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