gemAssetPricing: Compute Asset Market Equilibria for Some Simple Cases

Description

Compute the equilibrium of an asset market by the function sdm2 and by computing marginal utility of assets. This function is similar to the function gemSecurityPricing. The difference is that this function uses portfolio utility functions, while the function gemSecurityPricing uses payoff utility functions.

Usage

gemAssetPricing(S, uf, numeraire = nrow(S), ratio_adjust_coef = 0.1, ...)

Value

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

Arguments

S: an n-by-m supply matrix of assets.
uf: a portfolio utility function or a list of m portfolio utility functions.
numeraire: the index of the numeraire commodity.
ratio_adjust_coef: a scalar indicating the adjustment velocity of demand structure.
...: 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.

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

Examples

Run this code

# \donttest{
#### an example of Danthine and Donaldson (2005, section 8.3).
ge <- gemAssetPricing(
  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])),
  maxIteration = 1,
  numberOfPeriods = 500,
  ts = TRUE
)
matplot(ge$ts.p, type = "l")
ge$p

#### an example of Sharpe (2008, chapter 2, case 1)
secy1 <- c(1, 0, 0, 0, 0)
secy2 <- c(0, 1, 1, 1, 1)
secy3 <- c(0, 5, 3, 8, 4) - 3 * secy2
secy4 <- c(0, 3, 5, 4, 8) - 3 * secy2
# unit security payoff matrix
UP <- cbind(secy1, secy2, secy3, secy4)

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

ge <- gemAssetPricing(
  S = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  uf = list(
    function(portfolio) CES(alpha = 1, beta = wt, x = UP %*% portfolio, es = 1 / 1.5),
    function(portfolio) CES(alpha = 1, beta = wt, x = UP %*% portfolio, es = 1 / 2.5)
  ),
  maxIteration = 1,
  numberOfPeriods = 1000,
  numeraire = 1,
  ts = TRUE
)
matplot(ge$ts.p, type = "l")
ge$p
ge$p[3:4] + 3 * ge$p[2]

#### a 3-by-2 example of asset pricing with two heterogeneous agents who
## have different beliefs and predict different payoff vectors.
## the predicted payoff vectors of agent 1 on the two assets.
asset1.1 <- c(1, 2, 2, 0)
asset2.1 <- c(2, 2, 0, 2)

## the predicted payoff vectors of agent 2 on the two assets.
asset1.2 <- c(1, 0, 2, 0)
asset2.2 <- c(2, 1, 0, 2)

asset3 <- c(1, 1, 1, 1)

## the unit (asset) payoff matrix of agent 1.
UP1 <- cbind(asset1.1, asset2.1, asset3)

## the unit (asset) payoff matrix of agent 2.
UP2 <- cbind(asset1.2, asset2.2, asset3)

mp1 <- colMeans(UP1)
Cov1 <- cov.wt(UP1, method = "ML")$cov

mp2 <- colMeans(UP2)
Cov2 <- cov.wt(UP2, method = "ML")$cov

ge <- gemAssetPricing(
  S = matrix(c(
    1, 5,
    2, 5,
    3, 5
  ), 3, 2, TRUE),
  uf = list(
    # the utility function of agent 1.
    function(x) AMSDP(x, mp1, Cov1, gamma = 0.2, theta = 2),
    function(x) AMSDP(x, mp2, Cov2) # the utility function of agent 2.
  ),
  maxIteration = 1,
  numberOfPeriods = 1000,
  ts = TRUE
)
matplot(ge$ts.p, type = "l")
ge$p
ge$VMU

#### another 3-by-2 example.
asset1.1 <- c(0, 0, 1, 1, 2)
asset2.1 <- c(1, 2, 1, 2, 0)
asset3.1 <- c(1, 1, 1, 1, 1)

asset1.2 <- c(0, 0, 1, 2)
asset2.2 <- c(1, 2, 2, 1)
asset3.2 <- c(1, 1, 1, 1)

## the unit asset payoff matrix of agent 1.
UP1 <- cbind(asset1.1, asset2.1, asset3.1)

## the unit asset payoff matrix of agent 2.
UP2 <- cbind(asset1.2, asset2.2, asset3.2)

mp1 <- colMeans(UP1)
Cov1 <- cov.wt(UP1, method = "ML")$cov

mp2 <- colMeans(UP2)
Cov2 <- cov.wt(UP2, method = "ML")$cov

ge <- gemAssetPricing(
  S = matrix(c(
    1, 5,
    2, 5,
    3, 5
  ), 3, 2, TRUE),
  uf = list(
    # the utility function of agent 1.
    function(x) AMSDP(x, mp1, Cov1),
    function(x) AMSDP(x, mp2, Cov2) # the utility function of agent 2.
  ),
  maxIteration = 1,
  numberOfPeriods = 3000,
  ts = TRUE
)

ge$p
ge$D

#### a 5-by-3 example.
set.seed(1)
n <- 5 # the number of asset types
m <- 3 # the number of agents
Supply <- matrix(runif(n * m, 10, 100), n, m)

# the risk aversion coefficients of agents.
gamma <- runif(m, 0.25, 1)

# predicted mean payoff, which may be gross return rates, price indices or prices.
PMP <- matrix(runif(n * m, min = 0.8, max = 1.5), n, m)

# predicted standard deviation of payoff
PSD <- matrix(runif(n * m, min = 0.01, max = 0.2), n, m)
PSD[n, ] <- 0

# Suppose the predicted payoff correlation matrices of agents are the same.
Cor <- cor(matrix(runif(2 * n^2), 2 * n, n))
Cor[, n] <- Cor[n, ] <- 0
Cor[n, n] <- 1

# the list of utility functions.
lst.uf <- list()

make.uf <- function(mp, Cov, gamma) {
  force(mp)
  force(Cov)
  force(gamma)
  function(x) {
    AMSDP(x, mp = mp, Cov = Cov, gamma = gamma, theta = 1)
  }
}

for (k in 1:m) {
  sigma <- PSD[, k]
  if (is.matrix(Cor)) {
    Cov <- dg(sigma) %*% Cor %*% dg(sigma)
  } else {
    Cov <- dg(sigma) %*% Cor[[k]] %*% dg(sigma)
  }

  lst.uf[[k]] <- make.uf(mp = PMP[, k], Cov = Cov, gamma = gamma[k])
}

ge <- gemAssetPricing(
  S = Supply, uf = lst.uf,
  maxIteration = 1,
  numberOfPeriods = 2000,
  priceAdjustmentVelocity = 0.05,
  policy = makePolicyMeanValue(150),
  ts = TRUE
)

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

#### a 2-by-2 example with outside position.
asset1 <- c(1, 0, 0)
asset2 <- c(0, 1, 1)

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

uf1 <- function(portfolio) prod((UP %*% portfolio + c(0, 0, 2))^wt)
uf2 <- function(portfolio) prod((UP %*% portfolio)^wt)

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

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