Question

A perpetuity paying 1.000$ at the end of each month is replaced with an annuity paying X$ each month for 10 years. Calculate X if j4 = 7%

If the annuity paid 2.500$ at the end of each month, how long would the annuity last and what would be the size of the final, smaller payment?

Answer 1

Solution:

j4=7%

i = 7/4 % = 1.75% p.a = 0.0175

PV of perpetuity $=\dfrac{\text{Amount of continuous cash payment}}{\text{Interest rate or yield}}$

$=\dfrac{\$ 1}{0.0175}=\$ 57\dfrac17$

PV of annuity $=P[\dfrac{1-(1+r)^{-n}}{r}]$

Given, $P=\$X, r=0.0175,n=10$

So, PV of annuity $=X[\dfrac{1-(1+0.0175)^{-10}}{0.0175}]=9.10122X$

Now, $9.10122X=57\dfrac 17$

$\Rightarrow X=\$6.278$

Now, $P=\$2.5,n=?$

PV of annuity $=P[\dfrac{1-(1+r)^{-n}}{r}]$

$PV=\$ 57\dfrac17$

$57\dfrac17=2.5[\dfrac{1-(1+0.0175)^{-n}}{0.0175}] \\\Rightarrow 0.4=1-(1+0.0175)^{-n} \\\Rightarrow (1.0175)^{-n}=0.6 \\\Rightarrow -n\log 1.0175=\log 0.6 \\\Rightarrow n=29.44\approx 30$

Now, for n=30,

$PV=2.5[\dfrac{1-(1+0.0175)^{-30}}{0.0175}]=57.964$

Final payment will be $=57.964-57\dfrac17=\$0.823$