Interest Theory

Interest Theory#

Interest theory functions, that are in common actuarial and financial use, are reviewed. Interest rates are generally assumed to be fixed and constant.

Interest rates#

\(i\) is the amount earned on $1 after one year

  • effective annual interest rate

\(i^{(m)}\) is the nominal interest rate compounded m times per year

  • it is numerically equal to \(m\) times the effective interest rate over 1/m’th of a year.

\(d = \dfrac{i}{1 + i}\)

  • annual discount rate of interest

\(d^{(m)}\) is the nominal discount rate compounded m times per year

  • it is numerically equal to \(m\) times the discount rate over 1/m’th of a year.

\(v = \dfrac{1}{1 + i}\)

  • annual discount factor

\(\delta = \log(1+i)\)

  • continuosly-compounded rate of interest, or force of interest per year

Relationships between interest rates

\[\begin{split} \begin{align*} v^{t} & = e^{- \delta t} \\ & = (1-d)^{t} \\ & = (1+i)^{-t} \\ & = (1-\dfrac{d^{(m)}}{m})^{mt} \\ & = (1+\dfrac{i^{(m)}}{m})^{-mt} \\ \end{align*} \end{split}\]

Doubling the force of interest

Note that doubling the force of interest is not the same as doubling the rate of interest.

\(~^2\delta \leftarrow 2 \delta\)

\(~^2i \leftarrow 2 i + i^2\)

\(~^2d \leftarrow 2 d - d^2\)

\(~^2v \leftarrow v^2\)

Annuity certain

Present value of an annuity certain that pays at a rate of $1 per year for \(n\) years

\(\ddot{a}_{\overline{n|}} = \dfrac{1 - v^{n}}{d}\)

  • Annuity certain due: pays $1 at the beginning of the year

\(a_{\overline{n|}} = \dfrac{1 - v^{n}}{i} = \ddot{a}_{\overline{n+1|}} - 1\)

  • Immediate annuity certain: pays $1 at the end of the year

\(\overline{a}_{\overline{n|}} = \dfrac{1 - v^{n}}{\delta}\)

  • Continuous annuity certain: pays at a rate of $1 per year continuously.

Methods#

The Interest class implements methods to convert between nominal, discount, continuously-compounded and 1/m’thy rates of interest, and compute the value of an annuity certain.

from actuarialmath import Interest
import describe
describe.methods(Interest)

class Interest - Store an assumed interest rate, and compute interest rate functions
    
    Args:
      i : assumed annual interest rate
      d : or annual discount rate
      v : or annual discount factor
      delta : or continuously compounded interest rate
      v_t : or discount rate as a function of time
      i_m : or m-thly interest rate
      d_m : or m-thly discount rate
      m : m'thly frequency, if i_m or d_m are specified

    Methods:
    --------

    annuity(t, m, due):
      Compute value of the annuity certain factor

    mthly(m, i, d, i_m, d_m):
      Convert to or from m'thly interest rates

    double_force(i, delta, d, v):
      Double the force of interest

Examples#

The Interest class can be initialized with an assumed annual interest rate expressed in any one of the forms i, d, v, delta, i_m, d_m (where the latter two forms also require the argument m to specify the number of times compounded in a year). Then that rate can be retrieved in any other annual form as an attribute.

interest = Interest(i=0.05)
print(interest.d, interest.v, interest.delta, interest.i)

0.047619047619047616 0.9523809523809523 0.04879016416943205 0.05

The mthly static method converts between annual-pay and m-thly pay interest rates in any form.

delta = 0.05
i = Interest(delta=delta).i      # convert to annual interest rate
d = Interest(delta=delta).d      # convert to annual discount rate
i_m = Interest.mthly(i=i, m=12)  # convert to annual interest rate monthly-pay
d_m = Interest.mthly(d=d, m=12)  # convert to annual discount rate monthly-pay
print('Continuously-compounded annual rate of interest:', delta)
print('  Convert to annual interest rate:              ', i, Interest.mthly(i_m=i_m, m=12))
print('  Convert to annual discount rate:              ', d, Interest.mthly(d_m=d_m, m=12))
print('  Convert to annual interest rate (monthly-pay):', i_m)
print('  Convert to annual discount rate (monthly-pay):', d_m)

Continuously-compounded annual rate of interest: 0.05
  Convert to annual interest rate:               0.05127109637602412 0.051271096376023007
  Convert to annual discount rate:               0.04877057549928606 0.04877057549928587
  Convert to annual interest rate (monthly-pay): 0.05010431149342143
  Convert to annual discount rate (monthly-pay): 0.049895977858680496

The annuity class method computes the present value of an annuity that pays at the rate of $1 per year continuously or discrete (due, immediate, or m’thly)

import matplotlib.pyplot as plt
t = range(100)
plt.plot(t, [interest.annuity(t=s, due=True) for s in t])
plt.title(f'Present Value of Annuity Certain Due (i={interest.i * 100}%)')
plt.xlabel('Period (years)')
plt.ylabel('Annuity Due Factor')
print("Annuity Factors By Payment Frequency (perpetual, i=5%):")
print('1. Immediate:                 ', interest.annuity(t=Interest.WHOLE, due=False))
print('2. Immediate (quarterly pay): ', interest.annuity(t=Interest.WHOLE, m=4, due=False))
print('3. Continuous:                ', interest.annuity(t=Interest.WHOLE, m=0))
print('4. Due (quarterly pay):       ', interest.annuity(t=Interest.WHOLE, m=4, due=True))
print('5. Due:                       ', interest.annuity(t=Interest.WHOLE, due=True))

Annuity Factors By Payment Frequency (perpetual, i=5%):
1. Immediate:                  19.999999999999982
2. Immediate (quarterly pay):  20.371188429095998
3. Continuous:                 20.49593431428785
4. Due (quarterly pay):        20.621188429096076
5. Due:                        20.999999999999975
_images/a0dff2cf9ea9f53925344c235ba42eebb0f56afd390dd6a522cc9e1eb50408cd.png

The double_force static method takes an interest rate in any form, and converts to the same form after doubling the force of interest.

print("Example: double the force of interest i=0.05")
i = 0.05
d = Interest(i=i).d               # convert interest rate to discount rate
print('i:', i, 'd:', d)
i2 = Interest.double_force(i=i)   # interest rate after doubling force
d2 = Interest.double_force(d=d)   # discount rate after doubling force
print('i:', round(i2, 6), round(Interest(d=d2).i, 6))
print('d:', round(d2, 6), round(Interest(i=i2).d, 6))

Example: double the force of interest i=0.05
i: 0.05 d: 0.047619047619047616
i: 0.1025 0.1025
d: 0.092971 0.092971

SOA Question 3.10:

A group of 100 people start a Scissor Usage Support Group. The rate at which members enter and leave the group is dependent on whether they are right-handed or left-handed. You are given the following:

  • The initial membership is made up of 75% left-handed members (L) and 25% right-handed members (R)

  • After the group initially forms, 35 new (L) and 15 new (R) join the group at the start of each subsequent year

  • Members leave the group only at the end of each year

  • \(q^L\) = 0.25 for all years

  • \(q^R\) = 0.50 for all years Calculate the proportion of the Scissor Usage Support Group’s expected membership that is left-handed at the start of the group’s6 th year, before any new members join for that year.

print("SOA Question 3.10:  (C) 0.86")
interest = Interest(v=0.75)
L = 35 * interest.annuity(t=4, due=False) + 75 * interest.v_t(t=5)
interest = Interest(v=0.5)
R = 15 * interest.annuity(t=4, due=False) + 25 * interest.v_t(t=5)
print(L / (L + R))

SOA Question 3.10:  (C) 0.86
0.8578442833761983
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