# Modern Financial Theory

## Recap 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.

One of the purposes of finance is to 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)

Another purpose is to get the proper kind of financing for investments selected as good.

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 ai's)
• Probabilities: Pr {X = ai}
• Long series of past outcomes (the xi'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 variable; 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 any projected investment I (C0, C1, C2, C3, ..., Cn) there are two fundamental rates to take into consideration

• The first is a purely internal calculation with the cash flows of I.
• It is the generalisation of profitability.
• It is called to Internal Rate of Return of I

• 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 for instance Ross, Westerfield and Jordan, Fundamentals of Corporate Finance, sixth edition, pp501-502 for details)