How does one calculate the following loan problems? [closed]

Question

1.) Calculate the total interest to be paid on the loan during it's lifetime. Compute for BOTH compound interest and simple interest models.

Start Principal = $21,315.00, Date of Disbursement = 9/16/19, APR = 3.99%, Payment monthly: $333.41 on Day 16 of month, Compound Period = Daily (360 days/yr)

---

2.) Calculate the remaining term (number of payments until PV = 0) in months of the following loan. Compute for BOTH compound and simple interest models.

PV = $20789.41, PV Date = 11/1/19, Payment = $300 per month on day 16 of month, Compound Period = Daily (365 days/yr),

---

3.) Calculate the total remaining interest to be paid on the loan during the rest of its lifetime from it's present value. Compute for Both compound and simple interest models.

PV = $15,020.00 PV Date = 10/8/19 Payment = $208.25 per month on day 12 of month Compound Period = Monthly

Answer 1

Question 2 requires the more complex answer because the number of days in each month varies, although it can be calculated using a spreadsheet. Some of the other parts can be answered using formulae which I have attempted to do.

You have not stated your simple interest model so some suppositions are made in the answer to question 1. Comments welcome.

1.

Start Principal = $21,315.00
Date of Disbursement = 9/16/19
APR = 3.99%
Payment monthly: $333.41 on Day 16 of month
Compound Period = Daily (360 days/yr)

Using the following variables

s = principal
r = periodic rate
n = number of periods
d = periodic payment

Standard loan equation

d = r s (1/((1 + r)^n - 1) + 1)        Formula 1

and

s = 21315
r = 3.99/100 * 30/360
d = 333.41

∴ n = -(log(1 - (r s)/d)/log(r + 1)) = 71.9927

You could round this to 72 months, in which case the term is 6 years.

total interest = 72 d - s = 2690.52

Simple interest over 6 years would result in

total interest = s * 3.99/100 * 6 = 5102.81

However, if the term is unknown for the simple interest application it can be calculated since the interest plus principal should equal the sum of the payments

s r n + s = n d

∴ n = s/(d - r s) = 81.1884 months

∴ total simple interest = 81.1884 d - s = 5754.02 approximately

Some rather high figures using simple interest.

---

3.

PV = $15,020.00
PV Date = 10/8/19
APR = 3.99%
Payment = $208.25 per month on day 12 of month
Compound Period = Monthly

Find the balance on the 12th, after 4 days' interest.

s = 15020
r = 3.99/100/12

dailyrate = (1 + r)^(12/365) - 1 = 0.00010914
s = s (1 + dailyrate)^4 = 15026.56

Step back one month to use the regular formulae (with payment at month-end)

s = s/(1 + r) = 14970.22

∴ n = -(log(1 - (r s)/d)/log(r + 1)) = 82.2865 months

Using formula 2 to find the balance after 82 months

  x  =  82
∴ b  =  (d + (1 + r)^x (r s - d))/r  = 59.5334

total interest = 82 d + b (1 + r) - 15020 = 2116.23

I don't have time to check out a simple interest calculation for question 3, so I'm posting the calculations as they are. Hopefully they're of some help or serve as a conversation starter.

Formulae

1. Formula for periodic payment - loan payment formula

2. Formula for loan balance - inhomogeneous difference equation (Arne Jensen, Aalborg Uni.)