Respuesta :
Answer:
i = 3.25%
Explanation:
Ordinary simple annuity is one where the payments are made at the end of every interest period. The payments and the interest periods coincide. So we have $1,400 monthly payments for the period of 15 years, that means we have, 12 payments per month for 15 years and for 15 years we are gonna have 180 monthly payments of $1,400 and it is given that it at the end of every month. In exchange for those $1,400 per month payments we get $200,000. So we know from above that we are going to use Present Value formula. We know that the total amount we will be paying in 15 years is $1,400 times 180 monthly payments is equal to $252,000. The total interest paid during the period of 180 months will be the difference between the money in and money out which is $52,000.
PV = PMT [tex][\frac{1 - (1 + i)^{-n} }{i}][/tex]
where;
PV = Present Value = $200,000
PMT = The amount in each annuity payment = $1,400
i = Rate of Interest = ?
n = The number of payments = 15 x 12 = 180 payments
So, by putting in the values above in the formula we get;
200000 = 1400 [tex][\frac{1 - (1 + i)^{-180} }{i}][/tex]
Divide the whole equation by 1400 we get;
142.8571 = [tex][\frac{1 - (1 + i)^{-180} }{i}][/tex]
We have the value 142.8571 above and the significance of the 142.8571 says that if you were not paying any interest at all, if there was 0% interest rate, it will only take approximately 143 payments of $1,400 to pay off that $200,000 which means that 142.8571 (approx: 143) payments multiplied by $1,400 is equal exactly $200,000.
But the reality is that there is interest and with interest it will take 180 payments of $1,400. So, now we have to use brute force to solve the above equation for i. In order to do that we have to assume the value of i the one higher than 142.8571 and the other below 142.8571 so we can find the exact interest rate as follows:
So, lets assume if;
0.0025 => 144.8055
i => 142.8571
0.0030 => 138.9268
Above is our table, we got lower rate of 0.0025 at the top giving us a factor of 144.8055 and i per month factor which should give us a factor of 142.8571 and a higher rate of 0.0030 at the bottom giving us a factor of 138.9268. We can say that the higher the rate is the lower the present value factor is, So now let us come up with ratio of differences as follows:
[tex]\frac{(i - 0.0025)}{(0.0030 - 0.0025)} = \frac{(142.8571 - 144.8055)}{(138.9268 - 144.8055)}[/tex]
Solve this equation for i;
[tex]\frac{i - 0.0025}{0.0005} = \frac{-1.9483}{-5.8786}[/tex]
[tex]\frac{i - 0.0025}{0.0005} = 0.3314[/tex]
Multiplying the equation by 0.0005 we get and By adding 0.0025 on both sides we get;
i = 0.002666 compounded monthly
From above we can calculate our rate compounded annually as;
[tex](1 + 0.002666)^{12} = (1 + i)[/tex]
The above equation states that the effective rate (compounded annually rate) is 1 plus rate per month and in 1 year we will have 12 monthly periods and that is equal to 1 plus rate per year compounded only once, so 1 yearly period will have 12 monthly periods, so let us solve the above equation for i (the rate per year) as;
1.0325 = 1 + i
Subtracting 1 from both sides we get;
i = 0.0325
or i = 3.25%
