# 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]-R[1]C[-1])/R[1]C[-1],LN(RC[-1])-LN(R[1]C[-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:**Saturday 25th of February 2023 - 10:13 AM, [Australian Eastern Standard Time (EST)]