gemAssetPricingExample: Some Examples of Asset Pricing

Description

These examples illustrate how to find the equilibrium of an asset market by the function sdm2 and by computing marginal utility of assets (see Sharpe, 2008).

Usage

gemAssetPricingExample(...)

Value

A general equilibrium.

Arguments

...: 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.

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{
#### an example of Danthine and Donaldson (2005, section 8.3).
uf <- function(x) 0.5 * x[1] + 0.9 * (1 / 3 * log(x[2]) + 2 / 3 * log(x[3]))

ge <- sdm2(
  A = function(state) {
    VMU <- marginal_utility(state$last.A %*% dg(state$last.z), diag(3), uf, state$p)
    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")

    A <- state$last.A * Ratio
    prop.table(A, 2)
  },
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    10, 5,
    1, 4,
    2, 6
  ), 3, 2, TRUE),
  names.commodity = c("asset1", "asset2", "asset3"),
  names.agent = c("agt1", "agt2"),
  numeraire = "asset1",
  ts = TRUE
)

ge$p

#### an example of Sharpe (2008, chapter 2)
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

gamma.agt1 <- 1.5
gamma.agt2 <- 2.5

ge <- sdm2(
  A = function(state) {
    Payoff <- UAP %*% (state$last.A %*% dg(state$last.z))

    VMU <- marginal_utility(Payoff, UAP, list(
      function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / gamma.agt1),
      function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / gamma.agt2)
    ), price = state$p)
    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")

    A <- state$last.A * ratio_adjust(Ratio, coef = 0.05, method = "linear")

    A <- prop.table(A, 2)
  },
  B = matrix(0, 4, 2),
  S0Exg = matrix(c(
    49, 49,
    30, 30,
    10, 0,
    0, 10
  ), 4, 2, TRUE),
  names.commodity = c("asset1", "asset2", "asset3", "asset4"),
  names.agent = c("agt1", "agt2"),
  numeraire = "asset1"
)

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

#### an example of Xu (2018, section 10.4, P151)
asset1 <- c(1, 0, 0)
asset2 <- c(0, 1, 0)
asset3 <- c(0, 0, 1)
prob <- c(0.5, 0.5)
wt <- c(1, prob)
UAP <- cbind(asset1, asset2, asset3)

gamma.agt1 <- 1
gamma.agt2 <- 0.5

ge <- sdm2(
  A = function(state) {
    Payoff <- UAP %*% (state$last.A %*% dg(state$last.z))

    VMU <- marginal_utility(Payoff, UAP, list(
      # Here CRRA(...)$u, CRRA(...)$CE and CES functions are interexchangeable.
      function(x) CRRA(x, gamma = gamma.agt1, p = wt)$u,
      function(x) CES(alpha = 1, beta = wt, x = x, es = 1 / gamma.agt2)
    ), state$p)
    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")

    A <- state$last.A * Ratio
    prop.table(A, 2)
  },
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    1, 0,
    0, 0.5,
    0, 2
  ), 3, 2, TRUE),
  names.commodity = c("asset1", "asset2", "asset3"),
  names.agent = c("agt1", "agt2"),
  numeraire = "asset1",
  maxIteration = 1,
  ts = TRUE
)

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

## the same as above.
dst.agt1 <- node_new("util",
  type = "CD", alpha = 1, beta = c(0.5, 0.25, 0.25),
  "asset1", "asset2", "asset3"
)

dst.agt2 <- node_new("util",
  type = "CES", alpha = 1, beta = c(2, 1, 1), sigma = 0.5,
  "asset1", "asset2", "asset3"
)

ge <- sdm2(
  A = list(dst.agt1, dst.agt2),
  B = matrix(0, 3, 2),
  S0Exg = matrix(c(
    1, 0,
    0, 0.5,
    0, 2
  ), 3, 2, TRUE),
  names.commodity = c("asset1", "asset2", "asset3"),
  names.agent = c("agt1", "agt2"),
  numeraire = "asset1",
  maxIteration = 1,
  ts = TRUE
)

ge$p

#### an example with production.
asset1 <- c(1, 0, 0, 0, 0, 0)
asset2 <- c(0, 1, 0, 0, 0, 0)
asset3 <- c(0, 0, 1, 3, 1, 2)
asset4 <- c(0, 0, 4, 2, 6, 2)
asset5 <- c(0, 0, 1, 0, 2, 0)

# unit asset payoff matrix
UAP <- cbind(asset1, asset2, asset3, asset4, asset5)

muf1 <- function(x) 1 / x
muf2 <- function(x) 1 / x * c(0.4, 0.1, 0.2, 0.05, 0.2, 0.05)

ge <- sdm2(
  A = function(state) {
    Payoff <- UAP %*% (state$last.A[, 1:2] %*% dg(state$last.z[1:2]))

    VMU <- marginal_utility(Payoff, UAP, muf = list(muf1, muf2), price = state$p)
    Ratio <- sweep(VMU, 2, colMeans(VMU), "/")

    A <- state$last.A[, 1:2] * ratio_adjust(Ratio, coef = 0.15, method = "linear")
    A <- prop.table(A, 2)

    a.firm <- CD_A(alpha = 4, Beta = c(0.5, 0.5, 0, 0, 0), state$p)
    A <- cbind(A, a.firm)
  },
  B = matrix(c(
    0, 0, 0,
    0, 0, 0,
    0, 0, 0,
    0, 0, 0,
    0, 0, 1
  ), 5, 3, TRUE),
  S0Exg = matrix(c(
    1, 1, NA,
    1, 2, NA,
    1, NA, NA,
    NA, 1, NA,
    NA, NA, NA
  ), 5, 3, TRUE),
  names.commodity = c("asset1", "asset2", "asset3", "asset4", "asset5"),
  names.agent = c("consumer1", "consumer2", "firm"),
  numeraire = "asset1"
)

ge$p
ge$z

#### an example with demand structure trees.
asset1 <- c(1, 0, 0, 0, 0)
asset2 <- c(0, 1, 3, 1, 2)
asset3 <- c(0, 2, 1, 3, 1)

# the asset unit payoff matrix.
UAP <- cbind(asset1, asset2, asset3)

dst.consumer1 <- node_new("util",
                          type = "CES", es = 0.5, alpha = 1, beta = c(0.5, 0.5),
                          "x1", "u2"
)
node_set(dst.consumer1, "u2",
         type = "CES", es = 0.8, alpha = 1, beta = c(0.6, 0.4),
         "u2.1", "u2.2"
)
node_set(dst.consumer1, "u2.1",
         type = "CES", es = 1, alpha = 1, beta = c(0.8, 0.2),
         "x2", "x3"
)
node_set(dst.consumer1, "u2.2",
         type = "CES", es = 1, alpha = 1, beta = c(0.8, 0.2),
         "x4", "x5"
)

dst.consumer2 <- node_new("util",
                          type = "CES", es = 0.5, alpha = 1, beta = c(0.5, 0.5),
                          "x1", "u2"
)
node_set(dst.consumer2, "u2",
         type = "CES", es = 0.8, alpha = 1, beta = c(0.6, 0.4),
         "u2.1", "u2.2"
)
node_set(dst.consumer2, "u2.1",
         type = "CES", es = 1, alpha = 1, beta = c(0.2, 0.8),
         "x2", "x3"
)
node_set(dst.consumer2, "u2.2",
         type = "CES", es = 1, alpha = 1, beta = c(0.2, 0.8),
         "x4", "x5"
)

uf1 <- function(x) {
  names(x) <- paste0("x", seq_along(x))
  output(dst.consumer1, x)
}

uf2 <- function(x) {
  names(x) <- paste0("x", seq_along(x))
  output(dst.consumer2, x)
}

ge <- gemAssetPricing_CUF(
  S = matrix(c(
    3, 3,
    1, 0,
    0, 2
  ), 3, 2, TRUE),
  UAP = UAP,
  uf = list(uf1, uf2)
)

ge$p
ge$z
# }