Posts tagged with geometric series

Say I offer you a stream of payments.  You compute the present value and decide how much you’re willing to pay now for that stream of payments in the future.  This stream of payments comes from people paying their mortgages, so this transaction is called buying debt.

Now say somebody else has already bought the riskiest 80% … in other words, 80% of the payers (regular people) have to default before you lose a cent.  (In jargon, somebody else “bought the risk” … which implies higher returns iff the payers pay.)

You would probably suppose, quite naturally, that this is a “safe bet” — that it would take astronomically bad luck for a full 4/5 of the payers to default (fail to pay).  (And “safe bets” earn lowish returns.)

That supposition is the rub.  It sounds safe; therefore a salesman who knows that could trick you.  He could sell you something that’s quite likely to default all the way through — but sugar it up by saying that other investors have taken on the risk.  Nearly the whole thing would have to fail before you lost any money!

If you’re a big bank with billions that need to go somewhere, this sounds like an efficient way to make a safe, lowish return on your huge capital holdings.  Economically it sounds like we’ve divided up the financial jobs that need to be done.  Other noses have sniffed out the riskiness of this asset, and the bank has compiled many many small guys’ pocket change (mere \$10 000’s) into real money — which can serve as the stable base to finance this so-called structured asset.

Now you know what a structured asset is, as well as the root economics of the late aughties financial meltdown.

This is an amazing fact that comes up in so many applications.  It’s used in the valuation of companies, solution of equations, ……… any time you want to convert an infinite stream into something finite.

f is a proper fraction. (0 < f < 1)

$\large \dpi{200} \bg_white f^0+f^1+f^2+f^3+f^4+f^5+f^6+ \ldots = {1 \over 1-f}$

Or, in fancy notation:

$\large \dpi{200} \bg_white \sum_{i=0}^\infty f^{\,i} = {1 \over 1-f} , \quad 0

Or, in C++:

long big = 9999999999;float frac = .70;double total = 0;for ( i = 0; i < big; i++){  total += frac∗∗i;  }cout << total;                 # in this case, prints 1 / .3 = 10/3cout << total - 1/(1-frac)     # prints 0 for any value of frac

Isn’t it strange that adding together an infinite number of things can give you a finite answer?  The ancient philosopher Zeno thought that he could disprove reality through the following thought experiment

1. An arrow fired at a tree first covers half the distance to the tree.
2. Then it covers half the remaining distance to the tree.
3. Then it covers half the remaining distance to the tree.
4. Etc….so it only ever covers less than all the distance to the tree!  Because it just keeps adding halves of halves of halves of ….
5. So, since we see it hit the tree, but logically it cannot hit the tree, logically reality must be false!  (Motion is impossible, and we observe motion, so our observations are impossible.)

But calculus proves that:

$\large \dpi{200} \bg_white {1 \over 2} + {1 \over 4} +{1 \over 8} + {1 \over 16 } + {1 \over 32} + {1 \over 64} + {1 \over 128} + {1 \over 256} + \ldots = {^1\! / \!_2 \over 1- {1 \over 2} } = 1$

Take that, Zeno!

## dynamic stochastic economics

The above equation summarizes the theory of "optimal control", which is how Von Neumann and others figured out how to continually adjust the path of a missile so it connects with its moving target.

What does the missile equation have to do with economics? The analogy is to "the savings problem".

Money comes in at variable rates, it benefits you differently at different spending levels, investing can make you money, but spending now is fun; money is complicated!  So you have to continually balance the inflow and outflow of cash.

Here’s what the letters mean.

• z is income.
• c is consumption, aka spending.
• U is utility, aka satisfaction you derive from spending — not necessarily the same for all people, nor the same at all levels of spending! Going from starving to a sandwich is better than going from a sandwich to a steak, for example. Also you had better eat at least something this week, or you won’t be around next week to enjoy that cake you’d been saving up for. Dynamicalself-correlated utility.
• β is the time value of money.
• t is time, and most of these variables are indexed by time.
• k is capital, aka money in the bank or investment money.
• π is the probability of making z dollars.
• δ is the depreciation rate of capital. Everything falls apart, you know.

λ is the Lagrangian, something that’s used to balance the equation. It’s an equation-solving tool. Here’s the gist.

$\large \dpi{200} \bg_white \begin{matrix} &[A] &= &[B] \\ \quad &x &= &\lambda \cdot y \\ x \; - \, \lambda\ \cdot \!\!\!\! &y &= &0 \\ x \; - \, \lambda\ \cdot \!\!\!\! &y &= &L \end{matrix}$

You can also think of λ being the sensitivity parameter (in a slack vector sense). Maybe that’s not the best gist. Sorry. But the main point is—isn’t it weird that missile control theory would be used in describing savings behavior?

hi-res