What is the difference between the annuity amortization schedule (Spitzer) and linear payments?

In the annuity amortization schedule (Spitzer), the monthly payment is fixed throughout the life of the loan, whereas in linear payments the payment decreases over time because the principal is reduced by an equal amount each month. In the Spitzer schedule the ratio between principal and interest changes each month; in linear payments the principal portion is fixed.

Are linear payments cheaper than the Spitzer schedule?

In terms of total interest payments, linear payments are indeed cheaper. In practice, however, the higher payment at the start of the loan eats into the repayment budget for the other loans in the mortgage mixture, limits flexibility, and makes it harder to negotiate with other banks.

Why do banks not always offer linear payments?

Not all banks offer linear-payment loans. This is not necessarily bad for the borrower — a linear-payment loan limits the ability of banks to compete with one another and can make the other loans in the mortgage mixture more expensive.

VI: Annuity Amortization Schedule and Linear Payments

Repayment Methods Explained - Linear Payments and the Annuity Amortization Schedule (Spitzer)

How is the monthly payment on a loan determined? What does it depend on? Are linear payments better than the annuity amortization schedule (Spitzer)? Are the banks doing us wrong when they "push" the Spitzer schedule on us? This article answers these questions and gives a thorough explanation of the differences between the two methods.
The monthly payment on the loan we take depends on four variables: the interest rate, the loan duration, the loan amount, and the principal repayment method. There are two principal repayment methods — linear payments and the annuity amortization schedule (Spitzer).
There are actually other methods too, like the grace-period loan and the balloon/bridge loan — but we'll get to those in later articles.

Linear Payments

With linear payments, the principal (the debt to the bank) goes down by the same amount every single month.

Note! The monthly payment is not the same from one month to the next.

If we took a loan of 1,000,000 NIS for 20 years, and the principal has to come down by the same amount with each payment, then the monthly principal repayment is:

\frac{1{,}000{,}000~\text{₪}}{240} = 4{,}166~\text{₪}

Every month you repay a fixed amount of principal (roughly 4,166 NIS a month).

Here too, the interest payment is higher at the start (because the outstanding principal balance is high) and falls over time (as the principal shrinks).

Also called Linear payments
Only a small number of banks offer this amortization schedule

The following chart describes the monthly payment under linear payments for a loan of 1,000,000 NIS over 20 years:

Monthly Payment and Monthly Interest – Linear Payments Schedule

The Annuity Amortization Schedule (Spitzer)

The monthly payment stays the same throughout the entire life of the loan. With each payment, the split between the portion going to interest and the portion going to principal repayment shifts.

The payment is fixed unless the interest rate changes, or the principal changes due to CPI indexation or prepayment
As you move through the payments, the principal's share (of the total monthly payment) rises while the interest share falls

This is by far the more common method in the mortgage world. Under it, the monthly payment is fixed — throughout the life of the mortgage, every single month, we pay exactly the same amount we agreed on with the bank.

📐Deep dive — how the payment is set under the Spitzer amortization schedule

This section uses the example from the previous article.
We're going to borrow 1,000,000 NIS for 20 years at an interest rate of 3%.

Step 1: The problem with a fixed monthly payment, and what discounting means
For us as borrowers, repaying the same amount month after month is very appealing. For the lender, though, it's not so attractive.
Why? Because 100 NIS the lender receives today is worth more than 100 NIS the lender receives two years from now. That's why we have to discount the payments to find their true value (one that reflects the time value of money).

Discounting is a financial operation that lowers the value of future money the further away it is in time from the present. We reduce that value using a "discounting rate" — the rate that describes how much value our money loses each year.

\frac{\text{שווי הכסף כיום}}{(1+\text{ריבית להיוון)}^{N}} = \text{שנים}\text{ N }\text{שווי הכסף בעוד }

If we borrowed 1,000,000 NIS and the discounting rate is 5%, then:

1,000,000 NIS today is worth ... still 1,000,000 NIS
$\frac{1{,}000{,}000~\text{₪}}{1.05^0} = 1{,}000{,}000~\text{₪}$
1,000,000 NIS today is worth 952,380 NIS in one year
$\frac{1{,}000{,}000~\text{₪}}{1.05^1} = 952{,}380~\text{₪}$
1,000,000 NIS today is worth 907,029 NIS in two years
$\frac{1{,}000{,}000~\text{₪}}{1.05^2} = 907{,}029~\text{₪}$
1,000,000 NIS today is worth 863,837 NIS in three years
$\frac{1{,}000{,}000~\text{₪}}{1.05^3} = 863{,}837~\text{₪}$

Step 2: How to calculate the monthly payment under the Spitzer amortization schedule
Back to mortgages. We need to find a fixed monthly payment such that, when all the future payments are discounted at the discounting rate, the result equals the current principal.
In other words, we need the monthly payment (let's call it x) where the sum of the discounted repayments equals the loan amount.
Here's the equation we have to solve:

\sum_{i=1}^{\text{מספר החודשים}} \frac{\text{ההחזר החודשי}}{(1+\text{ריבית המשכנתא החודשית})^i} = \text{סכום הקרן}

And with the figures from our example:

\sum_{i=1}^{240} \frac{x}{1.0025^i} = 1{,}000{,}000

Step 3: How do you solve such a complicated equation?

The equation above is extremely hard to solve (it has many terms and the denominator differs from one term to the next) — there's no clean, elegant analytical solution. People who worked in this field in the past would buy books that had already pre-calculated the answers to these equations.
One of them, a German-language book, was published in Vienna in 1865 by the Jewish mathematician Simon Spitzer. Its title is:

Tabellen für die Zinses-Zinsen- und Renten-Rechnung mit Anwendung derselben auf die Berechnung von Anlehen, Construction von Amortisationsplänen, etc.
(Tables for compound interest and pension accounting with the same application to the calculation of loans, construction of amortization plans, etc.)

The book's 430 pages are full of tables that let you look up the monthly payment for a given loan term and interest rate. It was so thorough that it immediately caught on, and ever since, this type of loan has also gone by the name "loan on the Spitzer amortization schedule."
Today, thankfully, you can just use Excel to calculate the payment. The PMT function computes the monthly payment on a loan that follows the Spitzer amortization schedule.

Differences between linear payments and the Spitzer amortization schedule

Now we've reached a hot-button topic. Does the Spitzer amortization schedule have an advantage over a linear-payment loan? We'll use our central example to dig into this.

Monthly Payment and Monthly Interest – Annuity Amortization Schedule

The Spitzer amortization schedule

The first repayment on a Spitzer-schedule loan is 5,545.98 NIS. It's also the repayment in every other month. We calculated it with the following Excel formula:

PMT(3/1200, 240, 1000000)

The total interest payments come to 331,034 NIS. We calculated this with the Excel CUMIPMT function.

The CUMIPMT function only works for loans whose repayment doesn't change (for example, a fixed unlinked loan — KALATZ). Don't use it to build a mortgage mixture.

Linear payments

By contrast, the first monthly payment on a linear-payment loan is 6,666 NIS and the last is 4,177 NIS — because, as you'll recall, the payment falls along with the principal. The average payment is 5,421 NIS per month. The total interest and indexation cost is 301,250 NIS.

The total cost under linear payments is lower — so on the face of it, a linear-payment loan is cheaper. But is it really?

The drawback of including a linear-payment loan in the mortgage mixture

For the next section, we'll use this example:

We want to borrow 1,500,000 NIS, and the monthly payment we've chosen to service this debt is 8,000 NIS. We'll also assume the interest rate, over a 30-year term, is 4%.

Including a linear-payment loan in the mortgage mixture hurts the other loans

At the repayment we chose, we can't put the whole thing into a linear-payment loan, because the payment on 1,500,000 NIS would be 9,167 NIS per month — over the budget we set (8,000 NIS).

So we'll go with this compromise: say we decide that 1,000,000 NIS of the mortgage will be a linear-payment loan and 500,000 NIS will be a Spitzer-schedule loan. Let's calculate the monthly payment for each.

Financing plan for 1,500,000 NIS with a monthly payment of 8,000 NIS

Category	Spitzer	Linear payments
Loan amount	1,000,000 NIS	1,000,000 NIS
First-month monthly repayment	4,774 NIS	6,111 NIS
Monthly repayment remaining to service 500,000 NIS	3,136 NIS (8,000 - 4,774)	1,889 NIS (8,000 - 6,111)
Is it possible to take 500,000 NIS on the Spitzer schedule?	Yes	No

We can see that under linear payments we can't service the remaining debt. We could get around this by taking a CPI-linked loan, but that's a riskier loan and can also turn out expensive over a long term.

The problem with including linear-payment loans is that they "eat into" the repayment budget of the other loans (the ones not on linear payments). When that happens, you have to extend the other loans and/or convert them to CPI-linked loans. That's the significant drawback of the linear-payment schedule: the loans are cheap, but at the expense of the other loans that aren't on linear payments.

Including a linear-payment loan in the mortgage mixture hurts your negotiating position

Not all banks offer a linear-payment loan (and I hope you're already convinced this isn't because they're out to "hurt" us borrowers). If we ask one bank for a principle approval on a mortgage mixture with this amortization schedule and then take it to another bank that doesn't offer such a product, that bank simply can't compete.
That's a serious problem, because if it can't compete, you can't use it to challenge the first bank's offer. What's more, if the first bank knows its product is one of a kind, it can charge a premium for that exclusivity — which, of course, drags down the interest rates you get.

In an ideal world with no budget constraints, where there's no cap on the monthly payment, a linear-payment loan really is cheaper over the life of the loan — the total interest and indexation comes out lower. But who actually lives like that?

In reality, most of us have a monthly repayment limit, and squeezing in linear-payment loans comes at the expense of other tracks in the mortgage mixture — making them more expensive or limiting our room to negotiate with the banks.

So it's important to weigh not just the overall cost, but also the practical fit and flexibility of your mortgage mixture.

Want to see the difference between the repayment methods for yourself? In the mortgage calculator you can build a mortgage mixture and compare the Spitzer schedule against linear payments.

Good luck!