Core - math formulas Flashcards
Formula for group term life imputed income
Employees are taxed on the value of the employer - provided group term life insurance in excess of $50,000
This value is determined by Table 1 (rates vary by age)
Monthly imputed income = Table 1 rate x (coverage amount - $50K) / $1000 ] - employee contributions
typical retrospective refund formula
policyholder account balance = prior year’s balance + premium + investment earnings - charged claims - expenses - risk charge - increase in stabilization reserve - profit
Methods for calculating gross premiums
- block rating (short-term horizon) approach - claim costs are calculated for the rating period and premiums are calculated by adding on expenses and profit charges
Gross premium = [N x (1+ E^N) + E^F] / (1 - E^G)
N = net premium
E^N = percent of claim expenses
E^F = fixed expenses
E^G = percent of premium expenses + profit as a percent of premium - Asset share approach - involves long-term projections of various items, in order to determine the necessary premium
Steps for manual rating of disability coverage
hint: ben - orange drink, no pepperoni please
- determine the BASE RATES/ premium (base premium = base rate x benefit amount)
a. LTD: base rate - I x RSV / 12 where RSV = reserve at time 0
b. STD: base rate = I x D / 12 where D is the expected length of the claim - deduct OFFSET CREDITS - to get the net base premium
- DEMOGRAPHIC ADJUSTMENTS - adjust the net base premium to reflect the person’s salary, industry, occupation, and location
- NON-CLAIM adjustments - the prior steps give the final claim cost. add loadings for commissions, expenses, and premium taxes
- PLAN PROVISION adjustments - adjust for the definition of disability, maximum or minimum monthly benefits, pre-existing clause, and antiselection
- add PROFIT- can be a percent of premium or a needed ROI/ ROE
Formulas for benefit reserves and deferred acquisition cost reserves
- Definitions
x = issue age, t = policy duration, p = survival probability
v = 1/(1+i)
B = Benefits, S = claim cost per unit in force
DE = deferred acquisition expense - Benefit net premium = sum(v^t x tPx x Bt ) / sum (v^t x tPx)
- Benefit reserve
a. prospective: sum(v^t x tPx x Sx+t) - sum(v^t x tPx x BNP)
b. retrospective: sum(BNP x (1+i)^t/tPx - sum(Sx+t x (1+i)^t / tPx) - deferrable acquisition expense net premium (DENP)
= sum(v^t x tPx x DEt ) / sum(v^t x tPx) - deferred acquisition cost DAC = sum (DENP x (1+i)^t / tPx - sum(DEt x (1+i)^t / tPx )
Definitions of types of earnings
hint: nope ee
- net income = total revenue less total expenses
- operating earnings = profit realized from day to day operations, excludes taxes, interest income and expense, and extraordinary items
- pro forma earnings - revenue less expenses after omitting items the company believes might cloud perceptions of the true earning power of the business
- EBIT is earnings before interest and taxes
- EBITDA is earnings before interest, taxes, depreciation, and amortization
- EIATBS = earnings ignoring all the bad stuff
Definitions of types of cash flows
hint: Norwegian cod fish delights
- net cash flow = net income + noncash items
- cash flow from operating activities = net cash flow plus or minus changes in current assets and liabilities
- free cash flow = total cash available for distribution to owners and creditors after funding all worthwhile investment activities
- discounted cash flow = a sum of money today having the same value as a future stream of cash receipts and disbursements
Techniques for forecasting external funding needs
hint: please call cabs
- pro forma statement - a prediction of what the company’s financial statements will look like at the end of the forecast period. Is the recommended approach for most planning purposes and for credit analysis.
External funding required = total assets - (liabilities + owner’s equity) - Cash flow forecast - a forecast of sources and uses of cash. Straightforward and easily understood, but less informative than a pro forma statement.
External funding required = total uses - total sources - Cash budget - a forecast of cash receipts and disbursements. Is appropriate for short-term forecasting and the management of cash.
Ending cash = beginning cash + total cash receipts - total cash disbursements
External funding required = minimum desired cash - ending cash
Definition of sustainable growth rate
- the sustainable growth rate represents the limit on a company’s growth if there is no external source of capital
- g = R x ROEbop
R = earnings retention rate = 1 - dividends / earnings
ROEbop = earnings/ equity at beginning of period
3 Since ROEbop = PAT, then G = PRAT
P = profit margin
A = asset turnover ratio
T = financial leverage = assets to equity ratio
Therefore, increase to g, one of P, R, A, or T must increase
- Since ROA = profit margin x asset turnover ratio, g = RT x ROA
Formula for the Gordon Constant Growth Model
1. P/D = 1/(k-g) P = price D = dividends one year from now k = required rate of return or discount rate G = growth rate of dividends
- this formula shows that maximizing value (represented by the price to earnings ratio or P/D) is accomplished by maximizing growth (g)
The components of ROE (the Dupont formula)
ROE = total asset turnover x net profit margin x leverage ratio
asset turnover = revenue / assets; measure of how much investment required to meet requirements for the business
net profit margin = net income/ revenue; measure for every dollar of sales, what percent is profit; asset turnover x net profit margin = return on assets
leverage ratio = assets / equity; to what degree can creditors’ money magnify ROA
Common income statement ratios for health insurers
hint: a house overlooks Narnia
- administrative expense ratio = administrative expenses/ revenues
- health benefit ratio (or loss ratio) = health benefit expenses/ premium revenues
- for simple insured business (ie no non-premium revenues such as ASO fees)
a. operating profit margin = operating profits / revenues = 1 - health benefit ratio - administrative expense ratio - net margin = net income / revenues
a. net income = operating profits + investment income - interest expense - income taxes
Methods for coordinating benefits with Medicare
hint: SECS
C = covered expenses
M = Medicare payment
% represents the application of the employer’s benefit provisions
- standard coordination of benefits: plan payment = the lesser of (C * %) or (C-M). The plan pays the lesser of regular plan benefits or covered expenses - Medicare
- exclusion: plan payment = (C-M)*% exclude the Medicare payment, then apply the benefit formula of the secondary plan
- carveout: plan payment = (C*%) - M. Apply the benefit formula first, then subtract the Medicare payment
- Supplemental plans (Medigap)
Formula for accrued or prepaid expense for postretirement benefits
- an accrued expense for postretirement benefits results when the cumulative amount expensed exceeds the cumulative cash outlay
- a prepaid expense for postretirement benefit results when the cumulative cash outlay exceeds the cumulative amount expensed
- accrued expense = prior period accrued expense + net periodic postretirement benefit cost - contributions (plan funding and benefit payments)
Components of the IAS 19 benefit expense
hint: SIR POG
benefit expense = service cost + interest cost - expected return on assets + amortizations + other items
- service cost - the benefit earned by the employee during the plan year minus any employee contributions
- interest cost - equals the defined benefit obligation multiplied by the discount rate
- expected return on assets - equals the expected rate of return multiplied by the discount rate
- amortization of past service costs - these costs must be amortized over the average period until these benefits become vested
- amortization of gains and losses - this amortization amount is recalculated each year by dividing the amount to be amortized by the average future working lives of the population. can defer recognition until the amount exceeds 10% of the greater of the BDO or the fair value of assets
- other items - such as the effect of any curtailment or settlement
Cost formula (involving price tags)
cost = expected claims + expenses + credits - price tags
Shadow DAC adjustment =
shadow DAC adjustment = primary DAC balance - shadow DAC balance
primary DAC balance = DAC asset in the balance sheet for GAAP income statement purposes
shadow DAC balance = DAC balance as if the UHG&Ls had been recognized
External funding required =
external funding required = total assets - (liabilities + owners’ equity)
What is DPAC?
Deferred policy acquisition cost is the acquisition expense that is capitalized. Expenses are capitalized and amortized to match expenses to revenues for GAAP reporting. DPAC is not used for STAT reporting.
DPAC = adjusted aquisiition expense x UPR
adjusted acquisition expense = acquisition expense ratio - (expense loss ratio + acquisition expense ratio + maintenance expense ratio - 100%)
interest cost formula involving APBO
interest cost = (APBO + service cost) x d - benefits paid x d/2
value of gain or loss recognized under FAS 106
GoL = gain or loss = actual APBO - expected APBO
=assets BOY + assets contributed - benefit payments - asset EOY
expected APBO = (prior APBO + prior service cost)(1 + d) - prior benefit payments x (1 + d/2)
the amount that must be recognized is:
[GoL - 10% x max( APBO, assets) ] / remaining time to amortize
loss means actual APBO is greater than expected APBO
How to calculate pooled loss ratios for catastrophic claims greater than Y
- calculate LR for claims < Y
- pooled LR = (step 1) + (pooled claims > Y) / total premium
- pooled LR = (step 1) + (step 2)
How to calculate pooled loss ratios with a loss ratio limited to Y% prior to pooling charges
- total loss ratio = total claims / premium
- pooled claims = max (0, premium x (total loss ratio - Y%))
- pooling charge = pooled claims / total premium
- pooled loss ratio = min(Y, step 1) + pooling charge
How to calculate pooled loss ratio using credibility weighting
pooled loss ratio = (total loss ratio for subgroup x C) + (total loss ratio for aggregate group x (1-C))
where C = percentage credibility for each subgroup
How to calculate pooled loss ratio using multi-year avearging
pooled loss ratio = 5/9 x LR from current year + 3/9 x LR from 1 year ago + 1/9 x LR from 2 years ago
gross premium formula
G = ( N*(1 + EN) + EF ) / (1 - EG)
N = net premium EN = expenses related to net premium EF = fixed expenses EG = expenses related to gross premium
Sustainable growth rate formula involving ROE =
leveraged ratio =
sustainable growth = ROE x retention rate
leveraged ratio = ROE / ROA
Adverse selection formula
adverse selection = actual claims - actuarial value
Factors of business that contribute to growth based on DuPont formula
Growth = ROE = ROA x total leveraged ratio
factors:
total asset turnover = revenue / assets
-explains how much total investment is required to meet the requirements of the business
net profit margin = net income / revenue
-explains what return on sales the business achieves
total leverage ratio = total assets / shareholder’s equity
-explains to what degree the business can be operated by leveraging other people’s money
Needed rate change =
projected loss ratio / desired loss ratio - 1
Primary insurance amount (PIA) =
PIA = 90% x benpoint 1 + 32% x (bendpoint 2 - bendpoint 1) + 15% x (AIME - bendpoint 2)
bendpoints are adjusted for inflation each year
taxes paid =
taxes paid = provision for income taxes + increase in balance sheet tax asset accounts + decrease in tax liability accounts
goodwill asset =
price - replacement value
retention rate =
1 - dividend payout ratio
Fackler formula
(BR at time t-1 + NP)*(1 + i) = (BR at time t x (1 - q) + claim cost * (1 - q)
BR = benefit reserve NP = net premium
Base rate in terms of reserves
Base rate = reserve x incidence rate
unearned premium reserve =
premium paid x months remaining to be covered / total months to be covered
Various definitions for earnings
operating profit = premiums - benefits - expenses
operating income = EBIT = premiums - benefits - expenses + investment income
net income = pretax income - taxes
expected return on assets =
(assets x ROR) - (benefit payments x ROR / 2)
STD base rate =
monthly premium for STD =
STD base rate = I x D / 12
- I = incidence
- D = duration (in weeks)
Monthly premium = I x D x weekly benefit / 12
formula for net postretirement benefit cost
NPPBC = service cost + interest cost - expected return on assets + amortization of transition obligation + amortization of loss + amortization of prior service cost
NPPBC Amortization of loss =
(loss - corridor) / remaining service
loss = actual APBO - expected APBO
corridor = 10% x max (APBO, assets)
NPPBC expected ROA =
assets x ROR - benefit payments x ROR/2
loss ratio =
(incurred claims + certain taxes) / (earned premium - taxes - regulatory fees)
Under IAS 19, net liability =
defined benefit obligation - fair value of plan assets - unrecognized past service cost - unrecognized loss
Amount to recognize on balance sheet to recognize the pension for employees based on IAS 19
amount to recognize = DBO + unrecognized gain - unrecognized past service cost - value of plan assets
