Recap of the list of steps to introduce financial calculations

We left accounting to turn to finance

• Now there is a relationship between value, time and risk
• There is what's called "the time value of money".
• Meaning : a piece of paper promising a payment of $100 in one year, has a value today (a price). This value depends on the creditworthiness of the signatory, but at any rate is less than $100. (Right now, as of November 2006, in the US, at best, it is worth about $94,5.)

One of the purposes of finance : evaluate investments

• Physical investments (positive NPV's are possible)
• Financial investments :
  • either to just lend money to somebody else,
  • or to buy a small piece of somebody else's project
  • (NPV's of financial investments are theoretically always zero)

In finance, future cash flows are not sure (except for the future value of short term government bonds of prosperous countries)

• We need to take into account the risk (the uncertainty)
• The apparatus (i.e. the toolbox) is simple probability theory
• Future cash flows are outcomes of random variables

Random variables (therefore we need to talk about  the experiment which generates them)

• Possible values (the a_i's)
• Probabilities : Pr {X = a_i}
• Long series of past outcomes (the x_i's)
• The expectation of a RV (= the mean)
• Variability around the mean (Variance and Standard deviation)

We began by studying simple financial securities, or physical investments with just one future CF

Having two securities S and T (and for T we don't know the price), if S has the same risk pattern as T, we saw how to compute the price of T

• One of the central equations in finance : Price of T = mean future value of T "discounted" with the profitability of S
• Profitability of S is called the Opportunity Cost of Capital of T
• Investing into T "costs us the opportunity of investing into the comparable security S"

The procedure is called discounting ; it leads to Present Values

• The present value (PV) of a mean CF in one year is that CF properly discounted (i.e. discounted with its opportunity cost of capital)

We saw that the profitability of a securiy (or a one-year investment) is the figure r such that the NPV computed with r is zero

We used this property to define the "generalised profitability" of a multi CF investment : the exact name is IRR (Internal Rate of Return)

It is possible to find a security that has the same risk as a stream of future cash flows : then, the profitability of that security is called the Opportunity cost of capital of the investment

For a given stock, the series of past prices in the stock market over n years is not a series of n outcomes of one random variables ; the whole path is one outcome of a random walk

But the series of past profitabilities from one year to the next is a series of outcomes of the RV Profitability.

For a projected investment I (C₀, C₁, C₂, C₃, ..., C_n) there are two fundamental rates to take into consideration

• The first is a purely internal calculation (it is the generalisation of profitability) : it's called to Internal Rate of Return
• The second one relates to the rest of the world : it is the profitability of a security with the same kind of risk as I. The profitability of that security is called the Opportunity cost of Capital of I

Except for weird cases, an investment has a positive NPV (i.e. creates value right now) if and only if the IRR is greater than the Op Cost of Cap

What Op cost of cap to use for a given investment ?

Ans : if it is a "plain vanilla" investment in a given industry (like a capacity extension), use the Weighted Average Cost of Capital of firms in the industry (see Ross, Westerfield and Jordan, Fundamentals of Corporate Finance, sixth edition, pp501-502 for details)