import numpy as np
[docs]
def calc_sprime_c_td(Y_td, tau_td, i, pi):
r"""
Compute after-tax rate of return on savings invested in
tax-deferred accounts.
.. math::
s^{'}_{c,td} = \frac{1}{Y_{td}}log((1-\tau_{td})*e^{i*Y_{td}}+
\tau_{td}) - \pi
Args:
Y_td (scalar): number of years savings held in tax-deferred
retirement account
tau_td (scalar): effective marginal tax rate on investment
income from tax-deferred accounts
i (scalar): the nominal interest rate
pi (scalar): the inflation rate
Returns:
sprime_c_td (scalar): the after-tax return on corporate
investments made through tax-deferred accounts
"""
sprime_c_td = (1 / Y_td) * np.log(
((1 - tau_td) * np.exp(i * Y_td)) + tau_td
) - pi
return sprime_c_td
[docs]
def calc_s_c_d_td(sprime_c_td, gamma, i, pi):
r"""
Compute the after-tax return on corprate debt investments made
through tax-deferred accounts.
.. math::
s_{c,d,td} = \gamma(i-\pi) + (1-\gamma)s^{'}_{c,td}
Args:
sprime_c_td (scalar): the after-tax return on corporate
investments made through tax-deferred accounts
gamma (scalar): Fraction of debt owned through whole-life
insurance policies
i (scalar): the nominal interest rate
pi (scalar): the inflation rate
Returns:
s_c_d_td (scalar): the after-tax return on corprate debt
investments made through tax-deferred accounts
"""
s_c_d_td = gamma * (i - pi) + (1 - gamma) * sprime_c_td
return s_c_d_td
[docs]
def calc_s__d(
s_d_td, alpha_d_ft, alpha_d_td, alpha_d_nt, tau_int, tau_w, i, pi
):
r"""
Compute the after-tax return to debt investments.
.. math::
s_{j,d} = \alpha_{j,d,ft}((1-\tau_{int})i - \pi) +
\alpha_{j,d,td}s_{j,d,td} + \alpha_{j,d,nt}(i-\pi) - \tau_{w}
Args:
s_d_td (scalar): after-tax return on debt investments made
through tax-deferred acounts
alpha_d_ft (scalar): fraction of debt investments held in
full-tax accounts
alpha_d_td (scalar): fraction of debt investments held in
tax-deferred acounts
alpha_d_nt (scalar): fraction of debt investments held in
tax-free accounts
tau_int (scalar): marginal tax rate on interest income
tau_w (scalar): marginal tax rate on wealth
i (scalar): nominal interest rate
pi (scalar): inflation rate
Returns:
s__d (scalar): after-tax return on debt investments
"""
s__d = (
alpha_d_ft * (((1 - tau_int) * i) - pi)
+ alpha_d_td * s_d_td
+ alpha_d_nt * (i - pi)
- tau_w
)
return s__d
[docs]
def calc_g__g(Y_g, tau_cg, m, E_c, pi):
r"""
Calculate the real, after-tax annualized return on short or long-
term capital gains
.. math::
g_{icg} = \frac{1}{Y_{icg}}\ln\biggl[(1-\tau_{icg})e^{(\pi+mE)
Y_{icg}}+\tau_{icg}\biggr] + \pi
Args:
Y_g (scalar): number of years asset held before gains realized
tau_cg (scalar): tax rate on capital gains income
m (scalar): share of equity return retained by the firm and reinvested
E_c (scalar): expected, after-tax return on corporate equity
pi (scalar): inflation rate
Returns:
g__g (scalar): real, after-tax annualized return on capital
gains
"""
g__g = (1 / Y_g) * np.log(
((1 - tau_cg) * np.exp((pi + m * E_c) * Y_g)) + tau_cg
) - pi
return g__g
[docs]
def calc_g(g_scg, g_lcg, g_xcg, omega_scg, omega_lcg, omega_xcg, m, E_c):
r"""
Calculate the after-tax, annualized, real rate of return on all
capital gains
.. math::
g = \omega_{scg}\times g_{scg} + \omega_{lcg}\times g_{lcg} +
\omega_{xcg}\times mE
Args:
g_scg (scalar): the real, after-tax annualized return on short-
term capital gains
g_lcg (scalar): the real, after-tax annualized return on long-
term capital gains
g_xcg (scalar): the real, after-tax annualized return on
capital gains held until death
omega_scg (scalar): the fraction of capital gains that are
short-term
omega_lcg (scalar): the fraction of capital gains that are
long-term
omega_xcg (scalar): the fraction of capital gains that are held
until death
m (scalar): share of equity return retained by the firm and reinvested
E_c (scalar): expected, after-tax return on corporate equity
Returns:
g (scalar): the after-tax, annualized, real rate of return on
all capital gains
"""
g = omega_scg * g_scg + omega_lcg * g_lcg + omega_xcg * g_xcg
return g
[docs]
def calc_s_c_e_td(Y_td, tau_td, i, pi, E_c):
r"""
Calculate the after-tax return on investmentes in corporate equity
in tax-deferred accounts.
.. math::
s_{c,e,td} = \frac{1}{Y_{td}}\ln((1-\tau_{td})e^{(\pi+E)Y_{td}}
+\tau_{td}) - \pi
Args:
Y_td (scalar): years investments are held in tax-deferred
accounts
tau_td (scalar): marginal tax rate on investments in
tax-deferred accounts
i (scalar): nominal interest rate
pi (scalar): inflation rate
E_c (scalar): expected, after-tax return on corporate equity
Returns:
s_c_e_td (scalar): the after-tax return on investmentes in
corporate equity in tax-deferred accounts.
"""
s_c_e_td = (1 / Y_td) * np.log(
((1 - tau_td) * np.exp((pi + E_c) * Y_td)) + tau_td
) - pi
return s_c_e_td
[docs]
def calc_s_c_e(
s_c_e_ft, s_c_e_td, alpha_c_e_ft, alpha_c_e_td, alpha_c_e_nt, tau_w, E_c
):
r"""
Calculate the after-tax return on investments in corporate equity
.. math::
s_{c,e} = \alpha_{c,e,ft}\times s_{c,e,ft} + \alpha_{c,e,td}
\times s_{c,e,td} + \alpha_{c,e,nt}\times E - \tau_{w}
Args:
s_c_e_ft (scalar): after-tax return on investments in corporate
equity in fully-taxable accounts
s_c_e_td (scalar): after-tax return on investments in corporate
equity in tax-deferred accounts
alpha_c_e_ft (scalar): fraction of corporate equity investments
made through full-taxable accounts
alpha_c_e_td (scalar): fraction of corporate equity investments
made through tax-deferred accounts
alpha_c_e_nt (scalar): fraction of corporate equity investments
made through tax-exempt accounts
tau_w (scalar): marginal tax rate on wealth
E_c (scalar): expected, after-tax return on corporate equity
Returns:
s_c_e (scalar): the after-tax return on investments in
corporate equity
"""
s_c_e = (
alpha_c_e_ft * s_c_e_ft
+ alpha_c_e_td * s_c_e_td
+ alpha_c_e_nt * E_c
- tau_w
)
return s_c_e
[docs]
def calc_s(p):
"""
Compute the after-tax rate of return to savers, s. Calls other
`calc_s_x_y` functions to compute various rates of return.
Args:
p (CCC Specification Object): model parameters
Returns:
(tuple): return to savers and required return to pass-through
entities:
* s_dict (dict): dictionary of s for investments in
corporate and pass-through businesses and by type of
financing
* E_pt (scalar): required pre-tax return on pass-through
investments
"""
# Compute after-tax rate of return on savings invested in
# tax-deferred accounts
sprime_c_td = calc_sprime_c_td(
p.Y_td, p.tau_td, p.nominal_interest_rate, p.inflation_rate
)
# The after-tax return on corprate debt investments made through
# tax-deferred accounts
s_c_d_td = calc_s_c_d_td(
sprime_c_td, p.gamma, p.nominal_interest_rate, p.inflation_rate
)
# The after-tax return on corporate debt investments
s_c_d = calc_s__d(
s_c_d_td,
p.alpha_c_d_ft,
p.alpha_c_d_td,
p.alpha_c_d_nt,
p.tau_int,
p.tau_w,
p.nominal_interest_rate,
p.inflation_rate,
)
# The after-tax return on non-corporate debt investments made
# through tax deferred accounts
s_pt_d_td = s_c_d_td
# The after-tax return on non-corporate debt investments
s_pt_d = calc_s__d(
s_pt_d_td,
p.alpha_pt_d_ft,
p.alpha_pt_d_td,
p.alpha_pt_d_nt,
p.tau_int,
p.tau_w,
p.nominal_interest_rate,
p.inflation_rate,
)
# The after-tax real, annualized return on short-term capital gains
g_scg = calc_g__g(p.Y_scg, p.tau_scg, p.m, p.E_c, p.inflation_rate)
# The after-tax real, annualized return on long-term capital gains
g_lcg = calc_g__g(p.Y_lcg, p.tau_lcg, p.m, p.E_c, p.inflation_rate)
# The after-tax real, annualized return on capital gains held until
# death
g_xcg = calc_g__g(p.Y_xcg, p.tau_xcg, p.m, p.E_c, p.inflation_rate)
# The after-tax real, annualized return on all capital gains
g = calc_g(
g_scg, g_lcg, g_xcg, p.omega_scg, p.omega_lcg, p.omega_xcg, p.m, p.E_c
)
# The after-tax return on corporate equity investments made in fully
# taxable accounts
s_c_e_ft = (1 - p.m) * p.E_c * (1 - p.tau_div) + g
# The after-tax return on corporate equity investments made in
# tax-deferred acounts
s_c_e_td = calc_s_c_e_td(
p.Y_td, p.tau_td, p.nominal_interest_rate, p.inflation_rate, p.E_c
)
# The after-tax return on corporate equity investments
s_c_e = calc_s_c_e(
s_c_e_ft,
s_c_e_td,
p.alpha_c_e_ft,
p.alpha_c_e_td,
p.alpha_c_e_nt,
p.tau_w,
p.E_c,
)
# The after-tax return on corporate investments (all - debt and
# equity combined)
s_c = p.f_c * s_c_d + (1 - p.f_c) * s_c_e
# The required rate of return on non-corporate investments
E_pt = s_c_e
# The after-tax rate of return on non-corporate equity investments
s_pt_e = E_pt - p.tau_w
# The after-tax return on non-corporate investments (all - debt and
# equity combined)
s_pt = p.f_pt * s_pt_d + (1 - p.f_pt) * s_pt_e
# Return the after-tax rates of return on all types of investments
s_dict = {
"c": {"mix": s_c, "d": s_c_d, "e": s_c_e},
"pt": {"mix": s_pt, "d": s_pt_d, "e": s_pt_e},
}
return s_dict, E_pt
def calc_r(
u, nominal_int_rate, inflation_rate, ace_int_rate, f, int_haircut, E, ace
):
r"""
Compute firm nominal discount rates
.. math::
r_{m,j} = f_{m,j}\[i(1-(1-i_{hc})u_{j})] + (1-f_{m,j})
(E_{j} + \pi - ACE_{j})
Args:
u (array_like): tax rate on business entity income
nominal_int_rate (array_like): nominal interest rate
inflation_rate (array_like): inflation rate
ace_int_rate (array_like): rate of return on equity that can
be deductions under allowance for corporate equity
f (array_like): fraction of investments financed with debt
int_haircut (array_like): haircut to interest paid deduction
E (array_like): dictionary of pre-tax required rate of return
ace (integer): allowance for corporate equity indicator
Returns:
r (array_like): nominal discount rate
"""
r = f * (nominal_int_rate * (1 - (1 - int_haircut) * u)) + (1 - f) * (
E + inflation_rate - ace_int_rate * ace
)
return r
def calc_r_prime(nominal_int_rate, inflation_rate, f, E):
r"""
Compute firm nominal, after-tax rates of return
.. math::
r^{'}_{m,j} = f_{m,j}(i-\pi) + (1-f_{m,j})E_{j} + \pi
Args:
nominal_int_rate (array_like): nominal interest rate
inflation_rate (array_like): inflation rate
f (array_like): fraction of investments financed with debt
E (array_like): dictionary of pre-tax required rate of return
Returns:
r_prime (array_like): nominal after-tax rate of return
"""
r_prime = f * nominal_int_rate + (1 - f) * (E + inflation_rate)
return r_prime
