# 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.