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Theory behind asset valuation.
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If x then y
What would you pay for an opportunity to receive $y, if event x occurs?
This is a common structure.Flipping coinsDiceRouletteOptionsDerivatives
Coin Flipping
Fair coin = 50% probability of heads.If heads (x) then $100 (y) Value of the bet is $50v = px * y
v = valuepx = probability of event xy = payout amount
Coin Flipping Assumptions
Assumes that the parties are risk neutral, which means that the subjective value of a win is the same to them as the subjective value of a loss.
Assumes that the probability can be accurately calculated.
Assumes an instant payout of the $100.
Perception of Risk
A risk averse person perceives that a loss will inflict more pain than the pleasure created by a gain of the same size.
A risk seeking person perceives that a gain carries more pleasure than an equivalent loss inflicts pain.
A risk neutral person sees the pleasure and pain from a gain or loss of the same size as equivalent.
Assume a Risk Neutral World
For purposes of trying to determine a fair market value, we will assume risk neutrality.
In the real world, risk seekers may be willing to pay more, while risk avoiders would be willing to pay less.
Assume Probability is Calculable
Events in which probability is directly calculable are rare.
Casino’s like such events. Fair die = probability of each number is one in six. Cards = probability of drawing a particular card from a full deck is one in fifty-two. Roulette = probability of the ball landing on a particular number is one in thirty-eight.
Value of a Bet on Roulette
If a player bets on a single number in the American roulette game there is a probability of 1/38 that the player wins 35 times the bet, and a 37/38 chance that the player loses his bet. The expected value is: The value of the probable loss plus the value of the
probable win. −1×37/38 + 35×1/38 = −0.0526 (5.26% house
edge)
http://en.wikipedia.org/wiki/Roulette#Bet_odds_table_.28American_roulette.29
Assume Immediate Payout
The value of money changes through time.
For the equation (v = px * y) to be accurate, no time can pass between the event x and the payment of y.
If time passes, y must be adjusted to compensate for the time value of money.
Time Value of Money
People prefer a dollar today over a dollar at some point in the future.
In addition to a psychological preference for something now, risk is at the heart of the concept of the time value of money.
Time = Risk
Three risks associated with TIME.
The Risk Associated with the Lender. We prefer to have something now rather than later. In part this reflects our mortality. The greater the period of time the less certain it is that we will be in a condition that will allow us to enjoy our money. In fact, the longer the period that passes, the greater the chance that we will be dead and the money will be beyond our reach.
The Risk Associated with the Borrower. We understand that things happen and that a longer period of time makes it less and less certain that we will be repaid. Borrowers can lose jobs, become sick, go out of business, skip town or like lenders, die. As every loan collector knows, the longer the delinquency the less likely the repayment.
The Risk Associated with the Money Itself. We normally expect that there will be inflation, which will cause the value of each unit of money (such as each dollar) to buy less in the future. In some cases, hyperinflation makes the currency worthless. Even countries do not last forever. Wars, natural disaster, bad economic policies, revolution, coups, etc. can all destroy the value of the money itself.
Mathematical Time Machine
The equation that moves value through time:
(1+r)t
r = interest rate (for the period, i.e., week, month, year)
t = time (the number of periods)
Future Value (FV)
Future Value (FV) is the nominal value of an asset at some point in the future.
Present Value (PV) is the current nominal value of an asset.
FV=PV*(1+r)t
Present Value (PV)
PV = FV / (1+r)t
Interest Rate (r)
Interest Rate (r) reflects:Risk Free Rate = Normally the rate paid on
government bonds of approximately the same duration. This reflects the natural drift through time of nominal prices in an economy.
Risk = Risks associated with the asset as it moves through time (i.e., the probability of actually receiving the expected value in the future).
PV Examples
Low Risk = $1 million payment from the US government in 10 years. 3% compounded annually.Assume 3% reflects the risk free rate.
PV = $1 million / (1 + 0.03)10
PV = $1 million / (1.03)10
PV = $1 million / 1.3439
PV = $744,102.98
PV Examples
High Risk = $1 million from relatives in Greece in 10 years.
30% compounded annually.Assume 3% reflects the risk free rate.Assume 27% reflects risk.
PV = $1 million / (1 + 0.30)10
PV = $1 million / (1.30)10
PV = $1 million / 13.7858
PV = $72,538.40
Inverse Relationship
Value and Risk are inversely related.The higher the risk, the lower the PV of
the asset.
What is this car’s PV?
Measuring Risk
Harry Markowitz – 1952
Risk = Standard Deviation of Price Changes
Standard Deviation
Measures how spread out observations are, assuming that the observations are normally (bell curve) distributed.
Markowitz’s Big Idea
If the price of an asset like a share of stock or a bond fluctuates greatly, it is more risky than an asset with a stable price.
Standard Deviation is a measure of how much the price has fluctuated over a given period of time.
This is called “Volatility.”
Volatility Assumptions
The future will look like the past.Price fluctuations will be fairly uniform
over time.Assets have an inherent characteristic
called volatility (like an electron has a particular mass or charge).
Price volatility captures all risk associated with an asset.
Sampling and Volatility
Sampling is a way to try to understand something by observing selected parts.
We try to predict elections by asking only a few voters how they will vote.
Because we cannot know the future price of an asset, we try to predict the future by sampling from the past.
CAPM
A famous equation to convert this sampling into a specific interest rate is “the Capital Asset Pricing Model.”
Net Present Value (NPV)
What if your asset will produce multiple cash flows over a period of time?
NPV calculates the PV of each expected cash flow and adds the PVs together to calculate the present value of the asset producing those cash flows.
The interest rate used should reflect the risk associated with the cash flows.
Excel makes NPV easy to calculate.
Option Pricing
Option Pricing presented a very difficult valuation problem.
1973 – Fischer Black, Myron Scholes and Robert Merton published the Black-Scholes Option Pricing Formula.
1997 – Scholes and Merton won the Nobel Prize for the formula (Black passed away before 1997).
1998 - LTCM
In 1998, Scholes and Merton, as principals of Long Term Capital Management, applied their ideas and almost crashed the world economy.
Black-Scholes
Adapted a heat diffusion equation from quantum physics, Black-Scholes produced a solution for Option Pricing.
Coincident with Black-Scholes, The Chicago Board Options Exchange opened in 1973.
Black-Scholes and Randomness
Black-Scholes assumes that asset price movements are truly random.
Randomness means movement like the quantum jitters called “Brownian motion.”
You can see a two-dimensional simulation of Brownian motion at:
http://www.math.rutgers.edu/~sontag/336/brownian-applet.html
Randomness = Bell Curve
True randomness results in a bell curve distribution.
The following URL is a simulation of one-dimensional Brownian Motion, the results of which over numerous trial build into a bell curve distribution.
http://webphysics.davidson.edu/WebTalks/clark/onedimensionalwalk.html
Bell Curve = Probability Distribution
If the assumption of randomness is correct, then price changes form a bell curve distribution.
A bell curve distribution allows probabilities of an event to be easily calculated.
Normal Distribution Probability Calculator: http://stattrek.com/Tables/Normal.aspx
Black-Scholes Option Pricing
Try out Black-Scholes. This calculator allows you to see graphically how changes to inputs effect option prices.
http://www.hoadley.net/options/optiongraphs.aspx?
Problem with Black-Scholes
After the Fall of 2008, most experts acknowledge that price changes are not normally distributed. This is called the “Fat Tail Problem.”
The Fat Tail Problem is explained at:http://www.youtube.com/watch?v=UifzEpMjbsQ
Computer simulations can be superior to Black-Scholes for pricing options and derivatives.
Simulations
Computer simulations can also be used to value complex assets.
Boot-strap Simulation: Input market data Computer samples from past price data to
construct possible market trajectories. Computer runs 100,000 simulations market prices
over a specified time period, and them calculates the probability of the asset arriving at any particular price.
Critics
Particularly after the Financial Crisis of 2008, critics argue that statistical valuation techniques do not fully account for risk.
Risk tends to be underestimated, and thus options and other derivatives (which can be used to hedge against risk) tend to be sold too cheaply.
AIG’s sale of Credit Default Swaps (CDS) at cheap prices is an example.
Nassim Nicholas Taleb
Taleb is the best known of the critics. Born and raised in Lebanon during the
Lebanese Civil War, Taleb argues that Modern Finance Theory fails to properly account for the rare, high-impact events.
Black Swans
Taleb called rare, high-impact events “Black Swans.” For hundreds of years, every swan seen by a European was white. Millions and millions of sightings convinced Europeans that all swans were white. However, when the first Europeans went to Australia,
it took the sighting of one “black swan” to overturn millions of prior sightings.
Unknown Unknowns
As Rumsfeld used to say, it is the unknown unknowns that are going to get you.
There are known knowns. These are things we know that we know. There are known
unknowns. That is to say, there are things that we now know we don’t know. But there are also unknown unknowns. These are things we do not know we don’t know.
What if there are no Cash Flows?
Many assets do not have predictable cash flows that will allow any type of cash flow analysis.
Land, houses, cars, jewelry, etc. We have to rely on market price.
A Willing Buyer and a Willing Seller
Fair Market Value (FMV): Is the price that property would sell for on the open market. It is the price that would be agreed on between a willing buyer and a willing seller, with neither being required to act, and both having reasonable knowledge of the relevant facts. If you put a restriction on the use of property you donate, the FMV must reflect that restriction.
Appraisals
Appraisals are reports that seek to determine the FMV of an asset.
This is a form of sampling. The sample is comprised of
comparables, which are believed to be representative of the property at issue.
Problems
Sameness: Unique assets like land and houses and not uniform. (Comparing apples and oranges)
Sample size: Most appraisals are done with just a few comparables. The sample size is often too small to be statistically reliable.
Problems
Space: The farther the subject property and comparables are separated, the less meaningful the comparable.
Time: Likewise, the greater the separation in time, the less meaningful the comparable.
