Chapter 3 Return rates
3.1 Return rates – definitions
3.1.1 Total return
Total return, or gross rate of return, is the ratio of the revenue generated to the expenses incurred. In this script we denote it with a capital “caligraphic” letter \(\boldsymbol{\mathcal{R}}\).
\[\begin{equation} \boldsymbol{\mathcal{R}} = \frac{X_1}{X_0} \tag{3.1} \end{equation}\]
In the case of stock investments, the total return for the period \(t\) is determined based on the share price at the beginning of the period \(P_{t-1}\) and the price \(P_t\) at the end of the period as well as the dividend \(D_t\) paid at the end of the period:
\[\begin{equation} \boldsymbol{\mathcal{R}}_t = \frac{P_t+D_t}{P_{t-1}} \tag{3.2} \end{equation}\]
To some extent the return rate defined above is purely theoretical, as:
- it can be calculated even if we did not make a transaction (e.g. we did not sell a purchased share),
- it does not take into account transaction costs (e.g. commission for buying and selling shares),
- the dividend is paid on a date later than the date it was granted.
The total return for the last \(k\) periods is the product of single-period total returns (periods \(t-k+1\) to \(t\)):
\[\begin{equation} \boldsymbol{\mathcal{R}}_t(k) = \boldsymbol{\mathcal{R}}_t\cdot\boldsymbol{\mathcal{R}}_{t-1}\cdot\cdot\cdot\boldsymbol{\mathcal{R}}_{t-k+1} \tag{3.3} \end{equation}\]
If there are no dividends in the period under consideration, such a definition of the multi-period rate of return has an easy justification:
\[\begin{equation} \boldsymbol{\mathcal{R}}_t(k) = \boldsymbol{\mathcal{R}}_t\cdot\boldsymbol{\mathcal{R}}_{t-1}\cdot\cdot\cdot\boldsymbol{\mathcal{R}}_{t-k+1} = \left(\frac{P_t}{P_{t-1}}\right)\cdot\left(\frac{P_{t-1}}{P_{t-2}}\right)\cdot\cdot\cdot\left(\frac{P_{t-k+1}}{P_{t-k}}\right)=\frac{P_t}{P_{t-k}}, \tag{3.4} \end{equation}\]
If, however, dividends appear during the period, this formula assumes that they are invested immediately:
\[\begin{equation} \begin{split} \boldsymbol{\mathcal{R}}_t(k) = \boldsymbol{\mathcal{R}}_t\cdot\boldsymbol{\mathcal{R}}_{t-1}\cdot\cdot\cdot\boldsymbol{\mathcal{R}}_{t-k+1} = \\ \left(\frac{P_t+D_t}{P_{t-1}}\right)\cdot\left(\frac{P_{t-1}+D_{t-1}}{P_{t-2}}\right)\cdot\cdot\cdot\left(\frac{P_{t-k+1}+D_{t-k+1}}{P_{t-k}}\right). \end{split} \tag{3.5} \end{equation}\]
3.1.2 Rate of return
The (simple net) rate of return is the ratio of the income generated to the expenses incurred:
\[\begin{equation} R = \frac{X_1-X_0}{X_0} \tag{3.6} \end{equation}\]
For stock investments, analogously to equation (3.2):
\[\begin{equation} R_t = \frac{P_t+D_t-P_{t-1}}{P_{t-1}} \tag{3.7} \end{equation}\]
The connection between both rates of return is obvious:
\[\begin{equation} \boldsymbol{\mathcal{R}} = 1+R \tag{3.8} \end{equation}\]
The multi-period simple net rate of return typically assumes that dividends are immediately reinvested in the stock:
\[\begin{equation} R_t(k) = \boldsymbol{\mathcal{R}}_t(k) - 1 = \left(\frac{P_t+D_t}{P_{t-1}}\right)\cdot\left(\frac{P_{t-1}+D_{t-1}}{P_{t-2}}\right)\cdot\cdot\cdot\left(\frac{P_{t-k+1}+D_{t-k+1}}{P_{t-k}}\right) - 1 \tag{3.9} \end{equation}\]
In everyday language, as well as in the professional language of finance, the fact that we are talking about the gross or net rate of return can be inferred from the context. For example, “the investment earned a threefold return” means that the gross rate of return \(\boldsymbol{\mathcal{R}}\) = 3 (the net rate \(R\) = 2), while “we had a 20-percent return on our investment” means that the simple net rate\(R\) 0.2 (the gross rate \(\boldsymbol{\mathcal{R}}\) = 1.2).
3.1.3 Log returns
The log return, also known as the continuous compounding rate of return, denoted in this script by \(r\) can be defined as follows:
\[\begin{equation} r_t = \ln(\boldsymbol{\mathcal{R}}_t) = \ln(1+R_t) \tag{3.10} \end{equation}\]
With no dividends:
\[\begin{equation} r_t = \ln\left(\frac{P_t}{P_{t-1}}\right)=\ln P_t-\ln P_{t-1} \tag{3.11} \end{equation}\]
If the rates of return are relatively low, for example in the range (-0,2;0,2), then the values of the log rate of return and the simple net rate of return are similar.
The advantage of log rates is the simple relationship between single-period and multi-period rates of return:
\[\begin{equation} \begin{split} r_t(k) & = \ln\left(\boldsymbol{\mathcal{R}}_t(k)\right) = \ln(\boldsymbol{\mathcal{R_t}}\cdot\boldsymbol{\mathcal{R}}_{t-1}\cdot\cdot\cdot\boldsymbol{\mathcal{R}}_{t-k+1})=\\ &= \ln(\boldsymbol{\mathcal{R}}_t) + ln(\boldsymbol{\mathcal{R}}_{t-1}) + \cdot\cdot\cdot + \ln(\boldsymbol{\mathcal{R}}_{t-k+1})=\\ & =r_t + r_{t-1} + \cdot\cdot\cdot + r_{t-k+1} \end{split} \tag{3.12} \end{equation}\]
If we have log rates, we can obtain simple rates:
\[\begin{equation} \boldsymbol{\mathcal{R}} = e^r \\ R = e^r-1 \tag{3.13} \end{equation}\]
3.2 Margin trading
The initial margin is defined as the ratio of initial cash on the deposit account to the purchase value of the securities. For example, in the US, the Federal Reserve imposes a minimum of 50%. The initial margin percentage can be set higher by brokers.
The (required) maintenance margin is the minimum required ratio of cash on the deposit (“margin” account) and current market value of securities held. Margin serves as a buffer to protect brokers from losses. When the actual margin falls below the (required) maintenance level, the investor receives a margin call asking her to deposit additional cash or liquidate some of the securities. If this fails, the broker is authorized to restore compliance with the margin requirements through the latter method.
The leverage implied by margining (also known as gearing, or debt-financing of trades) increases volatility and risk, i.e. both the potential for the upside changes of the portfolio value (profits) and the downside changes (losses).
3.3 Short sales
In a typical investment (long positions, holding stocks), the investor would like to buy low and sell high – that is, make money on rising prices. In the case of short-selling, one borrows a security (takes out debt expressed in security units, like number of stocks) hoping to sell high and then buy it back low, thereby making money on falling prices. This can be viewed as borrowing money with the rate of interest equivalent to the rate of return on the security – the investor taking out a short position would like this rate of return to be as low as possible, preferably negative.
Securities are borrowed from a broker. Those taking a short position (short sellers) can close the position at any time by buying back the security (and covering fees and dividends).
In the case of short selling, maintaining appropriate deposits (“margins”) is also required.
3.4 Exercises
Exercise 3.1 In the biblical parable of the talents, three servants were given: the first – five talents, the second – two talents, the third – one talent. The first two doubled their possessions, the third – hid and brought back one talent. What were the gross and net rates of return they achieved? What was the total return of the owner of the property? What would have been the gross and net rates of return of the third servant (and property owner) if he had given the money at interest (20% for the entire period) to bankers during the owner’s absence?
Exercise 3.2 Show on the graph that the log rate of return is close to the simple net rate of return for values of the latter between -0.2 and 0.2.
Exercise 3.3 The price at time \(t_0\) is \(P_0=100\). What is the price \(P_2\), if (assuming no dividends):
\(R_1\) = 0.1 (10%), \(R_2\) = -0.1 (-10%)
\(r_1\) = 0.1, \(r_2\) = -0.1
Exercise 3.4 (Ruppert and Matteson 2015) Suppose the price of a stock at times 1, 2, and 3 are \(P_1\)=95, \(P_2\)=103 and \(P_3\) = 98. Find \(r_3(2)\).
Exercise 3.5 (Ruppert and Matteson 2015) The prices and dividends of a stock are given in the table below.
What is \(R_2\)?
What is \(R_4(3)\)?
What is \(r_3\)?
| \(t\) | \(P_t\) | \(D_t\) |
|---|---|---|
| 1 | 52 | 0.20 |
| 2 | 54 | 0.20 |
| 3 | 53 | 0.20 |
| 4 | 59 | 0.25 |
Exercise 3.6 Calculate log rate \(r_1\), (simple net) rate of return \(R_1\) and total return \(\boldsymbol{\mathcal{R}}_1\), if:
\(R_1\) = 0.1
\(R_1\) = -20%
\(r_1\) = 0.1
\(r_1 = 0.9\)
\(\boldsymbol{\mathcal{R}}_1=0.9\)
\(\boldsymbol{\mathcal{R}}_1=0.99\)
Exercise 3.7 (Linton 2019) Suppose initial margin requirement 50%, maintenance margin 25%. Investor buys 100 shares at the price of $100, i.e. pays $10,000 in total, borrowing $4,000. Own capital/equity is $6,000 and the initial margin is 60%.
What is the actual initial margin? Is it above the required level?
If the stock price fell to $70, what would the equity in the investment be? What would the actual maintenance margin percentage be in that event? Would it be above the required level?
At what price would a “margin call” occur?
At what price would the investment’s equity be negative (the broker would suffer a loss if actions were not taken)?
The interest rate on the loan is 5%, the price at the time of sale is 20% higher than the initial one. What is the rate of return on investment?
If the sale price is 20% lower than the initial price, what is the rate of return on investment?
Exercise 3.8 (Linton 2019) Investor borrows 100 shares from a broker who locates the securities, i.e. borrows from other clients or outside institutional investors. She sells the shares short at the current price $100 so that $10,000 is credited to her account. To satisfy the initial margin requirement of 50%, she must additionally deposit cash or securities (e.g. T-bills) worth at least $5,000 (5,000/10,000 = 50%) so that in total $15,000 are on the account.
What will be the investor’s profit, excluding fees, if the price per share is $75 when the position closes?
What will the rate of return be?
At what share price will a margin call occur if the required maintenance margin is 30%?