Annual cost of living increases

Section 3.4 Annual cost of living increases

Generally, an annuity is a sequence of equal payments over time. In reality, people like to use an annuity to fund retirement and one might want to allow for the payments to increase each year to keep up with the cost of living. Another alternative occurs perhaps when one might purchase a house and not be able to make the necessary payments at the beginning of the annuity but expects to earn more and therefore be able to pay more in later years of the loan. To analyze these situations takes a bit more care since the compounding period and the (annual) increases are not necessarily on the same time frames. To fix this, the idea is to convert the periodic payments in terms of years by converting, for example, the 12 monthly payments into a single but equivalent lump sum payment at the end of the year. So long as the lump sum payment is mathematically equivalent to the sequence of payments then all is equivalent. Then, we can tack on the desired cost of living increase to these annualized payments and use that rate of increase for the yearly annuity.

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Theorem 3.4.1. Yearly Annuity with Cost of Living Adjustment.

The Present Value of a yearly annuity initially in the amount of

$R_y$ which includes yearly cost-of-living adjustments is

$R_y \frac{1}{1+r_e} \left [ \frac{ \left ( \frac{1+a}{1+r_e} \right )^t-1}{ \left (\frac{1+a}{1+r_e} \right )-1} \right]$ with

$R_y$ = first year's payment,

$r_e$ = effective yearly rate, a = yearly cost-of-living increase, t = number of years.

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Proof.

To save some time, let's define $z = \frac{1+a}{1+r_e}\text{.}$ Notice that if the cost of living increase (a) is greater than the effective rate of interest $r_e$ then the net investment will grow with a interest rate related to $z-1 \gt 0\text{.}$ On the other hand, if the cost of living increase is less than the effective rate of interest, then the net investment wil grow at an interest rate $z-1 \lt 0\text{...}$that it, your periodic payments will not be keeping up with inflation. Notice in the formulas below that this quotient z essentially replaces the standard compound growth factor (1+r).

Starting with a payment of R at the end of the first year, increase this by a factor of (1+a) for each subsequent year gives a present value of

\begin{align*} P & = R_y (1+r_e)^{-1} + R_y (1+a)(1+r_e)^{-2} + R_y (1+a)^{2}(1+r_e)^{-3} + R_y (1+a)^{3}(1+r_e)^{-4} + ... + R_y (1+a)^{t-1}(1+r_e)^{-t}\\ & = R_y \left [ (1+r_e)^{-1} + (1+a)(1+r_e)^{-2} + (1+a)^{2}(1+r_e)^{-3} + (1+a)^{3}(1+r_e)^{-4} + ... + (1+a)^{t-1}(1+r_e)^{-t} \right ]\\ & = R_y \frac{1}{1+r_e} \left [ 1 + z + z^2 + z^3 + ... + z^{t-1} \right ]\\ & = R_y \frac{1}{1+r_e} \left [ \frac{1-z^t}{1-z} \right] \\ & = R_y \frac{1}{1+r_e} \left [ \frac{z^t-1}{z-1} \right] \end{align*}

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Obviously, one may want to receive these payments more often than once a year but account for yearly adjustments to the payments. In this situation, one should determine the size R of the m payments in the first year and then find the accumulated value of those payments at the end of one year. Taking that value as R in the theorem above gives the present value of the cost-of-living adjusted annuity stream. Of course one can take that present value and determine the value of the income stream at any other desired time.

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Example 3.4.2. Yearly Annuity with Cost of Living Increase.

Let's consider the value of making payments of $\$ 10$ per year in year one and then increasing the payments by a rate of a = 4% each year. Assuming an interest rate of $r_e$ = 3% compounded yearly for t = 8 years, what is the present value of this annuity?

$\begin{align*} z = \frac{1.04}{1.03} \approx 1.0097\\ P & \approx \$ 10 \frac{1}{1.03} \left [ \frac{1.0097^8-1}{0.0097} \right]\\ & = \$ 10 \frac{1}{1.03} 8.2769334 = \$ 80.36 \end{align*}$

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Example 3.4.3. Monthly Annuity with Annual Cost of Living Increase.

To allow for more frequent payments, say monthly, but with yearly adjustments to the amount of these payments one can determine the future value of the initial payments over one year. Then, use that value as a "yearly" payment adjusted for cost-of-living increase.

Consider the following: Initial monthly payments of $\$ 10$ per month with interest compounded at 3% and with 4% yearly cost-of-living adjustments for 8 years. First, compute the future value $R_y$ of a year of payments and then apply the theorem above.

$\begin{gather*} r_e = \left ( 1 + \frac{0.03}{12} \right )^{12}-1 \approx 0.0304159\\ R_y = \$ 10 \frac{(1 + \frac{0.03}{12})^{12}-1}{\frac{0.03}{12}} = 121.66\\ z = \frac{1.04}{1.0304159} \approx 1.00930119576 \\ P = \$ 121.66 \frac{1}{1.0304159} \left [ \frac{1.00930119576^8-1}{0.00930119576} \right] = \$ 975.88 \end{gather*}$

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The interactive cell below determines the effect of compounding and cost of living increases for each payment of an annuity.

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def round(x):
    return (floor(100*(x+0.005))/100).n()
    
@interact(layout=dict(top=[['R', 'rate'],['m','t'],['a']]))
def _(R = input_box(10,label='Yearly Payment',width=10),
           rate = input_box(0.03,label='Nominal Rate',width=6), 
           m = input_box(1,label='Periods per Year',width=4), 
           t = input_box(8,label='Years',width=4), 
           a = input_box(0.04,label='Cost-of-Living Adjustment',width=6) ):
    r = rate/m
    n = m*t
    A = 0
    for k in range(t):
        R1 = round(R*(1+a)^(k))
        print "Year ",k+1," has payment(s) of ", R1
        yearend = k*m
        print "yearend ",yearend
        for j in range(m):
            tempBeg = R1*(1+r)^(-(yearend+j+1))
            tempEnd = R1*(1+r)^(n-yearend-j-1)
            A += tempEnd
            print "  ",j+1,"th payment is worth at the start of time ",tempBeg
            print "  ",j+1,"th payment is worth at the end of time ",tempEnd
    P = A*(1+r)^(-n)
    print "Total value of payments at the start of time is ",P
    print "Total value of payments at the end of time is ",A