# Theory

Keywords: portfolio management, risk, risk aversion, utility function, certainty equivalent, risk-freerisk-free
The use of the word risk-free here refers only to the lack of volatility, not to the lack of risk of default or bankruptcy. For example, risk-free investments are savings accounts, term deposits, or AAA state bonds  which “guarantee” a fixed return.
Risk is the product of the probability of a negative event times its cost. As an example, consider a bet: you may gain an amount G with probability pG and you may lose an amount L with probability pL=1-pG. The risk R to lose L is given by: R = pL∙L .
A long-term investment strategy in which stock is bought and held without responding to market fluctuations. It is based on the assumption that over very long periods of time (10 years and more) stock markets deliver a positive average annual return that usually lies above the risk-free return of fixed deposits or bonds.
, Black-Scholes modelBlack-Scholes model
The Black-Scholes model is a mathematical model used to evaluate options. It was first published by Fischer Black and Myron Samuel Scholes in 1973. For the calculation of the fair value of an option, Black and Scholes assumed that the logarithms of the price changes in the underlying asset (e.g., the S&P 500 index would underlie a S&P 500 option) are independent random variables that satisfy a Gaussian or normal distribution.
, lognormal distribution, most probable returnmost probable return
For any investment other than risk-free assets (like savings accounts, fixed deposit accounts, or AAA state bonds) any estimates of future return are uncertain. Therefore the return should be seen as a random variable with a probability distribution. The most probable return is the return where the probability distribution has its maximum (the mode).
, modemode
The maximum of a distribution function. A probability distribution is characterized by several parameters (also called moments). The most important ones are the expectation value (or mean value) and the standard deviation (which measures the statistical spread about the mean). A Normal distribution (or Gaussian distribution) is given entirely by these two parameters. For distribution functions that are not symmetric about the mean value, there are two additional location parameters: the mode  and the median. The mode of a distribution is the value with the highest probability of occurrence. The median cuts the distribution in two halves. For a Normal distribution, the mean value is equal to the mode and also to the median.
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 500S&P 500
The S&P 500 index (Standard & Poor’s 500) is a stock index that covers the 500 largest US companies listed on the stock exchange. The weighting of the companies depends on their market capitalization. The rating agency Standard & Poor‘s decides which companies to include in the index. The S&P 500 index is considered an indicator for the changes of the total US stock market and is the more modern index when compared with the Dow Jones Industrial Average index. It represents about 75% of the US stock market capitalization and is one of the most highly observed stock market indices of the world. The classic S&P 500 index is a price index, while the S&P 500 Total Return Index is the associated performance index.
market, market index, fixed-term deposit, AAA bond, savings book, investment period, yield on stock, buy-and-hold strategybuy-and-hold strategy
A long-term investment strategy in which stock is bought and held without responding to market fluctuations. It is based on the assumption that over very long periods of time (10 years and more) stock markets deliver a positive average annual return that usually lies above the risk-free return of fixed deposits or bonds.
, risk-freerisk-free
The use of the word risk-free here refers only to the lack of volatility, not to the lack of risk of default or bankruptcy. For example, risk-free investments are savings accounts, term deposits, or AAA state bonds  which “guarantee” a fixed return.
Risk is the product of the probability of a negative event times its cost. As an example, consider a bet: you may gain an amount G with probability pG and you may lose an amount L with probability pL=1-pG. The risk R to lose L is given by: R = pL∙L .
return, volatilityvolatility
A synonym for standard deviation. In general, volatility is a measure of the variations of a time series around its mean value. For commercial paper, volatility is the variation of the price of the paper around its mean (the return) within a defined time period, usually one year. By means of the volatility, it is possible to estimate the gain or loss potential of a stock or a certificate. The larger the volatility, the higher is the investment risk. While historical volatility is calculated with prices of the past, implicit volatility is determined by the expected variations in the future. The current market prices of options are the basis for the calculation of the implicit volatility.
, loss probabilityloss probability
The probability that at the end of the investment period the investment will be less than at the beginning.
In contrast to the concept “Value at Risk” (VaR) that banks usually use and leaves out the worst x[%] of all results, we include all possible negative returns. In our use of "loss probability", PL, we set VaR=0, because we integrate the return distribution function from -100% to 0:
PL = -100%VaR(x)=0 p(r) dr = x , where r is the return in % and p(r) is the probability to achieve return r.
, 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 probabilityloss probability
The probability that at the end of the investment period the investment will be less than at the beginning.
In contrast to the concept “Value at Risk” (VaR) that banks usually use and leaves out the worst x[%] of all results, we include all possible negative returns. In our use of "loss probability", PL, we set VaR=0, because we integrate the return distribution function from -100% to 0:
PL = -100%VaR(x)=0 p(r) dr = x , where r is the return in % and p(r) is the probability to achieve return r.
would be for the portfolio as a whole (stocks + risk-freerisk-free
The use of the word risk-free here refers only to the lack of volatility, not to the lack of risk of default or bankruptcy. For example, risk-free investments are savings accounts, term deposits, or AAA state bonds  which “guarantee” a fixed return.
Risk is the product of the probability of a negative event times its cost. As an example, consider a bet: you may gain an amount G with probability pG and you may lose an amount L with probability pL=1-pG. The risk R to lose L is given by: R = pL∙L .
assets).

The basic assumption is that you buy the “market” (with ETFs, index certificatescertificates
In the financial world, especially in German speaking countries, a certificate is a bond that is issued by banks and mostly sold to private investors. From a legal point of view certificates are debt obligations, therefore, in addition to the risk of value fluctuations, there is also the risk of the issuer’s insolvency. If the issuer becomes insolvent, the invested capital is lost completely, as happened with the bankruptcy of Lehman Brothers Inc. in September 2008. With certificates, private persons can also invest in items that are difficult to access, like raw materials, or they can simulate more complex portfolio strategies. There exists a zoo of certificates: index certificates and tracker certificates (in the US known as ETNs or Exchange Traded Notes), ETCs or Exchange Traded Commodities (certificates on raw materials), and basket certificates. In addition, there are reverse certificates, discount certificates, bonus certificates, capital protection certificates, strategy certificates, leverage certificates, to name just a few which allow the simulation of certain portfolio strategies. Certificates are particularly popular in German speaking countries, while in the USA, Canada, Great Britain and Japan they are not sold (except of ETNs in the US). However, in some cases they can be simulated by accredited investments, like the combination of a call option and its underlying asset.
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-freerisk-free
The use of the word risk-free here refers only to the lack of volatility, not to the lack of risk of default or bankruptcy. For example, risk-free investments are savings accounts, term deposits, or AAA state bonds  which “guarantee” a fixed return.
Risk is the product of the probability of a negative event times its cost. As an example, consider a bet: you may gain an amount G with probability pG and you may lose an amount L with probability pL=1-pG. The risk R to lose L is given by: R = pL∙L .
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 probabilityloss probability
The probability that at the end of the investment period the investment will be less than at the beginning.
In contrast to the concept “Value at Risk” (VaR) that banks usually use and leaves out the worst x[%] of all results, we include all possible negative returns. In our use of "loss probability", PL, we set VaR=0, because we integrate the return distribution function from -100% to 0:
PL = -100%VaR(x)=0 p(r) dr = x , where r is the return in % and p(r) is the probability to achieve return r.
A long-term investment strategy in which stock is bought and held without responding to market fluctuations. It is based on the assumption that over very long periods of time (10 years and more) stock markets deliver a positive average annual return that usually lies above the risk-free return of fixed deposits or bonds.
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 modelBlack-Scholes model
The Black-Scholes model is a mathematical model used to evaluate options. It was first published by Fischer Black and Myron Samuel Scholes in 1973. For the calculation of the fair value of an option, Black and Scholes assumed that the logarithms of the price changes in the underlying asset (e.g., the S&P 500 index would underlie a S&P 500 option) are independent random variables that satisfy a Gaussian or normal distribution.
. 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 500S&P 500
The S&P 500 index (Standard & Poor’s 500) is a stock index that covers the 500 largest US companies listed on the stock exchange. The weighting of the companies depends on their market capitalization. The rating agency Standard & Poor‘s decides which companies to include in the index. The S&P 500 index is considered an indicator for the changes of the total US stock market and is the more modern index when compared with the Dow Jones Industrial Average index. It represents about 75% of the US stock market capitalization and is one of the most highly observed stock market indices of the world. The classic S&P 500 index is a price index, while the S&P 500 Total Return Index is the associated performance index.
market, which is of special importance as it is a lead market for all other world markets. We use the historical S&P 500S&P 500
The S&P 500 index (Standard & Poor’s 500) is a stock index that covers the 500 largest US companies listed on the stock exchange. The weighting of the companies depends on their market capitalization. The rating agency Standard & Poor‘s decides which companies to include in the index. The S&P 500 index is considered an indicator for the changes of the total US stock market and is the more modern index when compared with the Dow Jones Industrial Average index. It represents about 75% of the US stock market capitalization and is one of the most highly observed stock market indices of the world. The classic S&P 500 index is a price index, while the S&P 500 Total Return Index is the associated performance index.
data on a weekly basis, as one can show that there is no improvement in the results using data on a daily basis.

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