Asset returns

Let \(p_t\) be the price of an asset at time t, and assume no dividend. One-period simple return or simple net return.

\[R_t = \frac{P_t}{P_t-1} - 1 = \frac{P_t - P_{t-1}}{P_{t-1}} \]

Gross return

\[ 1 + P_t = \frac{P_t}{P_{t - 1}}\ or P_t = P_{t-1}(1+P_t) \]

Multi-period simple return or the k-period simple net return

\[R_t(k) = \frac{P_t}{P_{t-k}}-1\]

Gross return

\[1 + R_t(k) = \sum^{k -1 }_{j=0}(1+R_{t-j}) \]

Continues Compounding

from math import exp

def pay_interest(base, interest, payments):
    return base * ( base + interest / payments) ** payments


def pay_interest_continue(base, interest, number_of_years):
    return base * exp( interest * number_of_years)

pay_interest_continue(1, 0.1, 1)

1.1051709180756477

\[ R_t = log \]

Log return

\[r_{pt} = \sum_{i = 1}{n}w_ir_{it}\]

Excess return

\[Z_t = R_t - R_{0t}, z_t = r_t - r_{0t}\]

where \(r_{0t}\) denotes the log return of a reference asset (e.g. risk-free interest rate) such as shortterm U.S. Treasury bill return, etc..