Calculate interest paid at n years, interest compounded monthly vs. semi-annually

Starting with the standard loan equation based on the loan amount s equalling the sum of the periodic repayments d all discounted to present value.

s = principal
r = periodic rate
n = number of payments
d = payment amount

Formulae for s, d & n are:-

  s = (d - d (r + 1)^-n)/r
∴ d = r s/(1 - (1 + r)^-n)
& n = - log(1 - (r s)/d)/log(1 + r)

Adding values: assuming 5% effective annual interest for the whole 25 year amortisation period.

(The calculation for just 5 years is shown at the end.)

Semi-annual payments

  s = 500000
  n = 25*2 = 50
  r = (1 + 0.05)^(1/2) - 1 = 0.0246951 semi-annually
∴ d = r s/(1 - (1 + r)^-n) = $17,521.76 per half-year
total interest = d*n - s = $376,088.18

Monthly payments

  s = 500000
  n = 25*12 = 300
  r = (1 + 0.05)^(1/12) - 1 = 0.00407412 monthly
∴ d = r s/(1 - (1 + r)^-n) = $2,890.69 per month
total interest = d*n - s = $367,207.28
difference = 376,088.18 - 367,207.28 = $8,880.90

Note, 5% effective interest means the same rate is applied regardless of the compounding interval, so the difference in total interest is due only to the earlier repayments in the monthly amortisation.

The nominal rates equivalent to 5% effective annually are:-

((1 + 0.05)^(1/2) - 1)*2 = 4.93902 % compounded semi-annually
((1 + 0.05)^(1/12) - 1)*12 = 4.88895 % compounded monthly

Ref. Wikipedia: Effective rate calculation

The compounding interval should be specified for a nominal rate otherwise different intervals would mean different effective rates further affecting the total interest paid. E.g.

5% nominal semi-annually = (1 + 0.05/2)^2 - 1 = 5.0625 % effective
5% nominal monthly = (1 + 0.05/12)^12 - 1 = 5.11619 % effective

5% compounded monthly is actually a higher rate than 5% compounded semi-annually, so this wouldn't lead to a fair comparison.

Nevertheless, the difference in total interest with these rates is . . .

Semi-annual payments

  s = 500000
  n = 252 = 50
  r = 0.05/2 = 0.025 semi-annually
∴ d = r s/(1 - (1 + r)^-n) = $17,629.03 per half-year
total interest = dn - s = $381,451.42

Monthly payments

  s = 500000
  n = 2512 = 300
  r = 0.05/12 = 0.00416667 monthly
∴ d = r s/(1 - (1 + r)^-n) = $2,922.95 per month
total interest = dn - s = $376,885.06
difference = 381,451.42 - 376,885.06 = $4,566.36

This time the total interest amounts are both greater while the difference is less. But this scenario should never arise because only equivalent nominal rates should be compared.

To calculate for just 5 years the initial regular payment amount would probably be calculated by assuming 5% for the whole 25 years, to be adjusted after the 5 year term.

balance after x periods = (d + (1 + r)^x (r s - d))/r

Based on 5% effective annual interest.

Semi-annual payments

  s = 500000
  n = 252 = 50
  r = (1 + 0.05)^(1/2) - 1 = 0.0246951
∴ d = r s/(1 - (1 + r)^-n) = 17521.76
x = 52
balance = (d + (1 + r)^x (r s - d))/r = $442,112.22
interest paid = dx - (s - balance) = $117,329.86

  s = 500000
  n = 2512 = 300
  r = (1 + 0.05)^(1/12) - 1 = 0.00407412
∴ d = r s/(1 - (1 + r)^-n) = 2890.69
x = 512
balance = (d + (1 + r)^x (r s - d))/r = $442,112.22 (as above)
interest paid = dx - (s - balance) = $115,553.68
difference = 117,329.86 - 115,553.68 = $1,776.18

After 5 years the semi-annual payment scheme has cost £1,776 more.

(To find the difference it is not actually necessary to calculate the balance because (s - balance) is the same in both cases.)

Checking

Two online Canadian mortgage calculators calculated the regular monthly payment for a 25 year monthly amortisation at 5% as $2908. That works out at 5.0625 % effective, as calculated above for a semi-annual rate:

5% nominal semi-annually = (1 + 0.05/2)^2 - 1 = 5.0625 % effective

So Canadian rates appear to be quoted as nominal semi-annual for a monthly amortisation. I recall something like this is a quirk of Canadian mortgages. (Semi-annual payment is not an option on these calculators.)

i.e Monthly payments

  s = 500000
  n = 2512 = 300
  eff = (1 + 0.05/2)^2 - 1 = 0.050625
  r = (1 + eff)^(1/12) - 1 = 0.00412392 per month
∴ d = r s/(1 - (1 + r)^-n) = $2,908.02
total interest = dn - s = $372,407.48

The bi-weekly calculation is kind of funny

  s = 500000
  n = 2526 = 650
  eff = (1 + 0.05/2)^2 - 1 = 0.050625
  rmonthly = (1 + eff)^(1/12) - 1 = 0.00412392 per month
  r = rmonthly12/26 = 0.00190335
∴ d = r s/(1 - (1 + r)^-n) = $1,341.41
total interest = dn - s = $371,918.34

$2,908.02 ÷ 2 = $1,454.01

s = 500000
n = 2512 = 300
eff = (1 + 0.05/2)^2 - 1 = 0.050625
r = (1 + eff)^(1/12) - 1 = 0.00412392 per month
d = r s/(1 - (1 + r)^-n) = 2908.02

Redefine d & r and calculate n periods to repay

d = d/2 = 1454.01
r = (1 + eff)^(1/26) - 1 = 0.00190124
n = - log(1 - (r s)/d)/log(1 + r) = 558.435

Take the whole number of periods and calculate the balance

x = 558
b = (d + (1 + r)^x (r s - d))/r = 631.47