Respuesta :
Answer:
13.88 years
Explanation:
Using the formula for present value for an annuity with constant payment of $ 400000
PV = P ( 1 - ( 1+r)^-n) / r
where P = $ 400000, r = 6%/4 = 1.5 % = 0.015 and PV = $15 million
$15 million = $400000(1 - ( 1 + 0.015) ^-n / 0.015
$ 15 000000 × 0.015 / $ 400000 =(1 - ( 1 + 0.015) ^-n)
( 1 + 0.015) ^-n = 1 - 0.5625
1.015^-n = 0.4375
take log of both side
-n log 1.015 = log 0.4375
-n = log 0.4375 / log 1.015 = -55.52
n = 55.52
in years = 55.52 / 4 =13.88 years
Answer:
13.88 years
Explanation:
Since the quarterly payments starting from next quarter, the relevant formula to use is the present value (PV) of an ordinary annuity formula stated as follows:
PV = P × [{1 - [1 ÷ (1+r)]^n} ÷ r] …………………………………. (1)
Where;
PV = Present value or the office building cost today = $15,000,000
P = quarterly payment = $400,000
r = APR = 6%, 0.06 annually = 0.06/4 quarterly = 0.015
n = number of quarters = ?
Substitute the values into equation (1) to have:
15,000,000 = 400,000 × {1 - [1 ÷ (1 + 0.015)]^n} ÷ 0.015
15,000,000 × 0.015 = 400,000 × {1 - [1 ÷ (1.015)]^n}
225,000 ÷ 400,000 = 1 - [1 ÷ 1.015]^n
0.5625 = 1 - (0.985221674876847)^n
0.5625 - 1 = - (0.985221674876847)^n
- 0.4375 = - (0.985221674876847)^n
Both sides by minus one, we have:
(0.985221674876847)^n = 0.4375
Loglinearising both sides, we have:
n × ln(0.985221674876847) = ln(0.4375)
n = ln(0.4375) ÷ ln(0.985221674876847)
n = (- 0.826678573184468) ÷ (- 0.0148886124937505)
n = 55.52 quarters
We can now convert it to years by as follows:
n = 55.52 quarters ÷ 4 = 13.88 years
