# Loans: How do they work?

I wonder how loans work.

On the picture, you see 2 loans I took.

1. The first one has been running for 1 single tick, reducing by \$133
2. The second one has been running for 3 ticks already, reducing by \$64 each

I do not quite understand the loan system there…
I suppose the amount taken form the capital (the amount of the loan itself) is the one you pay back each day. Applying the advertised interest rate:

That means:

1. I should pay 120% of the daily paid back capital each and every day = ~\$160 -> Where does \$243 comes from?
2. I should pay 113% of the daily paid back capital each and every day = ~\$73. This one I get

However these numbers are round up, so that means some daily payments should disappear in the end and the repayment duration should be shorter… Is it?

Any point which is disturbing: loans duration.
What does ‘3 years’ or ‘12 months’ mean? You need to define ‘year’ and ‘month’ as part of the gameplay.

1. \$200000 * 120% = \$240000
\$240000 / \$243 = 988 days
\$240000 / \$160 = 1500 days
\$240000 / (\$133 * 120%) = 1504 days
With months of 31 days, you get 3 * 12 * 31 = 1116 days. With years of 365 days, you get 3 * 365 = 1095 days.
Where does these values come from?
2. \$25000 * 113% = \$28250
\$28250 / \$73 = 387 days
\$28250 / (\$64 * 113%) = 391 days
Even with months of 31 days, you cannot go higher than 12 * 31 = 372 days…

I guess the loan system seems to have quite a few quirks… or am I missing something?

I imagine this is based on how loans work in real life. Every day, the interest and payment are applied to the balance. To get the interest for the first day, we apply the compound interest formula (compounded daily):

200000(1+0.2/365) = \$110 (rounded to the nearest dollar)

That’s 200000k balance and 0.2 (20%) interest divided by 365.

So to get the balance after the first day, take the previous balance (\$200k), add the \$110 interest, and subtract the \$243 payment: 200000 + 110 - 243 = 199867

The consequence of this is that like loans in real life, the amount you reduce the balance by each day goes up over the life of the loan. At first, you are paying a lot of interest and barely paying off the balance at all, but by the end you are paying almost no interest, so your payment goes mostly towards paying it off.

Have you looked at the date in the lower right hand corner? Every tick in big pharma (where the conveyors move one tile) is one day in a 365 day long year. Years are also used gauge how much time you have to complete scenarios.

This follows from what I was talking about earlier. Because you pay interest on the current balance every day, the amount of the loan you actually pay off varies as the balance changes. To get the amount that you end up paying for each loan, the proper calculation is the cost per day multiplied by the period of the loan (where years and months correspond to actual years and months of game time). So for the loan on the left, it will end up costing you 2433365 or \$266085 over the life of the loan. This is more than 120% of your principal since the interest is compounded over 3 years, not one, and is less than 160% of your principal since the balance which you pay interest on is reduced over time.

It’s pretty much just how loans work in real life, which may be confusing if you aren’t familiar with economics.

Thanks for all this information.

The correct label for the interest would be N%/year, then.

Some things I am still wondering about:

1. How is the daily payment calculated?
Is it the lender’s choice? Are there ‘standard rules’ (other then being more profitable than a basic capital + interest formula)?
2. Since values are rounded (you said ‘to the nearest value’. Is is true, and not systematically up nor down?), that means the loan might be a little bit longer or shorter the more lengthy is the loan duration, is it not?

Another question (I cannot edit my last post anymore, could someone merge them?):

• III. Since the interest rate seems to be defined per year, how is the daily compounded interest adjusted when the loan duration is less than a year? It should mechanically be lower than the real one.
That would be good for the debitor but not for the creditor. I suspect bank do not give you a chance, so what trick do they have up their sleeve? Adjusting in this case (and only that way) the annual rate to the duration of the loan?

Yes, the interest could be labeled as annual interest compounded daily, but I think it’s generally assumed that interest is annual and that for loans it’s compounded every time a payment is made (often IRL it’s monthly compounding and monthly payments).

The daily payment is calculated from the interest rate, principal, and term such that the balance will run out exactly when the term runs out. The exact formula to calculate the payment is a bit complicated. Here it is in image form:

Running that formula for the loan on the left in your original picture:
PV = 200000
r = 0.2 / 365 = 0.0005479452
n = 3 * 365 = 1095

Then P = (0.0005479452 * 200000) / (1 - (1.0005479452)^(1095)) = \$243

Figuring that out would require some more testing, but in all the cases I have calculated so far the seem to round to the nearest value and/or round up. This would theoretically mean that the loan duration could be a little bit off unless there was some fudge factor that corrected it. I’d assume that the remaining term shows the actual time remaining (not the time based on the original calculation).

There’s no fundamental change when the duration is less than a year. In either case, the interest paid each day is the annual interest rate divided by 365. Longer loans make the bank more money since they are compounded over a longer period of time, but the daily interest is always one 365th of the annual interest. Also, as far as I am aware the shortest big pharma loans are 1 year long (shown as 12 months).

For the sake of clarity, allow me to correct some typos first:

That should be 200000 * 0.2 / 365) = \$110
200000 * (1 + 0.2 / 365) are the daily interests added to the capital for the first compound interests iteration.

This formula is wrong from a semantic point of view: r is both used as a value and a function there…
r should be a variable, and r(PV) should be written as r * PV

r * PV = 0.2 / 365 * 200000
Minux sign forgot on the exponent: P = (0.2 / 365 * 200000) / (1 - (1 + 0.2 / 365)^-(1095)) = \$243 (keeping ratio rather than multi-decimals, rounded, values preserves correctness).

We could also assume the rounding operation is only done on display, not inside calculations (which is the way you do it, otherwise you screw the results up).
Loads of assumption around here ;o)

I disagree with that:
If the loan lasts less than a year but the daily payment is computed on 365 days, the creditor loses loads of money, since the due interests are made up to vanish at the same time as the due capital: at the expiration of the loan.

If your loan lasts less than a year, your creditor will most probably recompute the daily payment to make it scale.
For example, take the second loan and modify it to be shorter: (\$25000, 13%, 3 months)
The daily payment will be :
0.13 / 90 * 25000 / (1 - (1 + 0.13 / 90)^-90) = \$296.42
Otherwaire, after 90 days, you will fall short of having paid back what is due… ;o)

You’re missing a parenthese in 200000 * (0.2 / 365) = \$110, otherwise you are correct, I forgot to adjust the formula to give only the interest rather than the total amount after interest. I just ignored the 200000 when I ran the calculation.

That’s sort of a grey area in notation because the notation “x(y)” can mean either “x * (y)” or “x of y”. In this case, since there is no function called r defined, it’s assumed that it means multiplication. It would be less ambiguous to write “r * (PV)” or even “(PV)r” but the way it is written is technically correct. That image was just the first thing I found when looking up the formula, there are probably nicer ways to write it.

Considering that I was going to round the result to 3 significant figures, I don’t think 7 significant figures is significantly less precise than 0.2 / 365.

Yes, I forgot the negative sign on the exponent. Although I would write it as (-1095) to make it clear that the negative sign applies to the 1095. There are cases where -(1095) will yield incorrect results. For instance, take -(5)^4 vs (-5)^4. The first yields -625 since the negative is applied after the exponent, the first correctly (if you want the 5 to be negative) yields 625 since the negative is applied before the exponent.

Well, I would expect the actual transactions which take place (such as interest and payments) would be discrete. That is, a certain number of dollars change hands. You can’t pay 0.5 dollars, just like you can’t pay 0.5 cents IRL. If that is the case, than those numbers must be rounded somehow.

Let’s roll with your hypothetical example of \$25k, 13%, 3 months. I would say that the correct formula for it would be:

((0.13 / 365) * 25000) / (1 - (1 + (0.13 / 365))^(-90)) = \$282 (rounded to the nearest dollar)

The total amount that you end up paying over 90 days is 282 * 90 = 25380. You do end up paying what is due - you paid back your \$25000 and \$380 extra. You don’t end up paying very much interest, but that’s because the money didn’t have very much time to accumulate interest. You probably also didn’t have time to do much with your \$25000 since it was paid back so quickly.

In your formula, you assume that the 13% interest is over a trimonthly period, but the general assumption is always that interest is annual, even when the period the money is invested or loaned is less than a year. The 13% doesn’t mean “you will end up paying us 13% more than we originally gave you”, it means that the current balance will accumulate interest at a rate of 13% per year. In fact, it doesn’t even mean that precisely, because if you let the balance sit for a year, it will be more than 13% larger thanks to the fact that interest is compound (that is, that interest is earned on the current balance, including any interest previously earned). What it really means, in this case, is that every day you will pay 0.03561643835% (13/365) interest on the current balance. The only way you will end up paying exactly 13% more than you were given is if you pay nothing but interest for a year (so the current balance stays at \$25k) and then pay the principal off all at once.

lol, I never thought my economics lessons would come in handy for explaining how a game works to a random guy on the internet.

Sub-minimal ‘real money’ values exists in banking transactions. Those are erased at the latest possible moment (ie when virtual money transaction convert into real money).

Every delta with the highest precision (high enough so it does not impact real money transaction, you do not need an infinite one) shall be remembered, one way or the other. The delta (either positive or negative) shall them be added to the last payment to make the balance even.

I would expect the payment schedule to be computed ahead with the highest precision, tand the round-up happening the latest possible (on display or when requiring payment such as on a bill).
With that, you can actually foresee this delta ‘money box’ and choose to round up/down transactions (or to increase/decrease their value) on the schedule to minimize its final value which will impact the final payment, hopefully making it vanish. That would help to avoid oddities such as a delta absolute value making the final transaction reverse (creditor owing money to debitor). Demanding more money than due is bad in every possible way.

Thanks for your help. Yup you did good not to sleep during these lessons ;o)