# How to annualize a loss of 100% for an investment?

Assume, I lend someone money against a certain fee or interest (charged at the day the loan gets repaid). The loan defaults and I get no fee or interest back. Hence, I have a return of -100% as all my starting capital is lost. How do I annualize this return? As I want to compare it with other annualized returns from other investments. I don't have an investment period to annualize the -100% over. Like when using the little inaccurate formula (return*365)/days . The "days" are missing to annualize my -100 %. Is there a best practice how to annualize complete losses to compare them with other annualized returns?

• Annualisation gets you an estimate for an annual loss or gain based on actual losses or gains over a different period. I don’t know what the best practices are, but from a layman’s perspective, projecting a 100% loss over any duration is still a 100% loss if there is no prospect of recovering the funds. – Lawrence Sep 2 '19 at 4:09
• @Lawrence thanks for the answer. I am not sure whether it is correct to say a 100% loss over 40 days is annualized still a 100% loss or to say it is a 912.5% loss ((-100*365)/40) (using the simple annualization) – sh_student Sep 2 '19 at 6:19
• If we go back to first principles, annualisation should mean something like “If I reinvested my earnings from the original sum invested and kept doing that for a whole year, and managed to get the same yield as the original investment each time, what would I get at the end of the year?” In that case, since you get zero at the end of the year, the writeoff is still -100% when annualised, even if it was written off 40 days after investment. – Lawrence Sep 2 '19 at 6:30

If your return for say, one month, is `r = -100 %` your future value on investment `a` is

`FV = a (1 + r) = a (1 - 1) = 0`

and the annualised return is `(1 + r)^12 - 1 = 0^12 - 1 = -100 %`

If your total return is `r` over `t` years then `r=q^t` where `q` is your annualized return. You can calculate it with `q=log(r)/t`.

The exact calculation of `t` depends on how exactly the investment works. If you are open for business every day then it could be `days/365`, or you may need to adjust for holidays and business days. It doesn't really matter how you measure time so long as you measure it in the same way as whatever benchmark you're comparing to.

If you have no time horizon, you cannot calculate an annualized return because there is no meaningful annualization to do. `t` would approach infinity and therefore `q` would approach 0. In reality, what you say makes no sense. The loan must have had some sort of maximum term, otherwise your debtor could in theory repay it after 100 or 1000 years, in which case the money you "lent" was not a loan, but a gift. If there was a max term, that max term is `t`.

I am not sure of your purpose, but you could consider taking it out of your annual returns and instead show it as “Default ratio”, hence leaving the non-defaulted loans in your return-calculations and then keeping a separate measure for default-ratio.

But this of course depends on the purpose.

• Assume, I have several investments. I want to annualize their return and then average everything to see how the several investments performed as a whole – sh_student Sep 2 '19 at 5:30