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: \(\begin{align} r_t^{Real} & = ln(1 + R_t^{Real}) \\ & = ln(\frac{P_t}{P_{t-1}} \cdot \frac{CPI_{t-1}}{CPI_t}) \\ & = ln(\frac{P_t}{P_{t-1}}) + ln(\frac{CPI_{t-1}}{CPI_t}) \\ & = r_t - \pi_t^{cc} \end{align}\)