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Basic Principles of Finance – Returns

Return

The most basic concept in finance is the return (gains or losses) expected from an asset.  Return is defined as:
$R_t=\frac{P_t}{P_{t-1}}-1=\frac{P_t-P_{t-1}}{P_{t-1}}$ or $1+R_t=\frac{P_t}{P_{t-1}}$

Present and Future Value

Derived directly from the previous definition, the Present Value PV of a future cash flow PF given the expected return of R is defined as:
$PV=\frac{PF}{1+R}$.
That is the current value of an asset is the expected value of that asset in the future, discounted by the expected rate of return.

Conversely, the future value of an asset is defined as:
$PF=PV*(1+R)$.

Multi-Period Return

The previous definition of return is for a single period between $t$ and $t-1$.  For multiple (k) periods, the return can be calculated as:
$1+R_t[k]=\frac{P_t}{P_{t-k}}=\frac{P_t}{P_{t-1}}*\frac{P_{t-1}}{P_{t-2}}*…*\frac{P_{t-k+1}}{P_{t-k}}=\prod_{j=0}^{k-1}(1+R_{t-j})$

The annualized rate of return is given by:

Annualized $R_t[k]=\Bigg[\prod_{j=0}^{k-1}(1+R_{t-j})\Bigg]^\frac{1}{k}-1=e^{\Big[\frac{1}{k}\sum_{j=0}^{k-1}(ln(1-R_{t-j})\Big]}-1$

The annualized returned is best calculated as a Taylor expansion:

Annualized $R_t[k]\approx\frac{1}{k}\sum_{j=0}^{k-1}R_{t-j}$

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