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

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

• Google "the compound interest formula" and solve for `r`. Plug in the independent variables and... voila you get the interest rate. Ditto the "the simple interest formula". – RonJohn Dec 30 '19 at 2:26
• @RonJohn I made an error. All 3 questions should have an APR of 3.99%. And now, how to calculate the answers? Because there is a monthly payment, this changes interest accrued over time. – FrozenLamb Dec 30 '19 at 11:01
• The formulae are out there. These are homework problems, so you must have a textbook of some sort. And if you don't, Google can tell you what the formulae are. Or you can use Excel. – RonJohn Dec 30 '19 at 11:06
• @FrozenLamb Welcome to the site, be aware that while homework questions are sometimes okay (see Homework section of the help page) You have to show some effort to solve it and can't just paste the assignment and expect the users of this site to do your work for you. – JohnFx Jan 1 at 18:36
• I'm voting to close this question as off-topic because homework questions cannot just be a copy-paste of the assignment and must show some effort on the part of the asker. – JohnFx Jan 1 at 18:37

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.

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

$s=\sum_{k=1}^{n}\frac{d}{(1+r)^k}=\frac{d-d(r+1)^{-n}}{r}$

$\therefore d=rs(\frac{1}{(1+r)^n-1}+1)$

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

and

$n=-\frac{log(1-\frac{r s}{d})}{log(r+1)}$

``````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

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