Basic Investment Models and Their Statistical Analysis

Introduction

Three cornerstones of quantitative finance are asset returns, interest rates, and volatilities. They appear in many fundamental formulas in finance. In this article, we consider their interplay and the underlying statistical issues in a classical topic in quantitative finance.

Asset Returns

One-period Net Returns and Gross Returns

Let denote the asset price at time . Suppose the asset does not have dividends over the period from time to time . Then the one-period net return on this asset is

which is the profit rate of holding the asset during the period. Another concept is the gross return , which is equal to .

Multiperiod Returns

One-period returns can be extended to the multiperiod case as follows. The gross return over periods is then defined as

and the net return over these periods is . In practice, we usually use years as the time unit. The annualized gross return for holding an asset over years is and the annualized net return is .

Continuously Compounded Return (Log Return)

Let . The logarithmic return or continuously compounded return on an asset is defined as

One property of log returns is that, as the time step of a period approaches 0, the log return is approximately equal to the net return:

Asset Prices and Returns

The mean and standard deviation (SD, also called volatility) of the annual log return are related to those of the monthly log return by

and

For daily returns, we consider only the number of trading days in the year (often taken to be 252). The convention above is for relating the annual mean return and its volatility to their monthly or daily counterparts. This convention is based on the i.i.d. assumption of daily returns.

Conclusion

Market data are actually much more complicated and voluminous than those summarized in the financial press. Transaction databases consist of historical prices, traded quantities, and bidask prices and sizes, transaction by transaction. These ‘high-frequency’ data provide information on the ‘market microstructure.’