Discounted Loss Ratio
Hi Graham,
I don't understand the formula for discounted LR, was hoping you could shed some light 
Why do we take out investment income from discounted reserves, doesn't the fact that reserves are discounted already mean they have been brought back to time 0 and investment income has been taken out, why do we subtract investment income again here?
Why do we need to include investment income earned on UPR in the denominator?
Thank you for looking into the questions.
Comments
PresentValue.Claims.(2016) / PV.Earned Premiums(2016)
earning interest against UEP is like earning interest on a PAYABLE - it's free money.
earning interest against UNPAID CLAIMS is like earning interest on a PAYABLE - it's free money.
both of these accounts represent funds held without justifiable reasons (not really the insurers money), but they get interest on them anyway.
It would be easier to consider you, ABC Construction, are building a project that was purchased and pre-paid by a government. ABC wants to determine what the cost/revenue ratio was in terms of present value. ABC was paid $100B upfront (Jan 1, 2010) and intends to spend $10B/year for 10-years - starting in 1 year (Jan 1, 2011) after they receive the government payment.
Initially (2010) they would calculate:
PV(10 x $10B)/$100B. (@ 5%) ==> 77.2B/100B <-- pv.cost / pv.revenue
We only want to calculate interest earned in bridge building operations (cost) side - which has a current present.value of $77.2B:
Interest (year 1) = $77.2B x 5% = 3.86B
In 2011:
PV of the future expenses would be higher: $81.1B (one year later).
Their ratio would therefor be adjusted for the historical interest ($81.1B - $3.86B)/$100B = $77.24B
Interest (year 2): ($81.1B - $10B) x 5% = $3.55B
(the $10B comes on on January 1, 20XX - you can't get interest on that)
At the end of the next year (year 2):
($74.6B - $3.86B - $3.55B + $10B) / $100B = $77.2B / $100B.
(Discounted.Future - SUM(Historic.Interest) + Historic.Payments) / Discounted Revenue
It should be more clear here that this discounted.future payments don't actually overlap with the Historical.Interest payments (or historic payments). And equation might make more sense as:
(Discounted.Future + [Historic.Payments - SUM(Historic.Interest)]) / Discounted Revenue
Although some of this [Historic.Interest] is used to adjusted the new "Discounted.Future" back to the original loss year.
Insurance is unique because the "Discounted Revenue" cannot be booked until earned. So interest income during this delayed process is also included as a benefit to earned premiums.
A discounted loss ratio considers the BENEFIT of interest on pre-paid premiums and a BENIFIT of savings as a result of FUTURE savings on interest (discounting). Once Discounted.Future = $0, some of these equations make a bit more sense: Paid claims over time discounted back to original loss year.
Claims
Take 1 claim for a given loss year, 2016, and paid in 2018: We have a unpaid reserve of $1,000 for 2 years that accrues 5% interest ($100 for the simple case).
The claim is eventually PAID @ $1,000.
PresentValue.Claims.(2016) = $900 <--- WANT (for discounted LR)
PresentValue.Claims.(2018) = $1,000 <-- HAVE PAID (cumulative loss)
Discounted LR should report: PresentValue.Claims.(2016) / PV.Earned Premiums
In 2018 we calculate:
PresentClaims.(2016) = PresentClaims.(2018) - $100.
If we just ADD up all the paid claims ($1,000) we wont be accounting for discounting.
Unearned Premiums
For premiums we want to include the benefits of TIMING (we are paid up front). And we are not calculating benefit of earned premiums (just unearned). This is really the whole point of a lot of these exercises:
Over time:
PresentPrem(2016) = PresentPrem(2017) - interest;
We want to include all of those PRIOR interest pieces we know we will collect on the premium (BENEFIT to insurance). And since premiums are all booked PRIOR to the current loss year they are UNDERSTATED. Eg. 2016 Policy with EP in 2017.
Bear in mind this is UEP (not EP) and is collecting interest throughout the policy term even though we haven't booked it as earned premium yet.
<< ADDING Interest to initial Earned Premiums ACCOUNTS for future benefits of delays between claim payment and premium collection >>
The discounted loss ratio measures the expected benefits in timing both for delayed payments on claims (unpaid claims reserves) and benefits of timing in terms of premium (unearned premium reserves):
(Expected Claims - Expected Interest on claims reserves) / (Earned Premiums - Interest Earned due to Early Payment)
UEP with Hyperinflation
It might be easier to work with some of the unearned premium concepts in a hyper-inflationary environment (100% inflation). Assume inflation always occurs as a jump on January 1, YYYY.
Costs are 100% higher in 2012 than 2011.
UEP = $1,000 (from premiums written in 2011).
We earn $1,000 interest on this. Claim payments against UEP will occur in 2012 (@ +100%).
UEP(2012) alone is not sufficient to cover CLAIMS(2012).
CLAIMS(2012) represent only losses that relate to UEP(2012).
CLAIMS(2011)+CLAIMS(2012)/(EP(2011)+UEP(2012)) would not be accurately accounting for discounting
CLAIMS(2011)+CLAIMS(2012)/(EP(2011)+UEP(2012)+UEP.Interest(2012))
WOULD account for interest and discounting to 2011
Note: in the hyperinflation example [UNPAID CLAIMS = $0] (there are no late-reported claims or development on opened claims). Interest on unpaid = $0. When dealing with claim development, we only want to discount the claims back to their EXPECTED loss date - not back to policy inception when premiums were received.
I had to work out an example in detail to make sure I properly understood what was going on here. The example is available in a pdf in the CIA.MfAD wiki article in the section at the bottom where the formulas for the undiscounted and discounted loss ratios are given:
Very briefly, the formula for the discounted loss ratio is given in the CCIR paper that has detailed instructions for completion of the annual statment. The formula is called Claims Ratio by Year of Accident and is on page VI-9, Line 31. It does indeed subtract the investment income from the numerator. This is the part I had to think about and is the topic of the example.
This type of question has also appeared 3 times on past exams, and this is indeed the formula they used to solve it, so that's the way you should do it.
But I'm sure you already know all that. You were asking for the reasoning behind the formula:
This is a topic that's worth thinking about, but for the exam I have to reluctantly say that the main thing is being able to solve the problem the way they want you to. Of course, it helps to understand the formulas intuitively, but you have to judge how much time to spend on that versus getting through the rest of the material.
Well, that's a long answer. Hope it helps!
Thank you Graham and Chrisboersma for the very detailed and thorough explanations, this is very clear to me and I understand why we include investment income on UPR in the numerator now. Thank you very much for the time and effort spent in answering my question!
Regarding to why we take out the investment income on discounted reserves, here's an explanation:
step 1: when we are calculating the APV of claim liabilities we are discounting the liability cash flows to year end (or valuation date), this backs out all the investment income we are expected to earn in the future years (on the existing book of liabilities).
Step 2: Next we subtract the investment income from this APV value to take out the investment income we earn during the current CY.
In short, Discounting to APV value takes out all the future years' investment income and subtracting one year of investment income takes out current year's investment income. This now makes a lot of sense to me. I do a agree with your point Graham, this is only one small piece of the iceberg and shouldn't spend too much time here. Hopefully this'll be useful to other users as well. Cheers