# Calculate which loan to pay first

I have 2 loans (mortgages) and wondering how to calculate which one to pay off first (there are some special conditions):

Loan A: 260.000€ over 35 years. 3,75% p.a. (4,1% p.a. effective), ~1180€/month payment, I can make additional payments of 10.000€/year on this loan (otherwise there is 1% penalty)

Loan B: 50.000€ over 30 years. (~150€/month payment)

This loan went directly to the first loan (so 260.000 - 50.000€ = 210.000€ left).

This is a subsidized loan with 0,5% annual interest for the first 20 years, and then 1,5% interest for the last 10 years. After (earliest) 10 years it is possible to pay the remaining loan off in a lump sum with a 25% discount. (e.g. after 10 years, there is around 50.000€ - 18.000€ = 32.000€ left, then I would only pay 32.000€ x 0,75 = 24.000€)

I want to know:

• Should I take advantage of the 25% discount after 10 years to pay off loan B, or better put all the money in loan A?
• How would I calculate this?
• (other question) How many years are left on loan A (35 years initial), after receiving the money from loan B (50.000€), assuming everything else stays the same.
• is the 25% discount on the interest or on the principal? Commented Jul 25 at 7:40
• So you get 25% of the remaining balance forgiven? Sounds like a sweet deal. Commented Jul 25 at 7:54
• My napkin math shows it's very close actually, but capturing the 8K discount would probably put you ahead, before even considering time value and inflation. Commented Jul 25 at 8:01
• A decade is a long time, politically. If the discount you are to receive is based on government intervention of some form [and not directly within the contract you signed for the loan itself], then between now and 10-20 years from now, is it possible that a new government might destroy the planned discount you would be relying on? If there is any risk of the discount disappearing, that would be a point in favor of not relying on it [and therefore, pay off just the higher interest loan first, even thought a potential discount exists]. Commented Jul 25 at 13:30
• It also depends on what interest you get on savings and how long you need to save for the 24k. I.e. how much money can you save or pay extra per month? That info is necessary. Commented Jul 25 at 18:42

Loan A: Since there are such strict restrictions on how much extra you can pay, and the interest rate is not that low, I would pay as much as possible on this loan, just because it's a guaranteed return of 4% whereas rates for saving accounts are unpredictable and has frequently gone below 4% for long periods of time. I'd want to get rid of this debt as fast as possible (without incurring the penalty) but I'm a very risk averse person who dislikes debt in general.

If things change in your life after you pay off this debt you might be able to take on new debt with similar conditions.

Loan B: After plugging this into a spreadsheet I think the most optimal strategy is the following.

First ten years, pay as little as possible. The interest rate is crazy low and you'll easily be able to find a saving account giving back more than 0.5%. That's basically free money.

At year 10, you will want to look at the current yield of a savings account. If it is 3.3% or higher above the interest rate of the loan, you make more money if you don't pay off the loan in one go, even with the discount. If the yield is much lower than 3.3% above the interest rate of your loan, and you think it will stay low for several years, you should pay off as much as you can as soon as you can.

The next year, and every year after that, you can re-evaluate the expected future yield and make a new choice of paying off the loan or waiting until the next year to make that decision.

Just note that any extra money you put into paying off loan B cannot be reversed, as you are unlikely to ever get such an amazing loan again.

Some math on loan B: You are essentially choosing between a one time return of investment of +33% (100% / 75%) or an average of 10 years where the return of investment is the savings account yield for that year above the interest rate of the loan for that year. So 10 years times 3.3% = +33% yield will break even in the long run.

Your last question: Approximately 23 years.

I plugged the new principal into a spreadsheet and after 23 years of paying the minimum, the principal will reach 0 indicating the loan is paid off. I did the calculations on a yearly basis for simplicity so the accuracy isn't great.

You can do this calculation by taking the minimum payment for a whole year, removing the interest from the payment, and then reducing the principal by the non-interest payment part. Then repeat this for each year until you reach 0.

• 33% over 10 years is 2.9%, not 3.3%. But the idea stands: loan B is the cheapest either way, so you don't pay it off unless you've maxed out payments on loan A. Commented Jul 26 at 9:12
• Paying out the loan isn't the only way to get a guaranteed 4% return. You can also just grab a 10+ year non-callable CD (e.g., via Vanguard). Commented Jul 26 at 14:09
• @brian where can I get that in Euros please? ... Commented Jul 26 at 15:00
• @DonQuiKong: I have no idea. However, I would expect CDs with similar real interest rates to exist for most first-world currencies (currencies with lower inflation will tend to have lower nominal interest rates). Mind you, that expectation mostly relies on the assumption that if they didn't exist, there would be arbitrage opportunities. Commented Jul 26 at 15:51
• @brian there is no guarantee that the dollar, which has higher "safe" interest currently, will yield the same return as an investment in €. We don't get 4% here. It's not arbitrage. It's not hedging. It's currency exchange. Commented Jul 26 at 21:20

Your loan situation is a bit uncommon, so I would break with tradition a bit. This only applies if these are the only debts you have and there is not a drastic reduction in interest rates.

I'd pay the minimum on Loan B. I would pay extra on Loan A up to the 10K limit per year. Ideally, I would pay that extra 10K on the first day I was eligible (that might be Jan 1, or the anniversary of the loan). I'd want to avoid that 1% fee.

If you could afford to do all this, the house will be in the clear in about 14 years, 5 months. It is an amazing thing to have a paid for home.

All other money would be put in a high yield savings account, or a fixed income security. The goal would be to have around 26.4K saved so the second I qualified for that 25% discount on loan B it would be paid off (Balance around year 10: ~35.2K, so with the discount around 26.4K).

Currently, my high yield savings account, is currently earning 4.5% and is not the best.

• Thanks, that was also my plan - just not sure about the actual numbers. Can you recommend a good saving accounts? Note: I am in the EU. Commented Jul 25 at 18:06
• This answer doesn't contain any maths and seems to be essentially a "I have no idea but do this". Commented Jul 25 at 18:37
• Oh and there is no 4,5% in the EU. Commented Jul 25 at 18:39
• @HectorLector there is a post on reddit that may be able to help you. Commented Jul 26 at 10:28
• @DonQuiKong you can achieve another level of enlightenment once you understand personal finance is more about behavior than math or knowledge; and, many use math to lie. Commented Jul 26 at 10:34

How would I calculate this?

With an irregular schedule like what you have, you'll likely have to run out the amortizations in a spreadsheet (or a program) to see which one comes out on top. Each month, calculate the amount of interest that is due for each loan (presumably `[annual rate/12] * remaining principal`), apply a monthly payment to each loan (hopefully not just the minimums), and applying the remainder (monthly payment minus interest) to the principal. You should see the interest and remaining principal accelerating downward down each month.

Mathematically it's generally best to pay off the highest interest balance first. That gets you out of debt more quickly if you pay off the same amount each month. There are (arguably) psychological benefits to knocking out small debts and focusing energy, but I don't think that is a concern here.

I would run 3 scenarios:

1. As much as possible to loan A
2. As much as possible towards loan B
3. Even split

and see how long it takes to pay them off in each case. It's hard to say without actually running it if the discount is worth the extra interest you'll pay over that time. My guess would be focusing energy on the higher interest loan will save you more interest that you'll save on the discount.

• What's the reason for those scenarios? Obviously loan b is either "take the 25% or not" but never anything else ... Commented Jul 25 at 18:41
• More precisely, loan b is "take 25% after N years" with N>=10. If you run a third intermediate scenario, it would have to be with N=20. Commented Jul 26 at 9:04

It all comes down to the best place to put your money at any one time. If you can make 6% in the stock market, you should put the 50,000 into the stock market instead of using it to help pay off your first loan for instance. Just an example as stocks are risky, but if you have a higher interest loan or credit card it would be better to pay that off first. The money you pay early on the first loan is basically a guaranteed 3.75% investment with no taxes.

I'm assuming the 1% penalty on paying more than 10,000 early on the first loan is a one time penalty for the overage amount (i.e. you pay 50,000 one year that is a penalty of 400 (40,000 * 1%). In this case there is no reason to worry about it. If you pay off 10,000 a year each year for 5 years that will save you 375 for each 10,000/year. That saves 375 * (1+2+3+4+5) over a five year period or 5,625. Paying 50,000 the first year saves 3.75% of 49,600 (50,000 - 400 penalty) over 5 years or 9,300.

That's 18,600 you save over the first 10 years. If you hadn't been paying off any principle on the 2nd loan it would have only accrued 1,250 in interest.

After 10 years you have roughly 32,000 remaining in the 2nd loan. If you pay that off with 24,000 to get the 25% bonus, that's a nice one-time bonus. But the 24,000 would save you 9,000 in interest on the first loan over the next ten years if applied to it instead. Assuming it wasn't a windfall and you were saving up to pay off that loan, that money could have been applied to the first loan earlier, saving even more interest.

In reality if you are using the 2nd loan to pay off principal on the first, you are adding a 150 a month payment to lower the rate of 50,000 of the loan by 3.25%. Use google sheets or excel and game it out. The `PMT(rate,periods,value)` (`PMT(0.0375/12,35*12,260000)`)function can be used to calculate payments, it shows 1,112.55 for your first loan.

• Paying first loan to term: 468,385 total paid over 35 years
• Paying an extra 150 a month on the first loan: 417,905 total paid over 27 years, 9 months, saving 50,500
• Using 2nd loan to pay off first: 371,000 total paid over 24 years, 9 months, saving 97,000
• Adds 150 a month to total payments
• After first loan is paid, save it's payment over the next 9 months to pay off the 2nd early
• Done in 24 years, 9 months for 371,000 total paid (that includes 2,500 bonus for paying off the last 10,000 of the 2nd loan early)

First some basics:

As long as loan A exists, paying loan A makes more sense than paying loan B if not looking at the discount (we'll get to that).

As long as you can get a yield of more than 2,75% p.a. (that's currently possible) on 1 year termed savings, paying more than 10k on loan A makes no sense. If interest drops below 2,75%, paying 1% is better than paying 3,75% so you can safely "invest" in paying off loan A even with more than 10k per year. Yes, while there's a penalty of 1%, the "penalty" of 3,75% is still more.

That far, the math is easy. Now, you have two possibilities for loan B. Either, pay more than the minimum only when the savings interest drops below the interest on the loan (0,5%/1,5%) and loan A is paid off,

or, save 24k to pay it off all at once.

Let's compare those cases.

You pay off loan B after X years. That gives you a return of 0,5% for the amount you paid off + the amount you didn't have to pay per year for as long as the loans run plus a return of 1/3 of the money you pay. We have to compare this return with the return of 3,75% of loan A for as long as loan A runs.

As the calculation on the bottom shows, if loan A runs for at least 8,9 more years, it accrues more interest by providing 3,75% than loan B by gifting you a third of what you pay off.

As soon as you enter the 1,5% period on loan B, the numbers change to 13 years. If the periods are mixed - somewhere in between.

The math doesn't depend on the amount you pay off, the number of years is independent.

Additionally, as interest on savings is probably dropping below 2,75% next year or so, saving up to pay off loan B loses you money, too. How much depends on how long you have to save the money, etc.

Bottom line:

Do not pay off loan B until approx. 9 to 13 years before the expected end of loan A. Don't save money over several years to pay down loan B unless interest on savings is high or you are getting close to the 9 to 13 years. The closer you get to the end of loan A, the more sense it makes to pay off loan B if you can.

Formula for return of paying towards loan A: amount * 0,33 + amount1,33 * 1,005^(years with 0,5%) + amount1,33 * 1,015^(years with 1,5%) - amount*2,66.

Formula for return of paying towards loan B: amount * 1,0375^(years)-amount

Exact number for only years with 0,5%:

8,9 years (click approx. numbers on real solution)

Exact number for only years with 1,5%:

https://www.wolframalpha.com/input?i=1*0%2C33+%2B+1%2C33*1%2C015%5E%28x%29-+1%2C33%3D1*1%2C0375%5E%28x%29-1%3B+solve+for+x

13 years

As the amount doesn't matter I inserted "1" as amount.

Very roughly: If your second loan is \$32,000 and you can pay it off with \$24,000 payment, you save \$8,000 instantly and 0.5% of \$32,000 = \$160 every year for the next ten years, \$320 after that.

If you make three overpayments of \$8,000 for your first loan, total \$24,000, you save 3.75% = \$900 interest every year. In ten years you save \$9,000 instead of \$9,600.

It’s not much difference. Not enough to worry about it. What would be better is overpaying the first loan as much as you can afford (but no credit card debt under any circumstances, and don’t forget to live). Also a good idea to have some money in a savings account that you don’t touch if it pays comparable interest, always good to have some money available instantly.