Total Inc. recently purchased a new office building costing $15 million. The firm financed this purchase at 6 percent APR with quarterly compounding. Quarterly payments starting from next quarter will be $400,000. How many years will it take the firm to pay off this debt

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

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

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