python - 在 numpy 利率函数中获取特定值集的负利率

Question

我是 numpy 的新手，正在使用 numpy.rate() 来计算固定利率每月还款贷款的年利率。

no_of_month = 24
payment = 8584
loan_amount = 50000
apr = rate(no_of_month,-payment,loan_amount,0.0) *12

apr 计算为 -22.816，但实际值应为 2.0102(在 LibreOffice Calc 中计算) )

对于 no_of_month = 23，apr 为 2.00079，但对于 no_of_month = 24，apr 是 -22.816

以下是我目前对为什么会发生这种情况的猜测(他们可能是错误的)

• 因为持续时间达到 12 的倍数(又一整年)
• numpy.rate 在一定范围内存在一些限制

我找不到任何相关资源。

根本原因是什么以及如何解决这个问题？

值的范围

no_of_month - 1 to 36
payment - 1000 to 1000000
loan_amount - 10000 to 10000000

所有组合都是可能的

Answer 1

np.rate 返回求解 x**24 多项式方程的利率。这有 24 个解决方案，其中一些可能是重复的，其中一些可能很复杂。对于此特定数据:

pv = 50000
payment = 8584 
mpr= np.rate(24, -payment, pv, 0.0)
np.pv(mpr, 24, payment)
# -49999.999999789325  This represents the 50000 pv 
mpr
# -1.901406995298687  #  mpr = -190.1% per month!

mpr1 = np.rate(24, -payment, pv, 0.0, guess = .15)
# guess lets you change the starting point for the search
mpr1
# 0.16750654293672343  # mar = 16.8% per month
np.pv(mpr1, 24, payment)
# -49999.99999999999

def apr(mpr, periods = 12):
    """  apr is ( 1 + monthly_rate ) ** 12 - 1 """
    return (1+mpr)**periods-1

apr(apr)
# -0.7122263079633477  apr = -71.2%

apr(mpr1)
# 5.4137477809069345  apr of 541.4%

即16.8% 和 -190.1% 都是方程的数学正确解。 -190.1% 在金融背景下没有多大意义。

分两个时期或许更容易理解。

loan_amount = 10
payment = 6
n_periods = 2

Solve 10 - 6r -6r**2
r = (6 +-sqrt(36-4*(-6)*10))/(2*-6)
r = -1.8844373105 and 0.8844373105

r = 1/(1+i) where i is the interest rate
i = 1/r - 1

r = -1.8844373105 
1/r-1
# -1.5306623862879625

r1 = 0.8844373105
1/r1-1
# 0.13066238627435212   

mpr = np.rate(2, -payment, loan_amount, 0.0)
mpr
# 0.13066238629183413

mpr1 = np.rate(2, -payment, loan_amount, 0.0, guess = -1.5)
mpr1
# -1.5306623862918336

在这种情况下，numpy 将求解两个速率，它们来自二次方程的两个根。

这可能没有帮助，但确实解释了为什么np.rate(和np.irr)可以解决意想不到的答案。

编辑:

我意识到，如果 r = 1/(1 +interest_rate)，则 r 只会有一个真正的正解。这是通常最具商业意义的解决方案。

import numpy as np
"""
    To simplify the analysis let 
        r = 1 / ( 1 + interest_rate )
        p = periodic payments
        n = number of periods
    then:
        pv = loan - p*r - p*r**2 - ... -p*r**n
        dpv/dr =  -p -2*p*r - ... -p*n*r**(n-1)
        If r > 0 and p > 0 dpv/dr is negative
        Therefore there is at most one solution to pv == 0 for r > 0.
        pv == loan when r == 0
        For large r -p*r**n will dominate and pv will be negative.
        Therefore there will be one positive real solution to pv == 0
"""

def polynomial_from(nper, loan, pay): 
    """ Create numpy array to represent the polynomial """
    return np.array([-pay]*nper+[loan])

# np.roots returns one root per nper.  Filter to real roots only.
def real_roots(poly):
    roots_ = np.roots(poly)
    return roots_[np.isclose(roots_.imag, 0)].real
    # return the real part of the roots- with a zero imaginary part

def feasible_rate(nper, loan, pay):
    poly = polynomial_from(nper, loan, pay)
    reals = real_roots(poly)
    r = reals[reals>0][0]   # r is the real root > 0
    return 1/r - 1

def apr(int_rate, nperiods = 12):
    return ( 1 + int_rate ) ** nperiods - 1

mpr = feasible_rate( 24, 50000, 8584 )
print( 'Monthly rate: {:%}, Annual Rate: {:%}'.format(mpr, apr(mpr)) )
# Monthly rate: 16.750654%, Annual Rate: 541.374778%