Answer to Question #181971 in Financial Math for Anamika

Question

1.Find the PV of an annuity consisting of 60 quarterly payments of Rs. 200, the first being made at the end of 4 years 9 months. Assume that the rate of interest is 12.3% p.a. compounded monthly.

2.Find the number of years for which an annuity due of Rs. 1,500 payable per annum accumulates to Rs. 30,000 at the effective rate of interest of 9.1% per annum.

3.How much should be deposited in a bank each year in order to accumulate Rs. 50,000 in 6 years, if the interest is calculated at a rate of 6% per annum compounded continuously?

4.A person buys a house for which he agrees to pay Rs. 5,00,000 now and Rs. 5,000 at the end of every month for 6 years. If the money is worth 8% compounded quarterly, what is the cash price of the house?

Answer 1

(1)

We have to calculate the PV of annuity that starts at time t = 4 years 9 months and then discount this back to t=0 to get the value as of today.

$PV of annuity = PMT \times\frac{1 − (1 + i)^{−n}}{i}$

PV=present value

PMT=amount in each annuity payment

I=interest

n=number of payment

Step 1

Effective Annual Rats of the rate of interest:

12.3% compounded monthly

$= (1+\frac{r}{m})^{m}-1$

$= (1+\frac{0.123}{12})^{12}-1$

$=0.130176595 = 13.02$ %

Value of annuity at 4 years 9 months from today:

$PV = 200\times[\frac{1−(1+\frac{0.1302}{4})^{-60}}{\frac{0.1302}{4}}]$

$PV=5245.91901$

Step 2

We need to discount this value of $5,245.92 to today's value,

4 years and 9 months is $4\times4 +3 = 19$ quarters

$PV0=\frac{ PV1}{(1+r)^{19}}$

$=\frac{ 5,245.92 }{(1+\frac{0..1302}{4})^{19}}$

$=2,854.68$

 The present value of the annuity today is $2,854.68

(2)

Amount payable per annum = 1500

Accumulated after n years = 30,000

Let years = n

Interest rate = 9.1% per annum

As per the future value formula,

 $= 1500\times\frac{[(1+0.091)^{n}-1]}{0.091} = 30000$

$=\frac{(1.091)^{n}-1)}{0.091} = 30000/1500$

$=1.091^{n} = 20\times 0.091 + 1$

$1.091^{n }= 2.82$

Now let

$n=10 years$

Left hand side,

1.091^10= 2.3891725 (less than 2.82 )

We have to increase the years 

Let n=12 years

$1.091^{12} = 2.8437866$ (more than 2.82)

Therefore, n will be between 10 and 12

Using interpolation,

$n = \frac{12 - (12-10)\times(2.8437866-2.82)}{2.8437866-2.3891725}$

  $=\frac{ 12 - 2\times0.02378662}{0.45461413}$

 $= 12- 0.1046453$

  $= 11.8953$

  $= 11.90 years$

3.

We need to use the future value of annuity formula for continuous compounding to solve this problem

Future value of annuity = $CF\times \frac{e^{rt }−1}{e^{r }−1}$

Where:

CF=Annual cashflow

r= rate of interest

t=time

Future value of annuity =50000

$Rate(r)=6\%=0.06$

$Time(t)=6$

we need to use exponential function e^x,

$50000=CF\times \frac{e^{0.06\times 6}-1}{e^{0.6-1}}$

$50000=CF\times \frac{1.43332941456-1}{1.06183654654-1}$

$50000=CF\times 7.0076587197$

$CF=\frac{50000}{7.0076587197}=7135.05$

Hence annual deposit is $7,135.05

 

(4)

Money’s value declines with the time due to inflation and other factors. Discounting considers the money’s value and computes the PV of future amount by considering appropriate rate or PVF.

 Computation of the cash price,

Since, the payment interval and compounding period are not the same. So, the effective rate will be computed.

Result of above table:-

Hence, cash price is $7,83,250.2488