EDIT: Using Tables for formulas butchered them due to text reflow, so converted formulas to images
Could someone with the financial and algebraic know-how please check my numbers? I don't know if I'm getting rounding errors, or if I'm not calculating this correctly:
The payment of annuity formula is:
Pmt = payment amount
Prin = Initial borrowed ammount
i = interest rate
n = number of terms in the loan.
So for those who need an example read below. If you're familiar with this, skip to the My Proposal
Let's say it's a $10,000 loan, 5% rate, 10 year term, payable once a year. Formula would look like:
Meaning that to pay off a $10,000, 5% loan over 10 pay periods (years), if you pay $1,295.05 every pay period, your loan balance will be $0 after the 10th payment (plus or minus rounding errors).
A VERY important aspect of this loan is that in an inflationary economy, the 10th payment of $1,295.05 is actually worth a lot less than the 1st payment. If we assume a 3% annual inflation, the 1st $1,295.05 payment at the end of the year is actually worth $1,295.05*(1+3%)^-1 = $1,257.33 in today's dollars, and the 10th payment is actually worth $1,295.05*(1+3%)^-10 = $963.64 in today's dollars (meaning at year 10, paying that $1,295.05 will feel like paying only $963.64 today). The idea is that the loan becomes easier to pay off as time goes on.
The big problem arises when the loan is given out in an deflationary currency, such as (future) Bitcoin. If we use that same level-payment annuity formula from above, but assume that instead of a 3% INflation, we have a 3% DEflation (i.e. -3% instead of +3%), the 1st payment will feel like $1,295.05*(1-3%)^-1 = $1,335.10, and the 10th payment will feel like $1,295.05*(1-3%)^-10 = $1,756.18. That's a bit worse than $1,295.05, and MUCH WORSE than $963.64. Now imagine stretching this out to 30 years, instead of 10, and the problem will be even worse! Now, sure, this will discourage borrowing, but from a lender's point of view, this will also SIGNIFICANTLY increase risks of default as time goes by, since people will have a much harder time making loan payments as their loan matures.My Proposal/Calculations
I propose that the formula be augmented as such:
t = period at which the payment is calculated (t is 1 to n)
n = number of terms in the loan
= payment amount for term t
= Outstanding principal at term t
i = interest rate of the loan
d = deflation rate
The first part is the annuity formula, the second part (after the *) is time value adjustment to present value using the 3% deflation rate.
So, as an example, using the loan above and assuming a 3% deflation, the calculation for the 1st payment will look like this:
After this payment, the outstanding principle is now $10,000*(1+5%)*(1+3%) - $1,740.43 = $9,074.57
Calculation for the 2nd payment will look like:
This will continue until the very last 10th payment, which will look like:
which will pay off the remaining outstanding principal, bringing the loan to $0
The excel table for this looks like this:
. Outstanding Int Prin Pmt Remain
1 $10,815.00 $800.00 $940.43 $1,740.43 $9,074.57
2 $9,814.14 $725.97 $939.84 $1,665.81 $8,148.34
3 $8,812.43 $651.87 $945.18 $1,597.05 $7,215.38
4 $7,803.43 $577.23 $956.37 $1,533.60 $6,269.83
5 $6,780.82 $501.59 $973.39 $1,474.97 $5,305.85
6 $5,738.27 $424.47 $996.24 $1,420.71 $4,317.56
7 $4,669.44 $345.41 $1,025.02 $1,370.42 $3,299.02
8 $3,567.89 $263.92 $1,059.84 $1,323.76 $2,244.13
9 $2,427.03 $179.53 $1,100.87 $1,280.40 $1,146.62
10 $1,240.07 $91.73 $1,148.34 $1,240.07 $-
Where Outstanding is previous Remain*(1+5%)*(1+3%), Pmt is my formula, and Remain = Outstanding - PmtProblems/Issues/Things I need help with
As you can see, this will calculate the payment structure that will front-load the first payment, and possibly make the last payment a lot more bearable (though still more difficult than in an inflationary system). If you run separate calculations, you'll also see that the lender is able to get both, the interest, and the deflation effect, from the lent money (i.e. the return is the same whether the lender lends the money, or sits on it and has it "magically" collect 5% interest). These are the problems I still see with this, or need help with:
1) This system calculates the accumulated interest on each outstanding principle as Prin * (1+i) * (1+d). Intuitively, I would think that 5% + 3% = 8%, and so I should just use Prin*(1+i+d), but when I do that, the last payment ends up overpaying, with the remaining outstanding balance at year 10 being -$1.69. Is my formula wrong somewhere, or is this a rounding error? I think either my formula is wrong, or the Prin*(1+i+d) assumption is wrong.
2) Having a recursive formula that depends on each previous calculation to work is annoying, to say the least. The original annuity formula is much cleaner and nicer by comparison. If anyone out there is good with algebraic simplifications and such, maybe you can help fixing/simplifying this thing?
3) The final payment, although lower, is still a tad higher than it would be in an inflationary system with level payments using today's dollars, though not by much. If anyone has a better suggestion than this formula/system, please share!