From time to time we have heard a number of statements about debt snowball vs debt avalanche strategies, such as avalanche is better mathematically or that snowball is an easier strategy to stick to.
I always thought the "only mathematically" comment was odd. We wouldn't say an income of $200K is better than $100K "only mathematically," for example. Finance is numbers. People seeking advice are wanting to know how to maximize and minimize various mathematical numbers, relating to the dollars they own or owe.
The other comment, about snowball being an easier strategy also struck me as odd. Somehow we've gotten into a situation in which people are convincing themselves that they are happier with fewer debt accounts rather than fewer dollars of debt. It's not wrong, and people should like the strategies that work for them. It just seemed odd to me because "reduce the number of debt accounts as quickly as possible" would not be a higher priority than "reduce the dollars of debt as quickly as possible."
Then there are statements that are just false, such as, snowball is better because once you pay off the first account, you're building up a snowball of extra payment capability to apply to the next account, in a way that avalanche can't match.
So, I thought I'd at least discuss the purely mathematical side of things now, with a theorem that definitely says which strategy is better "only mathematically." After, I'll have some observations on motivation, emotions, and ability to stick with a plan.
Okay, now some math. For each j = 1, 2, ..., n, let
D_j denote an amount of debt in dollars,
I_j denote an annual interest rate
m_j denote a minimum required monthly payment
Denote the total minimum monthly payment by M = sum_j m_j. Suppose the debt holder has income to support total fixed monthly payments of P, P > M, until all debt is exhausted.
The debt avalanche strategy, to begin, devotes M dollars to pay the minimum on each debt. The remaining P-M dollars are used to pay down the debt with the highest interest rate, i.e., D_i, where i = argmax_j I_j. When debt D_i is retired, the same strategy is followed for the n-1 remaining debts using the available P dollars/month.
The debt snowball strategy, to begin, devotes M dollars to pay the minimum on each debt. The remaining P-M dollars are used to pay down the smallest debt, i.e., D_d, where d = argmin_j D_j. When debt D_d is retired, the same strategy is followed for the n-1 remaining debts using the available P dollars/month.
Theorem: The debt avalanche strategy retires the total debt faster than the debt snowball strategy in all situations in which the strategies are different.
Proof:
Since the strategies are different, at some point we will have d != i.
Under snowball, after one month debts D_d and D_i become
D_d(1) = (D_d(0) * (1+I_d/12)) - m_d(0) - (P-M)
D_i(1) = (D_i(0) * (1+I_i/12)) - m_i(0)
where the parentheses denote the month. This accounts for the interest on outstanding principal, minimum payment, and extra payment beyond the minimum.
Under avalanche, we'll use prime notation, and after one month debts D_d and D_i become
D'_d(1) = (D_d(0) * (1+I_d/12)) - m_d(0)
D'_i(1) = (D_i(0) * (1+I_i/12)) - m_i(0) - (P-M)
For all other j not equal to d or i, the two strategies behave the same.
In month 2, after applying interest the debts d and i will total, for snowball and avalanche:
D = D_d(1) * (1+I_d/12) + D_i(1) * (1+I_i/12)
D' = D'_d(1) * (1+I_d/12) + D'_i(1) * (1+I_i/12)
And we can compute
D - D' = -(P-M) * (1+I_d/12) + (P-M) * (1+I_i/12)
= (P-M) * (I_i - I_d)/12
> 0
where strict inequality follows because I_i is a higher interest rate than I_d, by definition of the strategies.
Thus, after the first month, the outstanding debt is higher under snowball than avalanche. The same computation applies to second and subsequent months, wherein less debt is paid down in snowball by failing to devote the excess funds to the highest interest rate debt.
QED.
Observations.
I always thought the "only mathematically" comment was odd. We wouldn't say an income of $200K is better than $100K "only mathematically," for example. Finance is numbers. People seeking advice are wanting to know how to maximize and minimize various mathematical numbers, relating to the dollars they own or owe.
The other comment, about snowball being an easier strategy also struck me as odd. Somehow we've gotten into a situation in which people are convincing themselves that they are happier with fewer debt accounts rather than fewer dollars of debt. It's not wrong, and people should like the strategies that work for them. It just seemed odd to me because "reduce the number of debt accounts as quickly as possible" would not be a higher priority than "reduce the dollars of debt as quickly as possible."
Then there are statements that are just false, such as, snowball is better because once you pay off the first account, you're building up a snowball of extra payment capability to apply to the next account, in a way that avalanche can't match.
So, I thought I'd at least discuss the purely mathematical side of things now, with a theorem that definitely says which strategy is better "only mathematically." After, I'll have some observations on motivation, emotions, and ability to stick with a plan.
Okay, now some math. For each j = 1, 2, ..., n, let
D_j denote an amount of debt in dollars,
I_j denote an annual interest rate
m_j denote a minimum required monthly payment
Denote the total minimum monthly payment by M = sum_j m_j. Suppose the debt holder has income to support total fixed monthly payments of P, P > M, until all debt is exhausted.
The debt avalanche strategy, to begin, devotes M dollars to pay the minimum on each debt. The remaining P-M dollars are used to pay down the debt with the highest interest rate, i.e., D_i, where i = argmax_j I_j. When debt D_i is retired, the same strategy is followed for the n-1 remaining debts using the available P dollars/month.
The debt snowball strategy, to begin, devotes M dollars to pay the minimum on each debt. The remaining P-M dollars are used to pay down the smallest debt, i.e., D_d, where d = argmin_j D_j. When debt D_d is retired, the same strategy is followed for the n-1 remaining debts using the available P dollars/month.
Theorem: The debt avalanche strategy retires the total debt faster than the debt snowball strategy in all situations in which the strategies are different.
Proof:
Since the strategies are different, at some point we will have d != i.
Under snowball, after one month debts D_d and D_i become
D_d(1) = (D_d(0) * (1+I_d/12)) - m_d(0) - (P-M)
D_i(1) = (D_i(0) * (1+I_i/12)) - m_i(0)
where the parentheses denote the month. This accounts for the interest on outstanding principal, minimum payment, and extra payment beyond the minimum.
Under avalanche, we'll use prime notation, and after one month debts D_d and D_i become
D'_d(1) = (D_d(0) * (1+I_d/12)) - m_d(0)
D'_i(1) = (D_i(0) * (1+I_i/12)) - m_i(0) - (P-M)
For all other j not equal to d or i, the two strategies behave the same.
In month 2, after applying interest the debts d and i will total, for snowball and avalanche:
D = D_d(1) * (1+I_d/12) + D_i(1) * (1+I_i/12)
D' = D'_d(1) * (1+I_d/12) + D'_i(1) * (1+I_i/12)
And we can compute
D - D' = -(P-M) * (1+I_d/12) + (P-M) * (1+I_i/12)
= (P-M) * (I_i - I_d)/12
> 0
where strict inequality follows because I_i is a higher interest rate than I_d, by definition of the strategies.
Thus, after the first month, the outstanding debt is higher under snowball than avalanche. The same computation applies to second and subsequent months, wherein less debt is paid down in snowball by failing to devote the excess funds to the highest interest rate debt.
QED.
Observations.
- This is pedantic. It is pretty obvious that paying down the highest interest rate is the fastest way to pay down debt. Still, it may handy to be able to point people to a theorem to convince people, if "only mathematically."
- It's similarly straightforward to prove that the snowball strategy will reduce the number of debt accounts from n to n-1 faster than avalanche, if the strategies are different. So there's that, if that matters to you. (It comes at the cost of retiring the over all debt slower.)
- Neither strategy involves giving up or failing to pay P dollars each month. When people say snowball is better than avalanche, they mean that snowball is better than NOT sticking with avalanche. Sticking with avalanche is clearly better, financially, in the sense that the debt is paid off faster and cheaper.
- The theorem does not say anything about what is the best strategy for you. If you think you can't stick with the avalanche strategy, use a strategy that you can stick with. To me, the strategy I can stick with easiest is going to the one that retires debt the fastest and cheapest, which seems obvious to me, but I've heard many people say snowball is what works for them. This is not wrong.
Statistics: Posted by MoreTaxes — Fri Feb 02, 2024 8:00 pm — Replies 4 — Views 493