public:courses:finance:computational_finance:simple_returns

# Week1 - Simple Returns

• If we invest V dollars, for n years, with a simple interest rate of R, then the future value can be computed as: $FV_n = V \cdot (1+R)^n$
• Note that nowadays, the interest rate is usually around 1%.
• The present value can be computed as: $V = \frac{FV_n}{(1+R)^n}$
• The compound annual return is: $R = \left( \frac{FV_n}{V}\right)^\frac 1n - 1$
• The investment horizon is: $n = \frac{ln(\frac{FVn}{V})}{ln(1+R)}$

⇒ How long does it takes to double the investment V ? This doesn't depend on V, and we find: $n = \frac{ln(2)}{ln(1+R)}$. We have $ln(2) \approx 0.7$ and when R is very small $ln(1+R) \approx R$, so here we get $n \approx \frac{0.7}{R}$. With a typical interest rate of R = 0.01, we get the result n = 70 (years!) [This is known as the rule of the seventy]

• Compounding can occur m times per year (with m=2 or 12 or 365, etc…).
• In that case we define the periodic interest rate: $\frac Rm$. And the Future value is defined as: $FV_n^m = V \cdot \left( 1 + \frac Rm \right)^{m \cdot n}$
• We can also define continuous compounding, when m increases infinitely. This gives: $FV_n^\infty = \lim\limits_{m \to \infty} V \cdot \left( 1 + \frac Rm \right)^{m \cdot n} = V \cdot e^{R \cdot n}$ (note that $e \approx 2.71828$).
• The higher the value of m, the higher the future value $FV_n$ (marginally).
• The concept of multiple compound per year was originally introduced to get around some regulations (limitation on the interest rate that can be applied).
• What is the effective annual rate $R_A$ when we use multiple compounding per year ? $V \left( 1 + \frac Rm \right)^{m \cdot n} = V \cdot (1+R_A)^n \Rightarrow R_A = \left(1 + \frac Rm \right)^m -1$
• And if we use continuous compounding we get: $V \cdot e^{R \cdot n} = V \cdot (1+R_A)^n \Rightarrow R_A = e^R -1$
• We define $P_t$ as the price of an asset at the end of the month t.
• The holding period t is the time between when we by the asset, and when we sell it at price $P_t$.
• The holding period return is: $R_t = \frac {P_t - P_0}{P_0} = \%\Delta P$
• Note that the holding period is typically of t = 1 month. So over 1 month, we can define: $R_t = \frac{P_t - P_{t-1}}{P_{t-1}} = \frac {P_t}{P_{t-1}} - 1$
• The gross return is : $1+R_t = \frac{P_t}{P_{t-1}}$
• For a 2 month investment for instance we can define: $R_t(2) = \frac {P_t - P_{t-2}}{P_{t-2}} = \frac{P_t}{P_{t-2}} - 1$
• Then we can also write: $R_t(2) = \frac{P_t}{P_{t-1}} \cdot \frac{P_{t-1}}{P_{t-2}} - 1 = (1+R_t) \cdot (1+R_{t-1}) - 1$
• And the gross return relation is then: $1+R_t(2) = (1+R_t) \cdot (1+R_{t-1})$
• So $R_t(2) = R_t + R_{t-1} + R_t \cdot R_{t-1}$, which can give us: $R_t(2) = R_t + R_{t-1}$ if $R_t$ and $R_{t-1}$ are very small.
• In the general case of k month investment we get: $R_t(k) = (1+R_t) \cdot (1+R_{t-1}) \cdots (1+R_{t-k+1}) - 1$,
• Which can be written as: $1+R_t(k) = \prod\limits_{j=0}^{k-1} (1 + R_{t-j})$ (this is called a geometric average)
• A Portfolio is just a collection of asset that we want to invest into.
• For instance if we have a portfolio of 2 assets, we can invest V dollars in the assets A or B.
• We call the investment shares $x_A$ and $x_B$ the fraction of the money that is invested in A and B respectively.
• We assume here that we invest all of our money, so we have $x_A + x_B = 1$
• Then we can compute the rate of return $R_{P,t}$ of the portfolio itself:

\begin{align} V (1+R_{P,t}) & = V [ x_A (1+R_{A,t}) + x_B (1+R_{B,t})] \\ & = V [x_A + x_B + x_A \cdot R_{A,t} + x_B \cdot R_{B,t} ] \\ & = V [1 + x_A \cdot R_{A,t} + x_B \cdot R_{B,t}]\end{align}

• So we get $R_{P,t} = x_A \cdot R_{A,t} + x_B \cdot R_{B,t}$
• In the general case with a portfolio of n assets with investment shares $x_i$, we have the gross return: $1+R_{P,t} = \sum\limits_{i=1}^n x_i \cdot (1+R_{i,t})$ or the net return rate: $R_{P,t} = \sum\limits_{i=1}^n x_i \cdot R_{i,t}$
• We call $D_t$ the dividend payment between month t-1 and t, so that we get the total investment rate:

\begin{align} R_t^{total} & = \frac {P_t - P_{t-1} + D_t}{P_{t-1}} \\ & = \frac {P_t - P_{t-1}}{P_{t-1}} + \frac {D_t}{P_{t-1}} \\ & = \text{capital gain return} + \text{dividend yield} \end{align}

• We also have the gross return: $1+R_t^{total} = \frac{P_t + D_t}{P_{t-1}}$
• We try to compute the real rate of return as opposed to the nominal rate of return used so far.
• For that we can use the Consumer Price Index as provided every month by the bureau of labour and stats (in the states!):
• We can the adjust as: $P_t^{Real} = \frac{P_t}{CPI_t}$ and $R_t^{Real} = \frac{P_t}{P_{t-1}} \cdot \frac{CPI_{t-1}}{CPI_t} - 1$
• We define inflation as: $\pi_t = \frac{CPI_t - CPI_{t-1}}{CPI_{t-1}} = \% \Delta CPI$
• Then we can write: $R_t^{Real} = \frac{1+R_t}{1+\pi_t} - 1$
• Note that generally we have: $R_t^{Real} = R_t - \pi_t$
• We take the assumption here that the monthly return rate $R_m$ is always the same.
• In that case we can compute the compound annual gross return as: $1+R_A = 1 + R_t(12) = (1+R_m)^{12}$
• And the compound annual net return : $R_A = 1 + R_t(12) = (1+R_m)^12 - 1$
• This also work the other way, for instance if we have a 2 years investment horizon, then: $(1+R_A)^2 = 1 + R_t(24)$
• We define the continuously compounded return as: $r_t = ln(1+R_t) = ln(\frac {P_t}{P_{t-1}}) \Rightarrow R_t = e^{r_t} - 1$.
• Note that we always have $r_t \le R_t$.
• Common relations: $ln(0) = -\infty$, $ln(1) = 0$, $e^{-\infty} = 0$, $e^0 = 1$, $e^1 = 2.27828$, $\frac{d}{dx} ln(x) = \frac 1x$, $\frac{d}{dx} e^x = e^x$, $ln(x \cdot y) = ln(x) + ln(y)$, $ln(\frac xy) = ln(x) - ln(y)$, $ln(x^y) = y \cdot ln(x)$, $(e^x)^y = e^{x \cdot y}$,$e^x \cdot e^y = e^{x+y}$.
• $P_{t-1} \cdot e^r_t = P_t$
• With the First order Taylor serie development we know that: $f(x) = f(x_0) + \frac {d}{dx}f(x_0) \cdot (x - x0)$. This gives us for the ln function, if \R_t\) is small: $r_t = ln(1 + R_t) \approx R_t$.
• We define $p_t = ln(P_t)$. We are $r_t = ln(\frac{P_t}{P_{t-1}}) = ln(P_t) - ln(P_{t-1}) = p_t - p_{t-1}$
• $r_t(2) = ln(1+R_t(2)) = ln(\frac {P_t}{P_{t-2}}) = p_t - p_{t-2}$
• We also have $P_{t-2} \cdot e^{r_t(2)} = P_t$
• And $r_t(2) = ln(\frac{P_t}{P_{t-1}} \cdot \frac {P_{t-1}}{P_{t-2}}) = ln(\frac{P_t}{P_{t-1}}) + ln(\frac {P_{t-1}}{P_{t-2}}) = r_t + r_{t-1}$
• In the general case: $r_t(k) = \sum\limits_{j=0}^{k-1} r_{t-j}$
• We have $R_{P,t} = \sum\limits_{i=1}^n x_i \cdot R_{i,t}$
• So we have: $r_{p,t} = ln(1 + R_{p,t}) = ln (1 + \sum\limits_{i=1}^n x_i \cdot R_{i,t} \neq \sum\limits_{i=1}^n x_i \cdot r_{i,t} )$
• We define the cc inflation rate as: $\pi_t^{cc} = ln(\frac{CPI_t}{CPI_{t-1}})$
• Then we have: \begin{align} r_t^{Real} & = ln(1 + R_t^{Real}) \\ & = ln(\frac{P_t}{P_{t-1}} \cdot \frac{CPI_{t-1}}{CPI_t}) \\ & = ln(\frac{P_t}{P_{t-1}}) + ln(\frac{CPI_{t-1}}{CPI_t}) \\ & = r_t - \pi_t^{cc} \end{align}
• public/courses/finance/computational_finance/simple_returns.txt