means that you have to adjust the stock proportion from time to time: when the stock market goes up substantially, the proportion of stock increases as well and you must sell part of your stock in order to keep the ratio of stock to risk-free assets the same. Similarly you have to buy replacement stock, if the the market goes down, provided you have not exited the market altogether (as described in the section on market signals).

First we show the total return for various stock ratios as a function of the loss probability, assuming a buy-and-hold strategy and ignoring any market signals.

Second we show how you can improve your return on investment and, at the same time, decrease your risk by avoiding long term bear markets with the help of market signals.

The mathematical basis of this method of portfolio management is to model the stock market using a Gaussian distribution for the logarithmic price changes, as is done in the Black-Scholes model. Depending on the strategy – buy-and-hold or using market signals – different moments of the Gaussian distribution are used. For the following scenarios, we have applied the statistical characteristics of the US S&P 500 market, which is of special importance as it is a lead market for all other world markets. We use the historical S&P 500 data on a weekly basis, as one can show that there is no improvement in the results using data on a daily basis.

Portfolio Management without Market Signals

Portfolio Management with Market Signals

Glossary