Question

How much should each payment be for a 22-years term loan of $250,000 that charges 4.5% APR (compounded quarterly) if the payments are

a) Monthly,

b) Semi-annual,

c) Quarterly,

d) Annual.

Answer 1

Annual Interest Rate = 4.50%
Quarterly Interest Rate = 1.125%

Effective Annual Rate = (1 + Quarterly Interest Rate)^4 - 1
Effective Annual Rate = (1 + 0.01125)^4 - 1
Effective Annual Rate = 1.04577 - 1
Effective Annual Rate = 0.04577 or 4.577%

Amount Borrowed = $250,000

Answer a.

Monthly Interest Rate = (1 + Effective Annual Rate)^(1/12) - 1
Monthly Interest Rate = 1.04577^(1/12) - 1
Monthly Interest Rate = 1.00374 - 1
Monthly Interest Rate = 0.00374 or 0.374%

Period = 22 years or 264 months

Let Monthly Payment be $x

$250,000 = $x/1.00374 + $x/1.00374^2 + … + $x/1.00374^263 + $x/1.00374^264
$250,000 = $x * (1 - (1/1.00374)^264) / 0.00374
$250,000 = $x * 167.58135
$x = $1,491.81

Monthly Payment = $1,491.81

Answer b.

Semiannual Interest Rate = (1 + Effective Annual Rate)^(1/2) - 1
Semiannual Interest Rate = 1.04577^(1/2) - 1
Semiannual Interest Rate = 1.02263 - 1
Semiannual Interest Rate = 0.02263 or 2.263%

Period = 44 (22 years)

Let Semiannual Payment be $x

$250,000 = $x/1.02263 + $x/1.02263^2 + … + $x/1.02263^43 + $x/1.02263^44
$250,000 = $x * (1 - (1/1.02263)^44) / 0.02263
$250,000 = $x * 27.68091
$x = $9,031.49

Semiannual Payment = $9,031.49

Answer c.

Monthly Interest Rate = 1.125%

Period = 22 years or 88 quarters

Let Quarterly Payment be $x

$250,000 = $x/1.01125 + $x/1.01125^2 + … + $x/1.01125^87 + $x/1.01125^88
$250,000 = $x * (1 - (1/1.01125)^88) / 0.01125
$250,000 = $x * 55.67678
$x = $4,490.20

Quarterly Payment = $4,490.20

Answer d.

Annual Interest Rate = 4.577%

Period = 22 years

Let Annual Payment be $x

$250,000 = $x/1.04577 + $x/1.04577^2 + … + $x/1.04577^21 + $x/1.04577^22
$250,000 = $x * (1 - (1/1.04577)^22) / 0.04577
$250,000 = $x * 13.68587
$x = $18,267.01

Annual Payment = $18,267.01