Foundations of Finance

01Defining Finance and the Financial System 02Valuation and Management: The Two Challenges 03Stocks, Flows, and the Five Cash-Flow Decisions 04Time, Risk, and the Six Principles of Finance 05How to Learn Finance: Practice and Study Habits Problem set0/10 Problem set 20/10 Practice∞

Present Value

01Information and Price: An Auction and Fannie Mae 02Who Pays for the Fannie and Freddie Guarantee?03What Counts as an Asset: Patents, Secrets, Brands 04An Asset as a Sequence of Dated Cashflows 05The Value Operator and the Cashflow Timeline 06Dates as Currencies: Building Net Present Value 07Discount Factors: Market Prices for Future Dollars 08A Worked Net Present Value and Its Assumptions 09The Opportunity Cost of Capital and Present Value Problem set0/10 Problem set 20/10 MIT problem set0/5 Practice∞

01Does the Currency Change an NPV's Sign?02Moral Hazard and Contagion: Where Bailouts Stop 03The NPV Rule at Work, and Choosing the Rate 04The Perpetuity: Why Cash Forever Is Worth C/r 05Reading the Rate Off a Perpetuity's Price 06The Growing Perpetuity, and When Growth Outruns r 07Building the Annuity from Two Perpetuities 08The Annuity Discount Factor and Mortgage Payments 09APR, the Effective Annual Rate, and Compounding Problem set0/10 Problem set 20/10 MIT problem set0/13 Practice∞

The Growing Perpetuity, and When Growth Outruns r

What happens to C/(r − g) when growth catches the discount rate, and why is the answer not simply a big number?

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Your summary note

1. Payments $C$ , $C(1+g)$ , $C(1+g)^2$ discounted at $C(1+g)^2$
  Write out the discounted series for a payment growing at rate $g$ , and work the example with a first payment of $100 and $g$ of 5%, giving $105 the next year.
2. Deriving $\text{PV} = \frac{C}{r-g}$
  Multiply the series through by $\frac{1+r}{1+g}$ , subtract, reduce to $\text{PV} = \frac{C}{r-g}$ , and record what subtracting $g$ does to the denominator and to the value.
3. The condition $r > g$ and its two failure cases
  Set $g = r$ in the series and write down what every term becomes, then record what the closed form returns for $g > r$ and what the underlying series gives instead.
4. China's growth as the finite-horizon caveat
  Record China's roughly 10% average growth over the fifteen years to 2008 against 5.0% in 2025 as growth above $r$ that lasted a while, plus the formula's later use on bonds and stocks.
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