# Rate of return

## 1. Discrete returns

Let $P_t$ be the asset price at time $t$ and $P_{t-1}$ be the price in the prior period - day, week, month, ... For daily returns on stocks, $P_{t-1}$ is the closing price of the stock on the previous trading day.

Example calculations are provided in figure 1 - Excel Web App #1 - Sheet1, where a sample of stock price data is used.

### 1.1 Net return

• Also called "simple returns"
• Measured by the change in price $\Delta P$, divided by the price at the start of the period
• $R_t = \left(P_t - P_{t-1} \right) / P_{t-1} = \left( P_t / P_{t-1} \right) - 1$
• With dividends: $R_{d,t} = \left( P_t - P_{t-1} + D_t \right) / P_{t-1}$
• Rows 12 to 15 of figure 1

### 1.2 Gross return

• $R_{g,t} = P_t / P_{t-1} = R_t + 1$
• With dividends: $R_{g,d,t} = \left( P_t + D_t \right) / P_{t-1}$
• Rows 17 to 20 of figure 1

## 2. Log returns

• Uses continuously compounded returns
• Often written as log $r_t = \log P_t / P_{t-1}$, but uses natural logs. The LN function in Excel, and the LOG function in VBA
• Natural logs have base 2.71828. This is referred to as Euler's number, denoted by $e$
• Log return: $r_t = \ln P_t / P_{t-1} = \ln P_t-\ln P_{t-1}$
• $\ln P_t$ is called the "log price"
• With dividends: $r_{d,t} = \ln \left(P_t + D_t \right) / P_{t-1} = \ln (P_t + D_t) -\ln P_{t-1}$
• Rows 24 to 28 of figure 1

• Converting net returns to log returns: $r_t = \ln(1 + R_t)$
• Converting log returns to net returns: $R_t = e^{r_t} - 1$
• Rows 31 to 34 of figure 1

## 3. Returns method selector panel

• Cell D39 contains a Data Validation selector
• Validation criteria - Allow: List
• Validation criteria - Source: discrete, log
• Cell formula - range D43:D54 - =IF(Return_method="discrete",(RC[-1]-RC[-1])/RC[-1],LN(RC[-1])-LN(RC[-1]))

## 4. Cumulative returns

### 4.1 Cumulative discrete returns

• Denote $R_{c,t}$ as the cumulative discrete return as time $t$
• Cumulative discrete returns are multiplicative
• In vector form for the 12 month period: $\left(1+R_{c,t-12-1}\right) \times \left(1+R_{t-12}\right)-1, \left(1+R_{c,t-12}\right) \times \left(1+R_{t-11}\right)-1, \left(1+R_{c,t-11}\right) \times \left(1+R_{t-10}\right)-1, ...$
• $R_{c,t}$ for 12 months at 1 July 15 = 0.0072 (Cell E62). This is the same value as the Annual Discrete return for the single 12 month period - Cell E76: 0.0072

### 4.2 Cumulative log returns

• Denote $r_{c,t}$ as the cumulative log return as time $t$
• Cumulative log returns are additive
• In vector form for the 12 month period $\left(r_{c,t-12-1} + r_{t-12}\right), \left(r_{c,t-12} + r_{t-11} \right), \left(r_{c,t-11} + r_{t-10} \right), ...$
• $r_{c,t}$ for 12 months at 1 July 15 = 0.0072 (Cell E81). This is the same value as the Annual Log return for the single 12 month period - Cell E95: 0.0072

• This example was developed in Excel 2013 Pro 64 bit.

• Published: 10 August 2015
• Revised: Monday 10th of February 2020 - 12:21 PM, [Australian Eastern Standard Time (EST)]