Investing is based more on common sense than science. I am inspired by one of the smartest investors the world has ever seen: Warren Buffet. In his annual letter to his shareholders, you can read and learn about his investing philosophy. Soon you find out that his secret is based on attitude, common sense and simple arithmetic. The attitude you need for a successful long term investor you must find out yourself by reading his letters. I will tell here something more about the mathematics of investing.

It is clear that there is only one primary objective for an investor: *return*. Investing is converting cash into an asset, wait some time, and finally convert it back to cash for consumption. The investor expects that this waiting pays off. For each investment decision there are two important parameters to deal with: the expected *return* of the asset during the investment period and the amount of *risk* associated with the investment.

The value of an asset depends on the attractiveness of the asset for an investor. As we have seen this attractiveness depends on return and risk. The investors objective is to maximize return under the condition of an accepted level of risk. To determine the value of an asset an investor has to guess these two parameters: return and risk.

Normally assets are traded on active markets and there are bid and offer prices for these assets. These valuations imply that market participants have already assigned an expected return and risk to these assets. An investor only needs to judge whether these parameters are in line with his personal guess about the future. Our first step is therefore to understand how we can calculate the value of an asset, so that we are able to compare our calculations with market prices. As valuation is determined by two concepts: return and risk. We need to analyze these concepts in more dept.

There are two important principles in valuation:

*Time value*: The value of a future cash flow decreases exponential with time. $X$ dollar of purchasing power at $t=0$ could be exchanged against $X(1+i)^n$ dollar at $t=n$. This follows immediately from the primary objective of an investor to make return in exchange of delayed consumption of his purchasing power. So we find a time relationship: \[ X_0=\frac{1}{(1+i)^n}X_n=dX_n \] The rate $d$, which is less than 1 if $i>0$, is called a discount rate, and relates an amount of money in the future to the present.We can argue that in an economy where there are a lot of attractive investment opportunities competing with each other for cash from investors, investors will demand a higher return $i$ and therefor a lower value of $d$ results. In an economy where inflation is deteriorating future purchasing power investors will also demand a higher compensation due to inflation. So in general $d$ is related to economic growth and inflation prospects.

The chart below shows the time-value of money, as indicated by the yield on US-treasuries, over time in the US. We can see that from the lows of the 1940 it steadily rises to a top in 1983. Then is steadily decreases to the lows of 2011. These fluctuations play an important role in the valuation of assets and are long term trends. Investors have to accept these realities as the investment climate in which they live. It is therefore that investors profit is more dictated by the general market conditions than his personal wisdom.

*No arbitrage:*If asset $A$ and asset $B$ have the same cash flow pattern and risk then they have the same value. Suppose asset $A$ promises $5\% p.a.$ and pays of $100$ after 5 years. Its discounted value would be: \[ 100 *\frac{1}{1.05^{10}}=100*0.61=61 \] but suppose that investors are accepting a lower return on an equivalent asset $B$ of a 4% return, then the value of assets $A$ will be set in the market by discounting it with the market accepted higher discount rate, and so the value of $A$ increases: \[ 100 *\frac{1}{1.04^{10}}=100*0.68=68 \]

The mechanics of finding the value of an asset consists of: estimate future cash flows, choose the appropriate discount rate $i$, discount each future cash-flow to its present value, and sum up all discounted cash flows to get its current valuation. The appropriate discount rate $i$ is called the *required return, $R$ *. The required return is a price which depends on the time value of money and the price of the specific risk for the investment. The time value as well as the price for risk is determined by the market on the basis of the no-arbitrage principle. The market allocates assets in buckets of risk and then valuate the expected cash-flows of these assets with the appropriate price.

So, we could like in all textbooks on investing, define the required return on an asset as: \[ R=(1+\text{real RFR} )(1+\text{Expected Rate of Inflation})+\text{ RP}-1=\text{nominal RFR } + \text{ RP} \] where RFR=Risk Free Rate and RP=Risk Premium.

Most investments have cash flows not only at the end of the investment period, but also in between. In that case the investment return does not match the discount rate. The reason is that the intermediate cash flows are reinvested at other reinvestment rates. So, the actual investment return can only be measured retrospectively.In general risk is the loss of value due to a change in the valuation of an asset after the investment decision. The change of a valuation is caused by a change in the expected cash-flows and/or the required return.

The time-value of money includes a compensation for expected inflation. However actual inflation during the investment period can be different from the compensation at the time of our investment. This *inflation risk* is the only risk component of government bonds. So even risk free investments have a risk.

Also a change in the real RFR is a risk, *interest risk*,because it influences the valuation of the investment. However for fixed income investments (investments for which all future cash-flows have been contractually specified a priori), this risk is only an opportunity loss or gain. Therefore we define a *risk free asset* as an asset with predefined guaranteed series of cash-flows, that also compensate the actual inflation rate. Government bonds with very short maturities are approximately risk free assets. Government inflation protected bonds are approximations of longer term risk free assets. However these investments have a higher degree of risk related to the fixed real return at investment date.

If for a fixed income investment the cash-flows are not guaranteed there is an additional risk, called *credit or default risk*. This is the risk that the issuer will not pay the promised cash-flows. The compensation for this risk is part of the risk premium of the required return. related to credit risk is *country risk*. When a country defaults on its obligations, this can harm the performance of all other financial instruments in that country as well as other countries it has relations with. Country risk applies to all investments issued within a particular country.

If the underlying cash-flows of an investment are in a foreign currency then an additional risk arises, *currency risk*. Normally there is no compensation for this risk in the required return. So, if the investor wants to avoid this risk it needs to hedge the currency.

For an investment in equities, there are several differences compared to fixed income. First the cash flows (dividends) are not contractually defined but depend on the realized earnings of the company. Secondly there is no maturity. The investor needs to sell equities when he wants to convert it to cash and so the value $V_n$ always depends on the fair value at $t=n$. It is clear that the risks of an equity investment are significantly higher than for a bond investment. The causes for this risk can be either specific to the investment, *unsystematic risk*, or related to the market of equity securities as a whole, *systematic risk*. The first type of risk can almost entirely be reduced by diversification. A portfolio with 100 investment titles has less unsystematic risk than a portfolio with 1 title. Because it can be largely reduced by diversification, an investor does not receive an additional compensation via the risk premium for this type of risk. Systematic risk of an asset is measured as the covariance of its returns with the return of the aggregate market. This risk is rewarded via the risk premium.

Analyzing the different sources of risk is fine, but realize that the highest risk factor is the investor himself. The financial markets in which investors operate, give them unlimited opportunities to ruin themselves and only a few to survive. The main route to survive is managing risk.

There are two options. Either the investor accepts the market prices as the best estimates of return/risk estimates for assets and chooses a passive investment style. But with hindsight we know that markets can achieve the status of so called irrational exuberance — wishful thinking on the part of investors that blinds us to the truth of our situation.

To build a deeper understanding of valuations we must first understand some cash flow discounting techniques.

The technical details of discounting cash flows are left to the appendix. The remarkable thing about discounting cash flows is that when we have a perpetual flow of constant or increasing cash flows the calculation reduces to very simple formulas. For a constant perpetual cash-flow with a required return $r$: \[ V_0=\frac{C}{r} \] For a perpetual growing cash flow at rate $g$ and required return $r$: \[ V_0=\frac{C}{r-g} \] Now this last formula is interesting for valuation of an equity investment. An equity investment has no maturity and pays normally a dividend $D$. So let us assume a perpetual stream of dividends. The dividend is paid out of earnings $e$, let $\tau$ be the percentage of earnings paid out. The retained earnings have the ability to grow future earnings, let $g$ be a constant growth rate. Then $D_0=e_0\tau$ and $C=e_0\tau(1+g)$ Then applying above formula gives following valuation for this equity investment: \[ p_0=\frac{e_0\tau(1+g)}{r-g} \] It follows that the return is determined by $g$. \[ \frac{p_n}{p_0}=(1+g)^n \Leftrightarrow i=g \] Of course in reality at $t=n$ the parameters of $g$,$r$ and $\tau$ are likely to be different from $t=0$ so in practice $i$ will differ from $g_0$. Compare this with an investment in fixed income where the return,which we call yield,is just the market discount rate $i=r$. The success of an investment in equities depends on the success of the underlying companies whereas the success of an investment in fixed income is determined by the market conditions at the time of investment. It is because of this fundamental difference that investors like Warren Buffet achieve so good returns during a long period of time under very different macro-economic condtions.

We can also use this model to determine an acceptable $p/e$ ratio for an equity investment: \[ \frac{p}{e}=\frac{\tau(1+g)}{r-g} \]

We will now turn back to the technics of discounting cash-flows.

If $V_0$ is the value of the asset at the start and $V_n$ is the value at the end, and no cash is paid in between then the return $r$ of the investment $V_0$ is the growth rate defined by *compounding*:
\[
V_n=\left(1+r\right)V_{n-1}=\left(1+r\right)^n{V_0}
\]
This equation also learns us that the price we have to pay for an asset, $V_0$, follows from *discounting*:
\[
V_0=\left(1+r\right)^{-n}V_n
\]
The price an investor is willing to pay for an asset depends on the expected value of $V_n$ and the required return $r$.

We solve for $r$ as follows:
\[
r=\left[{\frac{V_n}{V_0}}\right]^{1/n}-1
\]
This type of return is called a *geometric mean* of the simple period returns $1+r_i=V_i/V_{i-1}$:
\begin{align*}
r&= \left[ {\prod\limits_{i = 1}^{n} {(1+r_i) } }\right]^{1/n} - 1
\end{align*}

Note that the logarithm of the geometric mean return is equal to the arithmetic mean of the logarithm of the simple returns: \begin{align*} \ln(1+r)&=\frac{1}{n}\ln{\left(\prod\limits_{i = 1}^{n} \left(1 + r_i\right)\right)} \\ &=\frac{1}{n}\sum\limits_{i=1}^{n}\ln \left(1+r_i\right) \end{align*}

The growth of an investment as a function of time $t$ (note that t is a discrete variable) is an *accumulator* function:
\[
a(t)=(1+r)^t \text{ with } t=1,2,3,...
\]
In this function $t$ is limited to an integer value. What if we have multiple compoudings in one period, let say $m$ times compounding in one period. The change of $a(t)$ in any compounding period is:
\[
a(t)\tfrac{r}{m}
\]
so the total growth for $t+\tfrac{1}{m}$:
\begin{align*}
a(t+\tfrac{1}{m})&=a(t)+a(t)\tfrac{r}{m} \\
&=a(t)(1+\tfrac{r}{m})
\end{align*}
the accumulator function becomes:
\begin{align*}
a(t)=(1+\tfrac{r}{m})^{mt}, \;t=0,\tfrac{1}{m},\tfrac{2}{m},\cdots
\end{align*}
The value of $t$ is now a rational number. However what if we let $m$ go to infinity.
\[
a(t)=\lim\limits_{m\to\infty}\left(1+\tfrac{r}{m}\right)^{mt},\;t \in \mathbb{R}
\]
This is called *continuous compounding*. We analyse this limit:
\[
\left(1+\tfrac{r}{m}\right)^m=e^{m\ln(1+\tfrac{r}{m})}
\]
We use the Taylor expansion for $\ln(1+\tfrac{r}{m})$:
\[
f(x+\delta x) = f(x) + f^\prime(x)\delta x+\frac{1}{2}f^{\prime\prime}(x)\delta x^2 +\cdots
\]
We take $f(x+\delta x)=\ln(1+\tfrac{r}{m})$ and this becomes:
\begin{align*}
\ln(1+\tfrac{r}{m})&=\ln(1)+1.\tfrac{r}{m}+.... \\
&=\tfrac{r}{m}+...
\end{align*}
For very small $\tfrac{r}{m}$, the higher order terms of this Taylor polynomial vanish and we have the approximation:
\[
\ln\left(1+\tfrac{r}{m}\right)=\tfrac{r}{m}
\]
and thus:
\[
\lim\limits_{m\to\infty}\left(1+\tfrac{r}{m}\right)^m=e^r
\]
and finally:
\[
a(t)=e^{rt}, \;t \in \mathbb{R}
\]
Alternatively one could derive this:
\begin{align*}
a(t)&=\lim\limits_{m\to\infty}\left(1+\tfrac{r}{m}\right)^{mt} \\
&=\left[\lim\limits_{m\to\infty}\left(1+\tfrac{r}{m}\right)^{\large\frac{m}{r}}\right]^{rt} \\
&=\left[\lim\limits_{m\to\infty}\left(1+\tfrac{1}{n}\right)^n\right]^{rt} \\
&=e^{rt}, \;t \in \mathbb{R}
\end{align*}

The instantaneous rate of growth or just rate of groth of a function $f(t)$ is defined by: \[ \begin{equation} \delta(t)=\frac{f'(t)}{f(t)} \end{equation} \] This is equivalent to taking the natural log and then differentiate with respect to $t$: \[ \begin{equation} \frac{d}{dt}\ln{\left(f(t)\right)}=\frac{f'(t)}{f(t)} \end{equation} \] So for the accumulator function we have: \begin{align*} \int_{0}^{t}{\delta(s)ds}&=\int_{0}^{t} d\ln\left(a(s)\right) \\ &=\ln\left(a(t)\right) \end{align*} from which follows an expression for the accumulator function in terms of the instantaneous growth: \[ a(t)=\exp\left(\int\limits_{0}^{t}{\delta(s)ds}\right) \] For the continous compounding accumulator function we have $\delta(s)=r$ a constant which confirms: \[ a(t)=\exp\left(\int\limits_{0}^{t}{rds}\right)=e^{rt} \]

As discounting is the inverse of compounding we have:
\[
d(t)=a^{-1}(t)
\]
So *discrete discounting*:
\begin{equation} d(t)=\left(1+\frac{r}{m}\right)^{-mt}, \;t=0,\frac{1}{m},\frac{2}{m},.... \end{equation}
and *continuous discounting* (note $\delta(s)=\tfrac{1}{r}$):
\begin{equation} d(t)=\exp\left(\int\limits_{0}^{t}{\frac{1}{r}ds}\right)=e^{-rt}, \; t \in \mathbb{R} \end{equation}

In a more general context we replace $V_n$ by a series of expected cash-flows $E(C_t)$ at $t=1,..n$ and the value $V_0$ is calculated as follows for the discrete case:
\[
V_0=\sum_{t=1}^n\frac{E(C_t)}{\left(1+r\right)^t}
\]
and
for the continuous case:
\[
\int_{0}^{t}E(C_t)e^{-rt}dt
\]
The process of calculating $V_0$ is called *discounting cash flows (DCF)* and $V_0$ is the *Net Present Value (NPV)* of the future cash flows.
The cash flows also contain the return of the terminal value of the investment if any.

The return $r$ is the *internal rate of return*, it is that single growth rate on invested capital such that the present value of all
future net cash-flows (receipts and payments) equal the initial investment layout. It is the rate that the compound return on the invested over the total investment period if all intermediate cash-flows receipts are reinvested at $r$ till the end of the investment period:
\[
V_n=V_0(1+r)^n+\sum_{t=1}^nE(C_t)\left(1+r\right)^{n-t}
\]
It is called internal since it only takes into account return on capital whenever it is invested in the investment. Once capital is withdrawn from the investment it does not influence the IRR anymore. Therefore it is sometimes called a money-weighted return.

We now analyze a few specific type of cash flow patterns.

The value of a *constant perpetuity* (infinite constantly stream of cash-flows):
\[
V_0\frac{C}{(1+r)}+\frac{C}{(1+r)^{2}}+\frac{C}{(1+r)^{3}}+.....=\frac{C}{(1+r)}(1+\frac{1}{(1+r)}+\frac{1}{(1+r)^{2}}+....)=\frac{C}{r}
\]
This last step follows from the sum of a geometeric series $|a|\less 1$
\begin{eqnarray*}
S & = & 1+a+a^{2}+...\\
-aS & = & \:\: a+a^{2}+...\\
(1-a)S & = & 1\\
S & = & \frac{1}{1-a}
\end{eqnarray*}
Substituting
\[
a=\frac{1}{1+r}
\]
gives the above result.

For the continuous case we have: \begin{align*} V_0&=\int_{0}^{\infty} C e^{-rt}\\ &=\lim_{y\to\infty}C\int_{0}^{y} C e^{-rt} \\ &=\lim_{y\to\infty}C\left[\frac{-e^{-rt}}{r}\right]_{0}^{y} \\ &=\lim_{y\to\infty}C\left[\frac{-e^{-ry}}{r}+\frac{1}{r}\right] \\ &=\lim_{y\to\infty}\frac{1}{r}(1-e^{-ry}) \\ &=\frac{C}{r} \end{align*}

The value of a *growing perpetuity* (infinite constantly growing stream of cash-flows):
\[
V_0=\frac{C}{(1+r)}+\frac{C(1+g)}{(1+r)^{2}}+\frac{C(1+g)^{2}}{(1+r)^{3}}+.....=\frac{C}{(1+r)}\left(1+\frac{(1+g)}{(1+r)}+\frac{(1+g)^{2}}{(1+r)^{2}}+....\right)=\frac{C}{r-g}\]

If we assume that $C=D_0(1+g)$ where $D_0$ is the dividend paid by a company at $t=0$ then the value of that company can be found as: \[ V_0=\frac{D_0(1+g)}{r-g} \] where $g$ is the annual growth rate and $r$ the required rate of return. This model is called the Dividend Growth model or Gordon Growth model.

The value for a *constant annuity* (a fixed stream of constant cash flows):
\begin{eqnarray*}
V_0 & = & \frac{C}{(1+r)}+\frac{C}{(1+r)^{2}}+...+\frac{C}{(1+r)^{n}}\\
& = & \frac{C}{(1+r)}+\frac{C}{(1+r)^{2}}+...+\frac{C}{(1+r)^{n}}+\left(\frac{C}{(1+r)^{n+1}}+...\right)-\left(\frac{C}{(1+r)^{n+1}}+...\right)\\
& = & \frac{C}{r}-\frac{C}{r(1+r)^{n}}=C(\frac{1}{r}-\frac{1}{r(1+r)^{n}})=CA_{r}^{n}\end{eqnarray*}

The value for a *growing annuity* (a fixed stream of growing cash flows):
\begin{eqnarray*}
V_0 & = & \frac{C}{(1+r)}+\frac{C(1+g)}{(1+r)^{2}}+...+\frac{C(1+g)^{n-1}}{(1+r)^{n}}\\
& = & \frac{C}{(1+r)}\left(1+\left(\frac{1+g}{1+r}\right)+...+\left(\frac{1+g}{1+r}\right)^{n}\right)\\
& = & \frac{C}{(1+r)}\left(\frac{1-\left(\frac{1+g}{1+r}\right)^{n}}{1-\left(\frac{1+g}{1+r}\right)}\right)=\frac{C}{r-g}-\frac{C}{r-g}\left(\frac{1+g}{1+r}\right)^{n}
\end{eqnarray*}
Note
\[
S=1+a+a^{2}+...+a^{n-1}\rightarrow S-aS=(1-a)S=1-a^{n}\rightarrow S=\frac{1-a^{n}}{1-a}
\]