#### How Insurers Pay Negative Interest Rates by S&C Messina

**An Analogy: Insurance Claims and Mortgage Payments**

It may not be intuitive at first how collecting insurance premiums and paying out claims is similar to borrowing money and paying interest.

An analogy, albeit an imperfect one, that may be helpful is the monthly mortgage payment. The mechanics of a mortgage and its monthly installments are something most of us are familiar with – you go to a bank and get a mortgage that matures in a certain number of years, you get charged an annual interest rate which translates into a fixed monthly payment to the bank – this monthly payment consists of both interest and principal pay-down.

__Below is an example of a mortgage with the following terms:__

- Principal = $100,000
- Maturity = 1 year
- Months = 12
- Annual Interest Rate = 5%
- Monthly Interest Rate = 5%/12 = 0.42%
- Monthly Mortgage Payment = $8,560.75

If you are the lender of this mortgage, you can calculate your monthly compounded rate of return which comes out to 5.116% (note: XIRR or daily compounded return is similar at 5.113%). In other words, **5.116% is how much it cost the borrower** to take out this $100,000 mortgage. It’s the cost of capital paid by the borrower.

If we look at the cash flows another way, we can see that during the year, the borrower **received a total of $100,000 and paid out a total of $102,728.98** ($8,560.75 x 12 = $102,728.98).

If we were to say this was similar to an insurer receiving premiums of $100,000 upfront at the beginning of the year and paying out total claims of $102,728.98 throughout the year, we would say the insurer had an underwriting loss of $2,728.98. Or in insurance industry parlance, the insurer had a **combined ratio of** **102.7%.**

If the claims were paid out monthly in equal, fixed amounts like we saw in the mortgage example above, we could say **the cost of capital for the insurer was same, at 5.116%.**

What happens if the insurer has an underwriting profit? If the claims were $400 less each month, this would be like the mortgage borrower paying $400 less in monthly installments than before or 8,160.75 per month. The cash flows could look something like this:

Here, the mortgage borrower doesn’t have to pay back $4,911.54 of the principal at maturity. As an analogy, the insurer took in $100,000 in premiums upfront and paid out a total of $97,928.98 in claims during the year. The insurer underwrote profitably and had a **combined ratio of 97.9%**.

What was the cost of capital for the insurer (and the mortgage borrower) to borrow this money? The cost of capital was **-3.779% **(note: XIRR or daily compounded return is similar at -3.777%). That’s right, the insurer was paid to borrow this money.

What happens if instead of paying $400 less in claims each month, the insurer’s claims losses were back-end loaded? In other words, the insurer similarly pays out $400 x 12 = $4,800 less in aggregate during the year, but this benefit occurs in the last month? The cash flows would look something like this:

The insurer pays out $3,760.75 in the final month. The insurer’s **combined ratio is the same at 97.9%** ($97,928.98 / $100,000 = 97.9%). This time, however, the **insurer’s cost of capital is even cheaper or more negative, at -3.939% **(note: XIRR or daily compounded return is similar at -3.937%).

Notice this would be similar to the mortgage borrower not having to pay the outstanding principal of $4,800.00 at maturity compared to not having to pay $4,911.54 at maturity in the prior example. Yet, the cost of capital is cheaper for mortgage borrower when more of the principal has been paid off in the final month.