fin-10:risk and return - actuarial considerations

CAS Seminar on RatemakingCAS Seminar on RatemakingLas Vegas, NeLas Vegas, Ne

March 11-13, 2001March 11-13, 2001

Moderator/PanelistModerator/Panelist

Robert F. WolfRobert F. WolfWilliam M. Mercer/MMC Enterprise RiskWilliam M. Mercer/MMC Enterprise Risk

PanelistsPanelists

Russ BinghamRuss BinghamHartford Financial ServicesHartford Financial Services

FIN-10:Risk and Return - FIN-10:Risk and Return - Actuarial ConsiderationsActuarial Considerations

AgendaAgenda

Overview Overview Net Present Value & Internal Rate of Return Net Present Value & Internal Rate of Return

Models - Characteristics and Considerations Models - Characteristics and Considerations Complete Rate of Return ModelComplete Rate of Return Model Compare and ContrastCompare and Contrast Questions and AnswersQuestions and Answers FIN-11: Parameter Estimation/ Current FIN-11: Parameter Estimation/ Current

ResearchResearch

Marginal Balance Sheet Marginal Balance Sheet ImpactImpact

Useful to Look at a hypotheticalBalance Sheet where all elementsare at market values, not statutoryaccounting values.

Assets Liabilities

Capital

Let K = Policyholder Supplied Funds.

Let S = Shareholder Supplied Funds

K

S

K+S


Let RA = Return on Assetssupplied by both policyholders and shareholders.

RL = Cost of Debt. BorrowingFrom Policyholders. Borrowing PHSF

RE = Cost of Capital. Using SHSF

K+S K

S

Returns RA

Costs RL

Costs RE


This relationship developsinto the generally acceptedview that an insurance companyis a levered trust.

K+S K

S

Returns RA

Costs RL

Costs RE

Levered Trust

(K+S)RA = KRL + SRE

MissionMission

Determine Fair Determine Fair PremiumPremium


Product Market Supply = F(cost of capital)Policyholders supply funds (premiums) in return for compensationfor adverse financial outcomes from fortuitous contingent events.PHSF Flow = Premiums in return for expenses and losses.

Financial MarketStockholder Invests S to Back Policyholder Supplied Funds (K)In Return, Equityholders Demand a Return (Re) on Stockholder Supplied Funds (S) (Dividends or futureappreciation of enterprise value)SRe

K

S

…….So Equityholders want SR.So Equityholders want SREE

Insurance Company Can Provide Insurance Company Can Provide Equityholders a Portion of Their Return by Equityholders a Portion of Their Return by

Investing S in portfolio of securities.Investing S in portfolio of securities....Have to return the remainder

S (RE-RA)

from Insurance Operations (i.e. Returns on Policyholder Supplied Funds)

Derivation of Derivation of Equilibrium Equilibrium

Underwriting ProfitUnderwriting Profit

…….because equityholder are taking the .because equityholder are taking the risk. risk.

They can achieve RThey can achieve RAA by investing in by investing in the same portfolio of securities on the same portfolio of securities on

their own.their own.

Why is RWhy is REE> R> RAA??

Fair Rate of ReturnFair Rate of Return

Solve for RL such that Stockholder demand in returns in excess of investment returns equates to the economic return on

Policyholder Supplied Funds

S (RE-RA)

SHSF

K(RA-RL)

PHSF=

Cost of Debt Capital ===> Cost of Debt Capital ===> Profit LoadProfit Load

…………If we solve for RIf we solve for RL, L, RRL L = R= RAA - - (S/K)(R(S/K)(REE- R- RAA))

RRLL should serve as Risk Adjusted Discount should serve as Risk Adjusted Discount Rate for Loss Reserves Rate for Loss Reserves

Risk Adjusted Rate < RRisk Adjusted Rate < RAA

Let RLet Ruu = Underwriting Profit Margin = Underwriting Profit Margin

RRUU = - K R = - K RLL/Premium/Premium

Insurance Company Earns Positive Economic Insurance Company Earns Positive Economic Returns on Underwriting if RReturns on Underwriting if RAA > R > RLL (R (Ruu> - > - (K/Premium) R(K/Premium) RAA ) )

Fair Rate of ReturnFair Rate of Return

Solve for RU such that Stockholder demand in returns in excess of investment returns equates to the economic return on

Policyholder Supplied Funds

(S/Prem) (RE-RA)

SHSF

(K/Prem) RA+RU

PHSF=

Solving for Return on EquitySolving for Return on Equity

RREE = (1 + K/S)R = (1 + K/S)RAA + (Prem/S)R + (Prem/S)RUU

ParametersParameters

RREE, R, RAA, K, S, K, S

What Should You Use for Each?What Should You Use for Each?

…….We get the usual leverage formula:

Investment Leverage

Underwriting Leverage

Considerations: Considerations: ParametersParameters

Cost of Capital (RCost of Capital (REE))– Dividend Growth Dividend Growth

ModelModel– CAPMCAPM– Cost of Holding Cost of Holding

CapitalCapital RBCRBC Best’sBest’s Undiscounted ReservesUndiscounted Reserves

Policyholder Policyholder Supplied Funds (K)Supplied Funds (K)

– Business as UsualBusiness as Usual Investment Income Investment Income

(R(RAA)-)-– New Money YieldsNew Money Yields– Imbedded YieldsImbedded Yields– Risk Free RateRisk Free Rate

How Much Capital (S)How Much Capital (S)– Allocated v. Allocated v.

ApportionedApportioned– Marginal Marginal

P = D (1+Growth)/(1+ Cost of Capital) + D(1+Growth)2/(1+ Cost of Capital)2+... = D(1+Growth)/(Cost of Capital - Growth)

Discounted Cash Flow Discounted Cash Flow ModelsModels

Two General TypesTwo General Types

Net Present Value Net Present Value

Internal Rate of ReturnInternal Rate of Return

Net Present Value ModelsNet Present Value Models

More Emphasis on Policyholder and Insurance More Emphasis on Policyholder and Insurance Company Flows (Myers/Cohn)Company Flows (Myers/Cohn)

Select IRR = Cost of CapitalSelect IRR = Cost of Capital NPV = CFNPV = CF11 / (1+IRR) + CF / (1+IRR) + CF22/(1+IRR)/(1+IRR)22+ ...+ ...

If NPV > 0, Good DealIf NPV > 0, Good Deal If NPV < 0, Bad DealIf NPV < 0, Bad Deal Set Premiums P, such that NPV = 0Set Premiums P, such that NPV = 0 Solve for RSolve for Ru u : P = : P = L(1+G) + FL(1+G) + F

1- V-R1- V-Ruu

Internal Rate of ReturnInternal Rate of Return

Policyholder Supplied Funds important only Policyholder Supplied Funds important only to extent it effects Shareholders and the to extent it effects Shareholders and the Insurance Company FlowsInsurance Company Flows

0 = CF0 = CF11 / (1+IRR) + CF / (1+IRR) + CF22/(1+IRR)/(1+IRR)22+ ...+ ...

Solve for IRRSolve for IRR If IRR > Cost of Capital then Good DealIf IRR > Cost of Capital then Good Deal If IRR < Cost of Capital, then Bad DealIf IRR < Cost of Capital, then Bad Deal Set Premiums P, such that IRR = Cost of Set Premiums P, such that IRR = Cost of

CapitalCapital

Discounted Cash FlowsDiscounted Cash Flows

ExamplesExamples

Certain and Uncertain CFsCertain and Uncertain CFs

DCF Model - Cashflows are DCF Model - Cashflows are CertainCertain

Frame 1: Cash Flows are Risk Free

Assumptions surplus to premium 0.5 Premiums 100loss ratio 75.0% capital 50expense ratio 25.0%risk free rate 6.0%

Investm InvestibleTime Surplus Premium Expenses Losses Income Funds

0 50.00 100.00 -25.00 125.001 -57.50 -75.00 7.50 0.002

Cash FlowsTime CFs NPV Profit Load = 0.0%

0 -50.00 -50.001 57.50 54.25

Sum 7.50 4.25

All CashFlows areRisk Free.

Hence all cashflowsare discountedat the risk free rate

CFs are at end of period.Loss Paid at end of year

Solve for LR such that Solve for LR such that NPV=0NPV=0

Frame 2 : Cash Flows are Risk Free

Assumptions surplus to premium 0.5 Premiums 100loss ratio 79.5% capital 50expense ratio 25.0%risk free rate 6.0%


0 50.00 100.00 -25.00 125.001 -53.00 -79.50 7.50 0.002

Cash FlowsTime CFs NPV Profit Load = -4.5%

0 -50.00 -50.001 53.00 50.00

Sum 3.00 0.00

Risky CashflowsRisky CashflowsFrame 3 : Cash Flows are Risky

Assumptions surplus to premium 0.5 Premiums 100loss ratio 75.0% capital 50expense ratio 25.0%risk free rate 6.0%Certainty Equiv LR 79.5%


0 50.00 100.00 -25.00 125.001 -53.00 -79.50 7.50 0.002

Cash FlowsTime CFs NPV Profit Load = 0.0%

0 -50.00 -50.001 53.00 50.00

Sum 3.00 0.00

By Definition, Inv Income Certainty Equivalent is risk-free rate

Indifference between a certain loss ratio of 79.5% and an uncertain LR of 75.0%

Another Hypothetical Another Hypothetical ExampleExample

AssumptionsAssumptions– Premiums of $100 Premiums of $100

Paid 80% Paid 80% @Inception, 20% a @Inception, 20% a year lateryear later

– Losses are paid at Losses are paid at the end of each of the end of each of the next three the next three years in years in proportions of proportions of 30%,20%, 10%30%,20%, 10%

– Expense Ratio is Expense Ratio is 20% of premium, 20% of premium, 75%of which is 75%of which is paid @inceptions, paid @inceptions, 25% a year later.25% a year later.

– Investment Yield is Investment Yield is 10.0%10.0%

– No Federal Income No Federal Income TaxesTaxes

– Reserve/Surplus Reserve/Surplus Ratio is 2.5Ratio is 2.5

Discount Cash Flows Discounted at the Discount Cash Flows Discounted at the Rate of the Cost of CapitalRate of the Cost of Capital

Frame 4 : NPV ExampleAssumptions

Premiums 100.00 80.00 20.00 Profit Margin -5.0%Losses 85.00 42.50 25.50 17.00 Loss Ratio 85.0%Expenses 20.00 15.00 5.00 Expense Ratio 20.0%Cost of Capital 15.0% IRR 15.0%Reserves/Surplus 2.5Inv Yield 10.0%

SurplusTime Premium Losses Expenses Flow Inv Inc CF NPV

0 80.00 0.00 15.00 40.00 -40.00 -40.001 20.00 42.50 5.00 -28.50 10.50 28.50 24.782 25.50 -16.15 5.95 16.15 12.213 17.00 -9.18 2.38 9.18 6.04

100.00 85.00 20.00 -13.83 18.83 13.83 3.03

Balance Sheet Items

Inv Other Reqd AddlTime Assets Assets LRsv UEPR Surplus Surplus

0 105.00 35.00 0.00 100.00 40.00 0.001 59.50 0.00 42.50 0.00 17.00 0.002 23.80 0.00 17.00 0.00 6.80 0.003 0.00 0.00 0.00 0.00 0.00 0.00

With a -5.0% profit load,NPV >0, therefore weshould write these policies

Solve for Premium such Solve for Premium such that NPV=0that NPV=0

Frame 5 : NPV ExampleAssumptions



0 76.70 0.00 14.38 38.35 -38.35 -38.351 19.18 42.50 4.79 -23.12 10.07 23.12 20.112 25.50 -16.15 5.95 16.15 12.213 17.00 -9.18 2.38 9.18 6.04

95.88 85.00 19.18 -10.10 18.40 10.10 0.00

Balance Sheet Items


0 100.67 33.56 0.00 95.88 38.35 0.001 59.50 0.00 42.50 0.00 17.00 0.002 23.80 0.00 17.00 0.00 6.80 0.003 0.00 0.00 0.00 0.00 0.00 0.00

One can write at an 88.7%LRto cover cost ofcapital

A -8.7%profit loadis floorbenchmark

Set Discounted Cash Flows Set Discounted Cash Flows to 0 and Solve for IRRto 0 and Solve for IRR

Frame 6 : IRR ExampleAssumptions



0 80.00 0.00 15.00 40.00 -40.00 -40.001 20.00 42.50 5.00 -28.50 10.50 28.50 23.642 25.50 -16.15 5.95 16.15 11.123 17.00 -9.18 2.38 9.18 5.24

100.00 85.00 20.00 -13.83 18.83 13.83 0.00

Balance Sheet Items


0 105.00 35.00 0.00 100.00 40.00 0.001 59.50 0.00 42.50 0.00 17.00 0.002 23.80 0.00 17.00 0.00 6.80 0.003 0.00 0.00 0.00 0.00 0.00 0.00

With a -5.0% profit load,the IRR = 20.5%therefore weshould write these policies as itexceeds cost of capital of 15.0%

Solve for Premium such Solve for Premium such that IRR=Cost of Capitalthat IRR=Cost of Capital

Frame 7 : IRR ExampleAssumptions



0 76.70 0.00 14.38 38.35 -38.35 -38.351 19.18 42.50 4.79 -23.12 10.07 23.12 20.11 23.122 25.50 -16.15 5.95 16.15 12.21 16.153 17.00 -9.18 2.38 9.18 6.04 9.18

95.88 85.00 19.18 -10.10 18.40 10.10 0.00

Balance Sheet Items


0 100.67 33.56 0.00 95.88 38.35 0.001 59.50 0.00 42.50 0.00 17.00 0.002 23.80 0.00 17.00 0.00 6.80 0.003 0.00 0.00 0.00 0.00 0.00 0.00

Again, -8.7%profit loadis floorbenchmark

One can write at an 88.7%LRto cover cost ofcapital.

Assumption VariationsAssumption Variations

If Premium Payment If Premium Payment Patterns are Revised Patterns are Revised From 80/20 Payouts From 80/20 Payouts to 45/45/10, then to 45/45/10, then IRR moves from IRR moves from 20.5% to 13.2%. If 20.5% to 13.2%. If revised from 80/20 revised from 80/20 to 100/0, the IRR to 100/0, the IRR moves from 20.5% moves from 20.5% to 23.8%to 23.8%

If Loss Payments If Loss Payments Revised From Revised From 50/30/20 to 50/30/20 to 20/30/50, the IRR 20/30/50, the IRR moves from 20.5% moves from 20.5% to 23.4%. If to 23.4%. If revised to 90/10, revised to 90/10, then the IRR then the IRR moves to 15.2%moves to 15.2%

Assumption VariationsAssumption Variations

If Less Surplus is If Less Surplus is Required, say Required, say reserves/surplus reserves/surplus ratio = 3, then ratio = 3, then IRR moved from IRR moved from from 20.5% to from 20.5% to 22.5%22.5%– More LeverageMore Leverage

If More Surplus is If More Surplus is Required, say Required, say Reserves/Surplus Reserves/Surplus Ratio = 2, then Ratio = 2, then IRR moves from IRR moves from 20.5% to 18.5% 20.5% to 18.5% – Less LeverageLess Leverage

Myers-Cohn (a Particular Myers-Cohn (a Particular NPV Application) NPV Application)

AssumptionsAssumptions

Insurance Company Invests Efficiently - RInsurance Company Invests Efficiently - Ruu Should Not Compensate for Inefficient Should Not Compensate for Inefficient Insurer Investment PortfoliosInsurer Investment Portfolios

Equityholders (SH) are Efficient Investors - Equityholders (SH) are Efficient Investors - RRuu Should not Compensate for Inefficient Should not Compensate for Inefficient EquityholdersEquityholders

S: Surplus Can be Imputed to a PolicyS: Surplus Can be Imputed to a Policy Underwriting Models Should Only Reflect Underwriting Models Should Only Reflect

Systematic Risk (i.e. Risk That is Systematic Risk (i.e. Risk That is Undiversifiable)Undiversifiable)

Myers-Cohn (a Particular Myers-Cohn (a Particular NPV Application) NPV Application)

AssumptionsAssumptions

… ….also directly considers the double .also directly considers the double taxation issue for shareholders and taxation issue for shareholders and considers is a cost born by considers is a cost born by PolicyholdersPolicyholders

Myers-Cohn EquationMyers-Cohn Equation

Net Present Value of Policy Net Present Value of Policy

==

Present Value of Collected PremiumPresent Value of Collected Premium

--

The Present Value of Loss and Loss Adjustment ExpenseThe Present Value of Loss and Loss Adjustment Expense

--

Present Value of Other ExpensesPresent Value of Other Expenses

--

Present Value of Tax on Underwriting ProfitPresent Value of Tax on Underwriting Profit

--

Present Value of Tax on Investment Income on Policyholder Present Value of Tax on Investment Income on Policyholder and Stockholder Supplied Fundsand Stockholder Supplied Funds

Capital Asset Pricing ModelCapital Asset Pricing Model

…….One Approach to estimate risk adjusted rate uses

Capital Asset Pricing ModelCapital Asset Pricing Model

Expected Return

Ris

k P

rem

ium

1.0 1.5 = Risk

20.1

R*= 15.4

Rf= 6.0

R* = rf + * (market risk premium)

Expected Return

Capital Asset Pricing ModelCapital Asset Pricing ModelQuick ReviewQuick Review

Investors are Risk AverseInvestors are Risk Averse Only Care About Mean and Variance Only Care About Mean and Variance

of Portfoliosof Portfolios E(RE(Rii) = R) = Rff + B + Bii (E(R (E(Rmm) - R) - Rff))

RRi i = Rate of Return on Asset i= Rate of Return on Asset i RRf f = Risk Free Rate= Risk Free Rate RRm m = Rate of Return on Market Portfolio= Rate of Return on Market Portfolio BBi i = Cov (R= Cov (Rii.R.Rmm)/Var (R)/Var (Rmm))

Estimation of Underwriting Estimation of Underwriting BetasBetas

BBee = (1+K/S)B = (1+K/S)Baa +(1/S)B +(1/S)Bu .u .

One Way to do it. See Derivation in Appendix.One Way to do it. See Derivation in Appendix.

Observations on EquilibriumObservations on Equilibrium

E(RE(Ruu) = -KR) = -KRf f + B+ Buu(E(R(E(Rmm)-R)-Rff))

Derivation in AppendixDerivation in Appendix Similar to Certainty-Equivalent FormulaSimilar to Certainty-Equivalent Formula Does Not Depend on Investment IncomeDoes Not Depend on Investment Income Does Not Depend on any Return on Equity Does Not Depend on any Return on Equity

TargetTarget Does Not Depend on Premium to Surplus Does Not Depend on Premium to Surplus

LeverageLeverage Risk Premium (E(RRisk Premium (E(Rm)m)-R-Rff) Fairly Stable) Fairly Stable

Estimating Underwriting Estimating Underwriting Betas directly has some Betas directly has some

issues…...issues…...

Line of Business ConsiderationsLine of Business Considerations State Betas Difficult to EstimateState Betas Difficult to Estimate Few Pure Property/Casualty Few Pure Property/Casualty

Insurers Publicly TradedInsurers Publicly Traded Prior Underwriting Profits Based Prior Underwriting Profits Based

Upon Prior Methodologies - Upon Prior Methodologies - NonapplicableNonapplicable

NPV v. IRRNPV v. IRR

Personal Bias - I like NPV better.Personal Bias - I like NPV better. Bad Experiences with IRRBad Experiences with IRR

– Case Study: Captive Feasibility and Case Study: Captive Feasibility and Tax DeductibilityTax Deductibility

Case StudyCase Study

Client is self-insuredClient is self-insured– Deducts Losses when Losses are paid.Deducts Losses when Losses are paid.– If accident year has 10 year pay-out, tax If accident year has 10 year pay-out, tax

deductions are amortized over ten yearsdeductions are amortized over ten years If pay a premium to insurer, tax is If pay a premium to insurer, tax is

deducted upon payment of premium deducted upon payment of premium (implicitly deducting for future paid (implicitly deducting for future paid losses in year one).losses in year one).

Case StudyCase Study

Client can Form a Captive Client can Form a Captive Insurance CompanyInsurance Company– If special conditions are met, If special conditions are met,

premium paid to Captive may be premium paid to Captive may be deducted.deducted.

Feasibility - Is this Worthwhile?Feasibility - Is this Worthwhile?

NPV PerspectiveCost of Capital 15.0%Captive Tax Deductions - current Max Investment = 26.7

CF NPV CF NPV CF NPV0 0 -20.0 -20.0 0 -20.0 -20.01 20 17.4 1 100.0 87.0 1 80.0 69.62 20 15.1 2 0.0 0.0 2 -20.0 -15.13 10 6.6 3 0.0 0.0 3 -10.0 -6.64 10 5.7 4 0.0 0.0 4 -10.0 -5.75 10 5.0 5 0.0 0.0 5 -10.0 -5.06 10 4.3 6 0.0 0.0 6 -10.0 -4.37 5 1.9 7 0.0 0.0 7 -5.0 -1.98 5 1.6 8 0.0 0.0 8 -5.0 -1.69 5 1.4 9 0.0 0.0 9 -5.0 -1.4

10 5 1.2 10 0.0 0.0 10 -5.0 -1.2

100.0 60.3 80.0 67.0 -20.0 6.7

$100 Million of Tax Deductions Taken as Losses are Paid

If you form a Captive, and certain conditions are met…..

…then you can take your deductions as premiums

Cash Flow Difference

If your investment in the Captive is no more than $26.7 Million, then form the captive.

At $26.7 Million, the NPV=0.

NPV ApproachNPV Approach

-10-505

10152025

NPV

5

15

25

35

Marginal Cost of Captive

Net Present Value As longs as As longs as marginal Cost of marginal Cost of the Captive is less the Captive is less than $26.7 than $26.7 Million... …Million... …

It’s a good deal.It’s a good deal.

…..Now Let’s do it the IRR Way

IRR PerspectiveIRR 9.3%Captive Tax Deductions - current Captive Tax Deductions - current

CF NPV CF NPV CF NPV0 0 -20.0 -20.0 0 -20.0 -20.01 20 18.3 1 100.0 91.5 1 80.0 73.22 20 16.7 2 0.0 0.0 2 -20.0 -16.73 10 7.7 3 0.0 0.0 3 -10.0 -7.74 10 7.0 4 0.0 0.0 4 -10.0 -7.05 10 6.4 5 0.0 0.0 5 -10.0 -6.46 10 5.9 6 0.0 0.0 6 -10.0 -5.97 5 2.7 7 0.0 0.0 7 -5.0 -2.78 5 2.5 8 0.0 0.0 8 -5.0 -2.59 5 2.2 9 0.0 0.0 9 -5.0 -2.2

10 5 2.1 10 0.0 0.0 10 -5.0 -2.1

100.0 71.5 80.0 71.5 -20.0 0.0

$100 Million of Tax Deductions Taken as Losses are Paid

Cash Flow Difference

If you invest $20.0 Million in captive, your accelerated tax deductions imply IRR of 9.3% which is < cost of capital of 15.0%

…….therefore bad deal?

Set NPV to 0

IRR ApproachIRR Approach

0.0%5.0%

10.0%15.0%20.0%25.0%30.0%35.0%

IRR

5

15

25

35

Marginal Cost of Captive

Internal Rate of Return

The Greater the The Greater the Cost of the Cost of the Captive the Better Captive the Better the IRR?the IRR?

……the better the the better the Deal?Deal?

……what’s going what’s going on?on?

IRR- Practical PitfallsIRR- Practical Pitfalls

Variation of the Classic IRR Pitfall Variation of the Classic IRR Pitfall “Oil Pump” Case“Oil Pump” Case

……and I fell right into it.and I fell right into it. Moral of the Story, if you can’t Moral of the Story, if you can’t

explain it intuitively, it’s probably explain it intuitively, it’s probably wrong.wrong.

NPV never served me wrong yet.NPV never served me wrong yet. …….Use NPV..Use NPV.

IRR - Practical PitfallsIRR - Practical PitfallsConsider two Choices - Cost of Capital =12.5%

Investment A Borrowing BTime 0 -100.0 Million

Time 1 + 50.0MillionTime 2 +75.0 Million

Time 0 100.0 Million

Time 1 -50.0MillionTime 2 -75.0 Million

IRR(A) = 15.1% IRR(B) = 15.1%

Good deal because 15.1%>12.5%

Bad Deal Because as Borrower, because you want

IRR<Cost of Capital

IRR PitfallsIRR Pitfalls

Property/Casualty Cash Flows may Property/Casualty Cash Flows may have >1 sign reversalhave >1 sign reversal– deposit premiums + audit premiumsdeposit premiums + audit premiums– retrospective premium adjustmentsretrospective premium adjustments– Agents BalancesAgents Balances– Are you borrowing/investing?Are you borrowing/investing?

Reinvestment Rate AssumptionReinvestment Rate Assumption

IRR v NPVIRR v NPVConsider two Investments

Investment A Investment BTime 0 -12.0 Million


Time 0 -12.0 Million


Using NPV, Investment B is better as long as discount rate <20.0%, otherwise Investment A is Better.

Why?

Because NPV(A) < NPV(B) if IRR<20.0%

NPV(B)<NPV(A) if IRR>20.0%

IRR v. NPVIRR v. NPVAgain Consider Two Investments

Investment A Investment BTime 0 -12.0 Million


Time 0 -12.0 Million


Now Using IRR, IRR(A) = 26.3% while IRR(B) = 25.0%

…Implies …..Investment A better than Investment B

True…..only if Cashflows in A are reinvested at a rate 26.3%

ConclusionConclusion

Moral of the Story, if you can’t Moral of the Story, if you can’t explain it intuitively, it’s probably explain it intuitively, it’s probably wrong.wrong.

NPV never served me wrong yet.NPV never served me wrong yet. …….Use NPV..Use NPV. IRR approach = NPV Approach if you IRR approach = NPV Approach if you

define the problem correctlydefine the problem correctly …………and that is Russ’s storyand that is Russ’s story

Appendix 1Appendix 1

One Way to derive BOne Way to derive BUU

Three BetasThree Betas

BBAA = Cov (R = Cov (RAA.R.Rmm)/Var (R)/Var (Rmm))

BBE E = Cov (R= Cov (REE.R.Rmm)/Var (R)/Var (Rmm))

BBUU = Cov (R = Cov (RUU.R.Rmm)/Var (R)/Var (Rmm))

One Approach: Use BOne Approach: Use Ba a and Band Be e and and back into Bback into Buu

Equity BetaEquity Beta

E(RE(Ree) = R) = Rf f + + BBee(E(R(E(Rmm) - R) - Rff))

Estimated via Estimated via Regressing Regressing historical Rhistorical Ree with Marketwith Market

Equityholders Equityholders Require a Rate Require a Rate of Return Based of Return Based upon Their upon Their Systematic RiskSystematic Risk

Unsystematic Risk Unsystematic Risk Can be Diversified Can be Diversified AwayAway

Still Need to Still Need to DetermineDetermine

– E(RE(Raa)?)?

– 1/S ?1/S ?– K ?K ?

Asset Portfolio BetaAsset Portfolio Beta

E(RE(Raa) = R) = Rff + B + Baa(E(R(E(Rmm) - R) - Rff))

Only Systematic Risk is Only Systematic Risk is Contemplated as Identified by BContemplated as Identified by BA.A.

Unsystematic Risk Can Be Unsystematic Risk Can Be Diversified AwayDiversified Away

BBAA = Estimated via Wtd average of = Estimated via Wtd average of elements in Portfolioelements in Portfolio

Decomposition of BetasDecomposition of Betas

BBe e = Cov ( = Cov ( RRe,e,RRmm)/Var (R)/Var (Rmm))

RRee = (1 + K/S)R = (1 + K/S)Raa + (1/S)R + (1/S)Ruu

Therefore,Therefore,

BBee = (1+K/S)B = (1+K/S)Baa +(1/S)B +(1/S)Buu

Derivation of Underwriting Derivation of Underwriting ProfitProfit

E(RE(Ree) = R) = Rf f + + BBe e (E(R(E(Rmm) - R) - Rff) )

= (1 + K/S)E( = (1 + K/S)E( RRaa )+ (1/S)E(R)+ (1/S)E(Ruu))

E( E( RRa a ) = R) = Rff + B + Baa(E(R(E(Rmm) - R) - Rff))

BBee = (1+K/S)B = (1+K/S)Baa +(1/S)B +(1/S)Buu

Therefore,Therefore,

E(RE(Ruu) = -KR) = -KRf f + B+ Buu(E(R(E(Rmm)-R)-Rff))

Appendix 2Appendix 2

Example of Myers/Cohn Example of Myers/Cohn ApproachApproach

Myers- Cohn Myers- Cohn Example GivenExample Given

GivenGiven

Expected Loss and Loss Adjustment Expense = Expected Loss and Loss Adjustment Expense = $10,000 , Paid in Equally at the end of Period 1 $10,000 , Paid in Equally at the end of Period 1 and Period 2, Respectivelyand Period 2, Respectively

Tax Rate on Investment Income = 25.0%Tax Rate on Investment Income = 25.0% Tax Rate on underwriting Profit = 35.0%Tax Rate on underwriting Profit = 35.0% Risk Free Rate = 8.0%Risk Free Rate = 8.0% Reserve Discount Rate = 5.0%Reserve Discount Rate = 5.0% Premium/Surplus Ratio = 2.00Premium/Surplus Ratio = 2.00 No Other ExpensesNo Other Expenses

Myers-Cohn Myers-Cohn ExampleExample

Premium Cash FlowsPremium Cash Flows Premium is Paid at Time 0Premium is Paid at Time 0 Present Value of Premium Flow = PPresent Value of Premium Flow = P

Premium Cash FlowsPremium Cash Flows

Myers-Cohn Myers-Cohn ExampleExample

Present Value of Loss and Loss Present Value of Loss and Loss Adjustment ExpensesAdjustment Expenses

5000/ (1.05) + 5000/ (1.05)5000/ (1.05) + 5000/ (1.05)22 = = 9297.0529297.052

Paid Loss Cash FlowsPaid Loss Cash Flows

Myers-Cohn ExampleMyers-Cohn Example

Present Value of Taxes on Investment Income at Present Value of Taxes on Investment Income at Time 0:Time 0:

(P + .5P)(.08)(.25) / (1.08) = .028P(P + .5P)(.08)(.25) / (1.08) = .028P


(P - .5P + .5P(.5))(.08)(.25)) / (1.08)(P - .5P + .5P(.5))(.08)(.25)) / (1.08)2 = 2 = .013P.013P


(P - 1P + .5P(0))(.08)(.25)) / (1.08)(P - 1P + .5P(0))(.08)(.25)) / (1.08)3= 3= 00

Tax on Investment Earnings Cash FlowsTax on Investment Earnings Cash Flows

Present Value of Taxes Paid on Present Value of Taxes Paid on Investment Income = .028P +.013P Investment Income = .028P +.013P

= .041P= .041P


Tax on Investment Earnings Cash FlowsTax on Investment Earnings Cash Flows


Present Value of Tax on Accrued Present Value of Tax on Accrued Underwriting Profit Underwriting Profit

P(.5)(.35) / 1.08 + P(.5)(.35)/1.08P(.5)(.35) / 1.08 + P(.5)(.35)/1.0822 - - (5000)(.35)/1.05 - (5000)(.35)/1.05(5000)(.35)/1.05 - (5000)(.35)/1.052 =2 =

.312P - 3253.968.312P - 3253.968

Tax on Underwriting Profit Cash FlowsTax on Underwriting Profit Cash Flows


P = 9297.052 + .041P + .312P - P = 9297.052 + .041P + .312P - 3253.9683253.968

P = 6043.08 + .353PP = 6043.08 + .353P P = 9340.155P = 9340.155 P = L / (1-RP = L / (1-Ruu) = 10000/ (1-R) = 10000/ (1-Ruu))

9340.155 = 10000/ (1- R9340.155 = 10000/ (1- Ruu))

RRu u = - .071 = - .071

Computation of Underwriting ProfitComputation of Underwriting Profit

fin-10:risk and return - actuarial considerations

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