Wealth Management Mathematical Tools & Techniques

Wealth management Mathematical Tools & Techniques

Normal Distribution and related measures

Holding Period Return

The holding return period formula was introduced previously when discussing time-weighted return measurement. The same formula applies when applied to frequency distributions

(Descriptions changed slightly).

Rt = [(Pt -Pt-1+Dt)/Pt-1]

Where:

Rt = holding period return for time period (t)

Pt = price of asset at end of time period t

Pt – 1 = price of asset at end of time period (t – 1)

Dt = cash distributions received during time t

Central Tendency

The term “measures of central tendency” refers to the various methods used to describe where large groups of data are centered in a population or a sample. Here it is stated another way: if one value or observation is pulled from a population or sample, what would be typically expected for the value to be? Various methods are used to calculate central tendency. The most frequently used is the arithmetic mean, or the sum of observations divided by the number of observations.

Weighted Mean

Weighted mean is frequently seen in portfolio problems in which various asset classes are weighted within the portfolio – for example, if stocks comprise 60% of a portfolio, then 0.6 is the weight. A weighted mean is computed by multiplying the mean of each weight by the weight, and then summing the products.

Where, maybe, stocks are weighted 60%, bonds 30% and cash 10%. Assume that the stock portion returned 10%, bonds returned 6% and cash returned 2%. The portfolio’s weighted mean return is.

Stocks (wtd) + Bonds (wtd) + Cash (wtd) = (0.6)*(0.1) + (0.3)*(0.06) + (0.1)*(0.02) = (0.06) + (0.018) + (0.002) = 8%

Range and Mean Absolute Deviation

The range is the simplest measure of dispersion, the extent to which the data varies from its measure of central tendency. Dispersion or variability emphasizes risk, or the chances that an investment will not achieve its expected outcome. If any investment has two dimensions – one describing risk, one describing reward – then one must measure and present both dimensions to gain an idea of the true nature of the investment. Mean return describes the expected reward, while the measures of dispersion describe the risk.