gemAssetPricing_CUF: Compute Asset Market Equilibria with Commodity Utility Functions for Some Simple Cases

Description

Compute the equilibrium of an asset market by the function sdm2 and by computing marginal utility of assets (see Sharpe, 2008). The argument of the utility function used in the calculation is the commodity vector (i.e. payoff vector).

Usage

gemAssetPricing_CUF(
  S = diag(2),
  UAP = diag(nrow(S)),
  uf = NULL,
  muf = NULL,
  ratio_adjust_coef = 0.05,
  numeraire = 1,
  ...
)

Value

A general equilibrium containing a value marginal utility matrix (VMU).

Arguments

S: an n-by-m supply matrix of assets.
UAP: a unit asset payoff k-by-n matrix.
uf: a utility function or a utility function list.
muf: a marginal utility function or a marginal utility function list.
ratio_adjust_coef: a scalar indicating the adjustment velocity of demand structure.
numeraire: the index of the numeraire commodity.
...: arguments to be passed to the function sdm2.

References

Danthine, J. P., Donaldson, J. (2005, ISBN: 9780123693808) Intermediate Financial Theory. Elsevier Academic Press.

Sharpe, William F. (2008, ISBN: 9780691138503) Investors and Markets: Portfolio Choices, Asset Prices, and Investment Advice. Princeton University Press.

Wang Jiang (2006, ISBN: 9787300073477) Financial Economics. Beijing: China Renmin University Press. (In Chinese)

Xu Gao (2018, ISBN: 9787300258232) Twenty-five Lectures on Financial Economics. Beijing: China Renmin University Press. (In Chinese)

https://web.stanford.edu/~wfsharpe/apsim/index.html

Examples

Run this code

# \donttest{
gemAssetPricing_CUF(muf = function(x) 1 / x)

gemAssetPricing_CUF(
  S = cbind(c(1, 0), c(0, 2)),
  muf = function(x) 1 / x
)

gemAssetPricing_CUF(
  UAP = cbind(c(1, 0), c(0, 2)),
  muf = function(x) 1 / x
)

#### an example of Danthine and Donaldson (2005, section 8.3).
ge <- gemAssetPricing_CUF(
  S = matrix(c(
    10, 5,
    1, 4,
    2, 6
  ), 3, 2, TRUE),
  uf = function(x) 0.5 * x[1] + 0.9 * (1 / 3 * log(x[2]) + 2 / 3 * log(x[3]))
)

ge$p

#### an example of Sharpe (2008, chapter 2, case 1)
asset1 <- c(1, 0, 0, 0, 0)
asset2 <- c(0, 1, 1, 1, 1)
asset3 <- c(0, 5, 3, 8, 4) - 3 * asset2
asset4 <- c(0, 3, 5, 4, 8) - 3 * asset2
# unit asset payoff matrix
UAP <- cbind(asset1, asset2, asset3, asset4)

prob <- c(0.15, 0.25, 0.25, 0.35)
wt <- prop.table(c(1, 0.96 * prob)) # weights

geSharpe1 <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  UAP = UAP,
  uf = list(
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 1.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 2.5)
  )
)
geSharpe1$p
geSharpe1$p[3:4] + 3 * geSharpe1$p[2]

## an example of Sharpe (2008, chapter 3, case 2)
geSharpe2 <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 49, 98, 98,
    30, 30, 60, 60,
    10, 0, 20, 0,
    0, 10, 0, 20
  ), 4, 4, TRUE),
  UAP = UAP,
  uf = list(
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 1.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 2.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 1.5),
    function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / 2.5)
  )
)

geSharpe2$p
geSharpe2$p[3:4] + 3 * geSharpe2$p[2]
geSharpe2$D

## an example of Sharpe (2008, chapter 3, case 3)
geSharpe3 <- gemAssetPricing_CUF(UAP,
  uf = function(x) (x - x^2 / 400) %*% wt,
  S = matrix(c(
    49, 98,
    30, 60,
    5, 10,
    5, 10
  ), 4, 2, TRUE)
)
geSharpe3$p
geSharpe3$p[3:4] + 3 * geSharpe3$p[2]

# the same as above
geSharpe3b <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 98,
    30, 60,
    5, 10,
    5, 10
  ), 4, 2, TRUE),
  UAP = UAP,
  muf = function(x) (1 - x / 200) * wt
)

geSharpe3b$p
geSharpe3b$p[3:4] + 3 * geSharpe3b$p[2]

## an example of Sharpe (2008, chapter 3, case 4)
geSharpe4 <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 98,
    30, 60,
    5, 10,
    5, 10
  ), 4, 2, TRUE),
  UAP,
  muf = function(x) abs((x - 20)^(-1)) * wt,
  maxIteration = 100,
  numberOfPeriods = 300,
  ts = TRUE
)

geSharpe4$p
geSharpe4$p[3:4] + 3 * geSharpe4$p[2]

## an example of Sharpe (2008, chapter 6, case 14)
prob1 <- c(0.15, 0.26, 0.31, 0.28)
wt1 <- prop.table(c(1, 0.96 * prob1))
prob2 <- c(0.08, 0.23, 0.28, 0.41)
wt2 <- prop.table(c(1, 0.96 * prob2))

uf1 <- function(x) CES(alpha = 1, beta = wt1, x = x, es = 1 / 1.5)
uf2 <- function(x) CES(alpha = 1, beta = wt2, x = x, es = 1 / 2.5)
geSharpe14 <- gemAssetPricing_CUF(
  S = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  UAP = UAP,
  uf = list(uf1,uf2)
)

geSharpe14$D
geSharpe14$p
geSharpe14$p[3:4] + 3 * geSharpe14$p[2]
mu <- marginal_utility(geSharpe14$Payoff, diag(5),uf=list(uf1,uf2))
mu[,1]/mu[1,1]
mu[,2]/mu[1,2]

#### an example of Wang (2006, example 10.1, P146)
geWang <- gemAssetPricing_CUF(
  S = matrix(c(
    1, 0,
    0, 2,
    0, 1
  ), 3, 2, TRUE),
  muf = list(
    function(x) 1 / x * c(0.5, 0.25, 0.25),
    function(x) 1 / sqrt(x) * c(0.5, 0.25, 0.25)
  )
)

geWang$p # c(1, (1 + sqrt(17)) / 16)

# the same as above
geWang.b <- gemAssetPricing_CUF(
  S = matrix(c(
    1, 0,
    0, 2,
    0, 1
  ), 3, 2, TRUE),
  uf = list(
    function(x) log(x) %*% c(0.5, 0.25, 0.25),
    function(x) 2 * sqrt(x) %*% c(0.5, 0.25, 0.25)
  )
)

geWang.b$p

#### an example of Xu (2018, section 10.4, P151)
wt <- c(1, 0.5, 0.5)
ge <- gemAssetPricing_CUF(
  S = matrix(c(
    1, 0,
    0, 0.5,
    0, 2
  ), 3, 2, TRUE),
  uf = list(
    function(x) CRRA(x, gamma = 1, prob = wt)$u,
    function(x) CRRA(x, gamma = 0.5, prob = wt)$u
  )
)

ge$p # c(1, (1 + sqrt(5)) / 4, (1 + sqrt(17)) / 16)

#### an example of incomplete market
ge <- gemAssetPricing_CUF(
  UAP = cbind(c(1, 1), c(2, 1)),
  uf = list(
    function(x) sum(log(x)) / 2,
    function(x) sum(sqrt(x))
  ),
  ratio_adjust_coef = 0.1,
  priceAdjustmentVelocity = 0.05,
  policy = makePolicyMeanValue(span = 100),
  maxIteration = 1,
  numberOfPeriods = 2000,
)

ge$p

## the same as above
ge.b <- gemAssetPricing_CUF(
  UAP = cbind(c(1, 1), c(2, 1)),
  muf = list(
    function(x) 1 / x * c(0.5, 0.5),
    function(x) 1 / sqrt(x) * c(0.5, 0.5)
  ),
  ratio_adjust_coef = 0.1,
  priceAdjustmentVelocity = 0.05,
  policy = makePolicyMeanValue(span = 100),
  maxIteration = 1,
  numberOfPeriods = 2000,
  ts = TRUE
)

ge.b$p
matplot(ge.b$ts.p, type = "l")

#### an example with outside position.
asset1 <- c(1, 0, 0)
asset2 <- c(0, 1, 1)

# unit (asset) payoff matrix
UAP <- cbind(asset1, asset2)
wt <- c(0.5, 0.25, 0.25) # weights

uf1 <- function(x) prod((x + c(0, 0, 2))^wt)
uf2 <- function(x) prod(x^wt)

ge <- gemAssetPricing_CUF(
  S = matrix(c(
    1, 1,
    0, 2
  ), 2, 2, TRUE),
  UAP = UAP,
  uf = list(uf1, uf2),
  numeraire = 1
)

ge$p
ge$z
uf1(ge$Payoff[,1])
uf2(ge$Payoff[,2])
# }