## Thursday, December 25, 2014

### Basic Investment Math - note #1

In this note, I describe some basic underlying mathematical principles that dictate investment strategies/ results which influence my portfolio construction and security selection.

One of, if not, the most important concepts in investing is the concept of compounding. A basic mathematical principal regarding compounding is the rule of 72, which is the approximate number of years an investment or portfolio will double at a given compound rate of return. For example, if an investor achieves an 8% annual rate of return his/her investment would double in value in approximately 9 years. An investor who achieves a 10% annual return would double his/her funds in just over 7 years, etc.

Real life in the markets is far messier than the math presented here as desired returns may or may not be available, may involve too much risk and may not be positive or smooth during an investment horizon. Besides overall macro and market conditions, portfolio construction itself will influence the future value of the portfolio over an investment horizon. At the risk of over generalization, for sake of discussion, I assume higher returns involve higher risk. The interplay between return and risk is a key concept. The basic math of risk and return suggests that steady portfolio returns with lower volatility mathematically tend to result in higher returns over time than portfolios designed for higher returns that have higher volatility bounded by the fact that portfolio construction involves attempting to achieve that the optimal combination of risk and reward given various parameters.

In other words, when comparing risk/reward characteristics over an investment horizon, portfolios losses are difficult to overcome. In the table below I illustrate four scenarios with each producing different returns at varying amounts of volatility. In each scenario I start with \$100 and stay invested for 5 years. In scenario 1 I achieve varying but positive returns in all 5 years. The 5 year compound average growth rate (CAGR) of 12.2% is the highest of the four scenarios. In scenarios 2 through 4 I assume negative returns in the models to show that the CAGR is lower than scenario 1 in each of the alternative scenarios. In scenario 2, I model only one slightly negative year (with the other 4 year returns unchanged) which results in a lower CAGR. In scenarios 3 and 4 I assume a series of higher returns combined with one and two losing years. Scenario 4 which includes two losing years produces the lowest return with the highest volatility. This is the worst case result of the four models.

Real life in the markets is far more complex than the math presented here. Outright losses are possible and common. Concepts that are important here include: (1) judging your ability as an investor to produce returns at a given level of risk; (2) the risk/return constraints/requirements of the portfolio in question; and (3) other holding period factors that come into play. Please see the table below for a summary of results. In real life numerous scenarios beyond these four exist.

In the table below PV is the present value of the portfolio, ARR is annual rate of return and CAGR is the annual compound average growth rate of the 5 year investment horizon. I have highlighted negative returns in red.

 Scenario PV Year 1 Year 2 Year 3 Year 4 Year 5 1 ARR 10% 20% 7% 5% 20% 100 110 132 141 148 178 CAGR 12.2% 2 ARR 10% 20% 7% -5% 20% 100 110 132 141 134 161 CAGR 10.0% 3 ARR 20% 20% -20% 20% 20% 100 120 144 115 138 166 CAGR 10.7% 4 ARR 20% -20% 20% -20% 20% 100 120 96 115 92 111 CAGR 2.2%