List of Symbols Used on Working Paper 149
| t | = | time
period t. |
| s, m | = | measures
of intensity of boom and depression states of the
economy. The value of m depicts the marginal utility of a
dollar for ( 0 < m < + ¥ ). The s variable is a
monotone transformation of m, where s < 0 depicts the
severity of a depression, s > 0 depicts the intensity
of a boom, and s = 0 depicts a market risk neutral state
when m = $1.00. |
| sw | = | an
actual state of
the world (economy). |
| sc | = | a
state of the world (economy) used as a basis to forecast
the C ( t, sc ) price of a comparison
asset. |
| sx | = | a
state of the world (economy) used as a basis to forecast
X ( t, s ) cash flows. |
| w | = | an
index depicting alternative states of the actual
world w. |
| c | = | an
index depicting alternative states of a marketed comparison
asset world c. |
| x, y | = | indices
depicting alternative states of a nonmarketed asset
worlds x or y. |
| p ( t, s ) | = | probability
that the economy is in state s at time t. This may also
be termed a state transition probability or a stochastic
process probability. |
| m ( t, s ) | = | marginal
utility of a dollar at time t when the economy is in
state s. The m ( t, s ) is also a measure of market
"risk preference" or boom / depression
"intensity." |
| C ( t, s ) | = | forecasted
price of a marketed comparison asset. |
| X ( t, s ) | = | forecasted
expected cash flow of a nonmarketed asset being valued
for investment purposes. |
| Y ( t, s ) | = | discounted
X ( t, s ) at a risk free discount rate. |
| d ( t, s ) | = | discount
factor at the risk free rate in a risk neutral state s =
0. |
| m | = | expected
value at t = 0 of all Y ( t, s ) discounted cash flows
over all t and s. |
| mm | = | expected
value at t = 0 of all Y ( t, s ) discounted cash flows after
adjusting Y ( t, s ) amounts for
market (systematic) risk. |
| s 2 | = | variance
of t = 0 net present values about the m expected value. |
| s 2 m | = | variance
of t = 0 net present market risk
adjusted net present values about m m
expected value. |
| a ( t, 0 ) | = | arbitrage
probability equal to p ( t, 0 ) state transition
probability in a risk neutral world (economy). |
| S ( t, s ) | = | state
(contingent) claims price that confounds p ( t, s ) state
probabilities, m ( t, s ) market risk preferentials, and
d ( t, s ) discount factors. The S ( t, s ) state prices
are also termed (market) risk adjustment present value
factors (RAPVFs). |
| sc ( C ) | = | the
state of the world (economy) c that generated the
comparison asset price forecast C. |
| s 2m ( x ) | = | s2m
assuming all states of the world (economy) used in
forecasting X ( t, s ) cash flows. This is generally
known because sx must be known. |
| s 2m ( c ) | = | s2m
assuming all states of the world
(economy) used in forecasting C ( t, sc )
comparison asset prices. This is generally unknown if sc is unknown. |
| mm ( t, s ) | = | the
market risk adjusted net present value of the X ( t, s )
cash flow. |
| Sc ( t, 0 ) | = | the
S ( t, s ) state price derived from comparison asset
price C ( t, s ) if it is assumed s = 0 is a risk neutral
state. |