How To Write A Financial Model

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1 Formulae Ad Tables The Icfai Uiversity Press

2 Formulae ad Tables Board of EditorS : Prof. V R K Chary, CFA Mr. S Sarkar, CFA Mr. Prakash Bhattacharya, CFA Ms. V D M V Lakshmi, CFA ISBN : The Uiversity, All rights reserved. This book cotais iformatio obtaied from authetic ad highly regarded sources. Although every care has bee take to avoid errors ad omissios, this publicatio is beig sold o the coditio ad uderstadig that the iformatio give i this book is merely for referece ad must ot be take as havig authority of or bidig i ay way at the editors, publishers or sellers. Neither this book or ay part of it may be reproduced or trasmitted i ay form or by ay meas, electroic or mechaical, icludig photocopyig, microfilmig ad recordig or by ay iformatio storage or retrieval system, without prior permissio i writig from the copyright holder. Trademark otice: Product or corporate ames may be trademarks or registered trademarks, ad are used oly for idetificatio ad explaatio without itet to ifrige. Oly the publishers ca export this book from Idia. Ifrigemet of this coditio of sale will lead to civil ad crimial prosecutio. Published by Uiversity Press, 5, Nagarjua Hills, Hyderabad, Idia Phoe : (+9)(040) , 369, 370, 37, 373, 374 Fax : (+9)(040) 3355, icfaibooks@icfai.org, ssd@icfai.org Website : First Editio : 004 Prited i Idia Bulk Discouts Books are available at special quatity discouts whe purchased i bulk by libraries, colleges, traiig istitutios ad corporates. For iformatio, please write to Istitutioal Marketig Divisio, Books, 5, Nagarjua Hills, Hyderabad, Idia Phoe : (+9)(040) , 88, 89, 90, , 369, 370, 37, 373, 374 Fax : (+9)(040) 3355, icfaibooks@icfai.org, ssd@icfai.org, ifo@icfai.org.

3 Cotets Preface A. Formulae Sectio I: Actuarial Priciples ad Practice Measuremet of Iterest Itroductio to Auities Demography 8 Survival Models 9 Mortality Tables 9 Assurace ad Auity Beefits 0 Premiums for Assurace ad Auity Plas 3 Credibility Theory 6 Loss Distributios ad Risk Models 7 Policy Values 9 Surplus ad it s Distributio 9 Sectio II: Ecoomics 0 Supply ad Demad Aalysis 0 Cosumer Behavior ad Aalysis Productio Aalysis Aalysis of Costs Market Structure: Perfect Competitio 3 Market Structure: Moopoly 4 Market Structure: Oligopoly 4 Measuremet of Macro Ecoomic Aggregates 4 The Simple Keyesia Model of Icome Determiatio 5 Icome Determiatio Model icludig Moey ad Iterest 5 Moey Supply ad Bakig System 6 The Ope Ecoomy ad Balace of Paymets 6 Moder Macro Ecoomics: Fiscal Policy, 7 Budget Deficits ad Govermet Debt Sectio III: Fiacial Maagemet 8 Time Value of Moey 8 Risk ad Retur 9 Valuatio of Securities 3 Fiacial Statemet Aalysis 3 Fiacial Forecastig 33 Leverages 34

4 Cost of Capital 35 Capital Structure 36 Divided Policy 37 Estimatio of Workig Capital Needs 38 Ivetory Maagemet 39 Receivables Maagemet 39 Cash Maagemet 4 Capital Expediture Decisios 4 Sectio IV: Fiacial Risk Maagemet 43 Corporate Risk Maagemet 43 Futures 43 Optios 45 Swaps 48 Sesitivity of Optio Premiums 48 Value at Risk 49 Sectio V: Iteratioal Fiace 50 The Foreig Exchage Market 50 Exchage Rate Determiatio 50 Iteratioal Project Appraisal 5 Iteratioal Equity Ivestmets 5 Short-term Fiacial Maagemet 5 Sectio VI: Ivestmet Bakig ad Fiacial Services 53 Moey Market 53 Rights Issues 53 Lease Evaluatio 53 Hire Purchase 55 Cosumer Credit 56 Housig Fiace 57 Veture Capital 57 Sectio VII: Maagemet Accoutig 58 Cost-Volume-Profit Aalysis 58 Stadard Costig ad Variace Aalysis 59 Sectio VIII: Portfolio Maagemet 6 Capital Market Theory 6 Arbitrage Pricig Theory (APT Model) 6 Asset Allocatio 6 Delieatig Efficiet Frotiers 6 Portfolio Aalysis 6

5 Portfolio Performace 64 Bod Portfolio Maagemet 65 Sectio IX: Project Maagemet 66 Appraisal Criteria 66 Risk Aalysis i Capital Ivestmet Decisios 66 Applicatio of Portfolio Theories i Ivestmet Risk Appraisal 67 Social Cost Beefit Aalysis 67 Optios i Ivestmet Appraisal 67 Project Schedulig 68 Project Moitorig ad Cotrol 68 Sectio X: Quatitative Methods 70 Basics of Mathematics 70 Calculus 70 Iterpolatio ad Extrapolatio 7 Cetral Tedecy ad Dispersio 73 Probability 75 Probability Distributio ad Decisio Theory 76 Statistical Ifereces 78 Simple Liear Regressio ad Correlatio 79 Multiple Regressio 80 Time Series Aalysis 80 Idex Numbers 8 Quality Cotrol 8 Chi-Square Test ad Aalysis of Variace 83 Sectio XI: Security Aalysis 85 Bod Valuatio 85 Equity Stock Valuatio Model 87 Techical Aalysis 87 Warrats ad Covertibles 88 Real Assets ad Mutual Fuds 88 Sectio XII: Strategic Fiacial Maagemet 89 Capital Structure 89 Decisios Support Models 89 Workig Capital Maagemet 90 Firms i Fiacial Distress 9 Valuatio of Firms 9 Mergers ad Acquisitios 9

6 B. Tables 93. Iterest Rate Tables: 95 Future Value Iterest Factor Future Value Iterest Factor for a Auity Preset Value Iterest Factor Preset Value Iterest Factor for a Auity 0-0. Stadard Normal Probability Distributio Table t Distributio Table Area i the Right Tail of a Chi-Square (χ ) Distributio Table F Distributio Table Cotrol Chart Factors Table Table for Value of Call Optio as Percetage of Share Price 0-8. Table for N(x) Table for Relatioship betwee Nomial ad Effective Rates of Iterest ad Discout 4-5 C. Formulae Idex 6

7 FORMULAE

8 I. Actuarial Priciples ad Practice. Measuremet of Iterest i. Future Value of a lump sum (Sigle Flow) FV PV( + i) FV Future value of the iitial flow years hece PV Iitial cash flow i Aual Rate of Iterest Life of ivestmet ii. Doublig Period iii. iv. 69 Iterest Rate Future value of a lump sum with icreased frequecy of compoudig FV PV( + i m ) m FV Future value after years PV Cash flow today i Nomial Iterest Rate per Aums m Number of times compoudig is doe durig a year Number of years for which compoudig is doe The relatioship betwee Effective vs. Nomial Rate of Iterest r i m (+ ) m r Effective rate of iterest i Nomial rate of iterest m Frequecy of compoudig per year v. Accumulated value of a Auity ( + i) FVA A i FVA s Accumulatio at the ed of years A Amout deposited/ivested at the ed of every year for years i Rate of iterest (expressed i decimals) Time horizo or umber of istallmets s Accumulated value of a auity vi. Sikig Fud factor i Rate of iterest i ( + i) Number of years

9 Formulae ad Tables vii. viii. ix. Preset Value Iterest Factor of a Auity, i Rate of iterest Number of years Capital Recovery Factor A i( + i) ( + i) i Rate of iterest Number of years Preset Value of a Perpetuity a i i Rate of iterest.. Itroductio to Auities a ( + i) i( + i) ( v ) i. Preset Value of a Immediate Auity Certai, a i a Preset value of a Auity v Preset value of the th paymet payable at the ed of the th year /( + i) ii. Preset Value of a Deferred Auity Certai m a v iii. iv. m Defermet period v + i i Rate of iterest m a Accumulated Value of a Deferred Auity Certai, ( + i) s m Defermet period Number of Auity Istallmets i Rate of iterest s Accumulated value of a Auity Preset Value of a Auity Due, a&& ( + i a ) a Preset value of a Immediate Auity Certai The umber of auity istallmets i The rate of iterest m

10 v. Accumulated Value of a Auity Due, vi. s&& ( + i)s s Preset value of a Immediate Auity Certai The umber of auity istallmets i The rate of iterest Actuarial Priciples ad Practice Preset value of a deferred auity due of Re. oe p.a. for a term of years certai ad the defermet period is beig m years m && a m v && a vii. viii. ix. v i + i The rate of iterest && a Preset value of a Auity due Accumulated value of a deferred auity due of Re. oe p.a. for a term of years certai ad the defermet period is beig m years m && s ( + i)s i The rate of iterest s The accumulated value of a auity Preset value of a immediate perpetuity, a i The rate of iterest Preset value of a perpetuity due, && a d i d The rate of discoutig v.i + i x. Preset value of a deferred Perpetuity with defermet period of m years, where the first paymet is to be made immediately o completio of m years xi. m && a m v i i The rate of iterest v + i Preset value of a deferred Perpetuity with defermet period of m years, where first m v paymet is made oe year after completio of m years i i The rate of iterest v i + i 3

11 Formulae ad Tables xii. Preset Value of a Immediate Icreasig Auity a. (Ia) && a v i a a + v i a&& The preset value of a auity due a The preset value of a auity certai Number of istallmets i The rate of iterest v + i b. Preset value of a icreasig auity due ( Ia ) a&& The preset value of a auity due Number of istallmets i The rate of iterest v + i c. Accumulated value of a icreasig auity due ( Is) ( ) s + i && && s + && i a && && a + && s&& The Accumulated value of a auity due Number of istallmets i The rate of iterest v i Ia i + i xiii. Preset Value of a Immediate Icreasig Perpetuity, ( ) i The rate of iterest xiv. Preset Value of a Icreasig Perpetuity Due, (I a&& ) d The rate of discoutig i The rate of iterest i + i d 4

12 Actuarial Priciples ad Practice xv. xvi. xvii. xviii. The Preset Value of a Icreasig Auity wherei the cosecutive periodical a v auity paymets are i a Arithmetic Progressio A a + D i A The paymet at the ed of first year D The commo differece a The preset value of a Auity certai The umber of istallmets v + i i The rate of iterest The Preset Value of a Icreasig Auity wherei the cosecutive periodical auity paymets are i a Geometric Progressio R v A ( + i) R v + i R The commo multiple i The rate of iterest The umber of istallmets A The amout of first istallmet Accumulated Value of Icreasig Immediate Auity by Re. Oe per aum (Is) s s + i s Accumulated value of a Auity certai Number of auity istallmets i The rate of iterest The Accumulated Value of a Icreasig Auity wherei the cosecutive periodical auity paymets are i a Arithmetic Progressio s A. s + D i A The amout of first istallmet D The amout of commo differece The umber of istallmets s The accumulated value of a auity certai i The rate of iterest 5

13 Formulae ad Tables xix. The Accumulated Value of a Icreasig Immediate Auity wherei the cosecutive periodical auity paymets are i a Geometric Progressio ( ) + i R A ( + i) R A The amout of first istallmet R The commo ratio i The rate of iterest The umber of istallmets xx. xxi. Preset Value of a Immediate Auity of Re. p.a. for a term of years uder which paymets are made p times a year (p) a a i i (p) i The rate of iterest per aum a The preset value of a Auity certai i (p) p (+ i) p Accumulated Value of a Immediate Auity of Re. p.a. for a term of years uder which paymets are made p times a year (p) s s i (p) i v i The rate of iterest per aum v + i s The Accumulated value of a Auity certai i (p) p (+ i) p xxii. Preset Value of a Auity Due of Re. p.a. for years uder which paymets are made p times a year (p) a&& a i i (p) i + p i The rate of iterest per aum a The preset value of a Auity certai i (p) p (+ i) p 6

14 Actuarial Priciples ad Practice xxiii. Accumulated Value of a Auity Due of Re. p.a. for years uder which paymets are made p times a year (p) s&& s i i (p) i + p i The rate of iterest per aum s The accumulated value of a Auity certai i (p) p (+ i) p xxiv. A immediate auity for years where paymet of r are made at each iterval of r years, beig a exact multiple of r ad the umber of paymets beig r xxv. a. Preset value of the above Auity a (/ r) s r a The preset value of a Auity certai for years s The accumulated value of a Auity for r years r b. Accumulated value of the above auity s ra rs (/ r) s r s The preset value of a Auity certai for years s The accumulated value of a Auity for r years r Preset value ad accumulated value of a auity due for years where paymets of r are made at iterval of r years, beig exact multiple of r a. Preset value: && a a (/ r) r a r a The preset value of a Auity certai for years a The preset value of a Auity for r years r b. Accumulated value: && s (/ r) s r a r s The preset value of a Auity certai for years a The preset value of a Auity for r years r xxvi. Capital Redemptio Policies a. The Amout of Aual Premium P s + s + The Accumulated value of a Auity certai for a period of + years at a rate of iterest of i per aum 7

15 Formulae ad Tables b. Sigle Premium A v ( + i) i The rate of iterest per aum The umber of years xxvii. Average Iterest Yield o the Life Fud A The fud at the begiig B The fud at the ed of the year I A + B I I The iterest eared durig the year after paymet of tax xxviii. Office premium A A + l A 3. Demography A Pure premium l Premium loadig factor. i. Crude Death Rate D x000 P ii. iii. D stads for total umber of deaths i a give year, ad P stads for the size of the mid year populatio. Fertility Rates: a. Crude Fertility Rate (CFR) Number of births durig a specified period Total umber of mid-year populatio of wome b. Geeral Fertility Rate (GFR) c. Age-Specific Fertility Rate, at age y (ASFR y ) Marriage Rates: a. Crude Marriage Rate (CMR) b. Geeral Marriage Rate (GMR) Number of births durig a specified period Total umber of mid-year populatio of wome aged betwee 5-49 Number of births i a specified period to wome aged y years Total umber of mid-year populatio of wome aged y years 5 Number of marriages take place durig a specific period Total umber of mid-year populatio M x 000 P + M stads for the total umber of marriages solemized i a give period ad P 5+ stads for the mid-year populatio of age 5 years or more 8

16 c. Age-Specific Marriage Rate at age y (ASMR y ) Actuarial Priciples ad Practice Number of people married at age y durig the year Total umber of mid-year populatio at age y iv. Migratio Rate of ay area r Number of people movig i ad out of area r i a specified period Total umber of populatio i area r at the begiig of the time period v. Depedecy ratio Ecoomically iactive populatio Ecoomically active populatio 4. Survival Models i. The estimated probability of deaths i a iterval computed per uit time, ii. iii. F i i P P h i i+ i F Respective probability desity i the ith iterval P i Estimated cumulative proportio survivig at the begiig of the ith iterval (at the ed of the iterval i ) P i + Cumulative proportio survivig at the ed of the ith iterval h Width of the ith iterval i Expoetial Distributio e λt F(T) λ e ( / m)t m λ Costat death rate i terms of deaths per uit of measuremet m Mea time betwee deaths T Operatig time, Life or age i hours, cycles, etc. Weibull Distributio f(t) β β T β e (T/η) η η f(t) 0,T 0,β 0 ad η > 0 η Scale parameter β Shape parameter (or slope). 5. Mortality Tables i. The probability that a perso of age x years dies withi oe year q x Number of deaths betwee age x ad x + d x lx lx+ Total umber of persos livig at age x l l x x 9

17 Formulae ad Tables ii. iii. iv. The probability that a perso of age x years survives aother oe year p x Number of survivors to age (x + ) Total umber of persos livig at age x Expectatio of life at age x is give by: e x N N x + x + l x w l t t x+ w Termial age Cetral Death Rate: m x qx q x q x The probability that a perso of age x years dies withi oe year x+ v. The probability that a perso of age x years survives aother years vi. vii. p x No. of persos livig at age x + No. of persos livig at age x l l x+ The probability that a perso of age x years dies withi the ext years m q x Total o. of persos dyig betwee ages x ad x + m Total o. of persos livig at age x x l l x l l x x + m The probability that a perso of age x years will die withi years followig No. of deaths betwee ages x + m ad x + m + m years from ow m q x No. of persos livig at age x l l l x+ m x+ m+ x 6. Assurace ad Auity Beefits i. The preset value of a term assurace of Re..00 payable o death durig a year period is give by A x: (vd x + v d x + + v 3 d x v d x + ) l x l x x Age of the perso Number of years the policy is i force i The rate of iterest per aum v l + i d x The umber of deaths betwee age x ad x + l x Total umber of persos livig at age x 0

18 Actuarial Priciples ad Practice ii. iii. iv. The preset value of beefit of Re..00 payable to a isured agaist a pure edowmet policy for years take at a age x is give by: l A v x : l x + x x Age of the perso Number of years the policy is i force i The rate of iterest per aum v l + i l x Total umber of persos livig at age x l x+ Total umber of persos livig at age x + The preset value of beefit of Re..00 payable to a isured agaist a edowmet assurace policy for years take at a age x is give by: A A + A x: x: x: x: A The preset value of beefit i a Term Isurace Policy x: A The preset value of beefit i a Pure Edowmet Policy The preset value of a icreasig whole life assurace o the life of a perso aged x at etry where the sum assured is Re..00 i the first year, Rs..00 i the secod year, Rs.3.00 i the third year ad so o, is give by: (IA) x (vd x + v d x+ + 3v 3 d x+ + 4v 4 d x ) l x x Age of the perso Number of years the policy is i force i The rate of iterest per aum v l + i d x The umber of deaths betwee age x ad x + l x Total umber of persos livig at age x v. Commutatio Fuctios: a. D x v x l x b. C x v x+ d x c. M x C x + C x+ + C x d. R x M x + R x+ vi. Preset value of the assurace beefits to the isured i terms of the commutatio fuctios are as follows: a. Temporary Assurace Policy, A x: M x M D b. Whole Life Assurace Policy, A x D (M x) x x x+

19 Formulae ad Tables vii. viii. ix. c. Pure Edowmet Assurace Policy, d. Edowmet Assurace Policy, A x: e. Double Edowmet Assurace Policy: DA x M M + D x x+ x+ D x A x: f. Icreasig Temporary Assurace Policy: R R M x x+ x+ (IA) x: D x D D x+ x M M + D x x+ x+ R g. Icreasig Whole Life Assurace Policy, (IA) x D h. Special Edowmet Assurace Policy that provides icreasig death beefit ad icreasig survival beefits: (IA) x: R R M + D x x + x + x + i. Deferred Temporary Assurace Policy: x: x t t A A : A + D x:t j. Deferred Whole Life Assurace Policy, t A x A x A x Preset value of a immediate auity for life of Re..00 to a auitat of age Nx+ x years is give by a x D x Preset value of a immediate auity due for life of Re..00 to a auitat of age x years is give by a&& x + a x a x N D x+ x Preset value of a deferred life auity for Re..00 to a auitat of age x years for a defermet period of t years is give by t a x N x + t + D x x. Preset value of a deferred life auity for Re..00 due to a auitat of age x years for a defermet period of t years is give by: xi. t a&& x N x + t D x Preset value of a temporary immediate life auity for life of Re..00 to a auitat of age x years for a term of years is give by N N x + x + + a x: D x D x x x x: t

20 Actuarial Priciples ad Practice xii. Preset value of a deferred temporary immediate life auity for life of Re..00 to a auitat of age x years for a term of years to be started after a defermet period of t years is give by xiii. xiv. xv. t a&& a x: x:+ t a x:t Preset value of a icreasig life auity i terms of commutatio fuctio S x is give by: (Ia) && x S D x x Preset value of a icreasig life auity i terms of commutatio fuctios is give by: ( ) x: S S N Ia [ ] D x x+ x+ x Preset value of a life auity with m umber of paymets i a year is give by: (m) m + Dx+ a a + x: x: m Dx 7. Premiums for Assurace ad Auity Plas i. The amout of level aual premium to be paid by a perso of age x at the begiig of each year to have a term assurace pla for years: ii. iii. iv. x: P M N M x x + N x x + The amout of level aual premium to be paid by a perso of age x at the begiig of each year to have a pure edowmet assurace pla for years: x: P N D x + N x x + The amout of level aual premium to be paid by a perso of age x at the begiig of each year to have a edowmet assurace pla for years: P x: M M + D N N x x+ x+ x x+ x: P + P x: The amout of level aual premium to be paid by a perso of age x at the begiig of each year to have a whole life assurace pla: P x M N x x v. The amout of level aual premium to be paid by a perso of age x at the begiig of each year to have a limited paymet assurace pla for a limited period of t years: vi. Px t M x Nx N x + t If the paymet of premiums is limited to a shorter period t where t < years i a edowmet assurace pla the the level premium is deoted by P t x: M x Mx+ + Dx+ t N N x x+ t P x: t 3

21 Formulae ad Tables vii. viii. ix. The preset value of a decreasig term assurace policy A S(Mx R x+ + R x+ + ) D x S The amout of sum assured i the first year S The amout by which the amout of sum assured decreases i every year R x, M x ad D x are the differet commuicatio fuctios. If P is the et aual premium limited for a fixed period t years Where t 3, the P S( Mx R x+ R x+ + ) N N x x+ t S The amout by which, the amout of sum assured is reduced every year Childre s Deferred Assuraces: a. Aual premium for the Childre s Deferred Whole Life Assurace pla is give by: v x x A x && a + v && a is the age of the child. b. Aual premium for the Childre s Deferred Edowmet Assurace pla, maturig at a age m is give by: v x A :m x + x :m && a v a c. Additioal Aual premium payable durig the defermet period to get the premium waiver beefit i the evet of death of the father durig the defermet period, correspodig to the basic aual premium of Rs.00, give by: 00 && a && a y ( ) x y: x The age of the father o the date of commecemet of the policy. Net sigle premium for a immediate auity of Re..00 per aum payable i arrear every year for years certai ad thereafter durig the life time of the auitat of age x at etry is give by: a + D D x+ x x (a ) x+ x. Net aual premium P payable for t years for the deferred auity of Re. per aum payable m times i a year for years certai ad thereafter durig the lifetime of the auitat of age x at etry with a defermet period of t-years is give by: P a (m) Dx + + t m + a x + + t + D m x + t && s t 4

22 Actuarial Priciples ad Practice xi. Calculatio of premiums whe frequecy of paymet is m times a year: a. Let (m) P x represets the et premium per aum payable for a whole life (m) Px assurace at the ed of the year of death of (x). A premium of is m payable at the commecemet of each m th period of a year which (x) eters. (m) (m) Px Px a&& x m (Px + d) m P x The amout of aual premium (m) a&& x The preset value of auity due where the premiums are paid m times a year i d Discout factor + i i The rate of iterest per aum b. For a edowmet assurace Re. o (x) for a term of years for which premiums are payable m times, we have, P (m) x: P x: m P + d m x: x: { } P Level aual premium c. For whole life limited paymet policy we have, Px t Px(m) t m P + d m x:t Px { } t Level aual premium d. For limited paymet edowmet policy : (m) P x: Px: t m P + d m x:t ( ) xii. P x: t Level aual premium Premiums for additioal risks: a. The sum assured is subject to a iitial debt of td that reduces by D every year. The additioal premium payable for a whole life assurace policy will D(tM x R x + + R x + t + ) be: Px Px N x M x, R ad N x are the commutatio fuctios for additioal risks. 5

23 Formulae ad Tables b. The sum assured is subject to a iitial debt of td that reduces by D every year. The additioal premium payable for a edowmet assurace policy will be: P x: P x: D(t M R + R ) x x + x + t + N + N x x + xiii. Calculatio of Office premium: a. Whole life assurace policy: ( I K ) S Px + + K P a && x ( I K ) K && a x P Office premium P x Level aual premium a&& x The Preset value of a immediate auity due for life of Re..00 to a auitat of age x years I ad I Iitial expeses which are expressed per uit of premium ad per uit of sum assured respectively K ad K Reewel expeses equal to which are expressed per uit of premium ad per uit of sum assumed respectively b. Edowmet Assurace Policy: P 8. Credibility Theory ( I K ) S P x: + + K && a x: ( I K ) K && a x: P Office premium P x: Level aual premium && a x: The Preset value of a immediate auity due for life of Re..00 to a auitat of age x years for a term of years I ad I Iitial expeses which are expressed per uit of premium ad per uit of sum assured respectively K ad K Reewel expeses equal to which are expressed per uit of premium ad per uit of sum assumed respectively. i. Whe Normal approximatio is applied to the Poisso distributio the, the probability (P) that observatio X is withi ± k of the mea µ is give by: P Ф(k ) Number of claims Ф stads for ormal distributio 6

24 Actuarial Priciples ad Practice ii. iii. iv. The stadard for full credibility for severity is give by N 0 CV s 0 The full credibility stadard for frequecy CV s The coefficiet of variatio for the claim size distributio Process variace for pure premium is give by: f S S f Var (PP) µ σ + µ σ µ f Mea of the claim frequecy distributio µ s Mea of the claim severity distributio σ f Variace of the claim frequecy distributio S σ Variace of the claim severity distributio The expected umber required for full credibility of pure premium S F 0 ( + CV ) 0 The ratio betwee the mea pure premium ad the stadard deviatio of pure premiums CV S The coefficiet of severity v. If the Poisso assumptio does ot hold good, geeral formula for the stadard for full credibility is give by: vi. f f S s F {y /k }( σ /µ + σ /µ ) k Allowace for the variace of the observed sampled frequecy rate y Stadard ormal variatio µ f Mea of the claim frequecy distributio µ s Mea of the claim severity distributio f σ Variace of the claim frequecy distributio S σ Variace of the claim severity distributio B U && HLAMANN Credibility is give by N Z N + k N is the umber of observatios ad k is the B u&& hlama credibility parameter. 9. Loss Distributios ad Risk Models i. Poisso Distributio: ii. e a. P (N r) r! r 0,,,... b. Mea c. Variace Logormal Distributio: a. The PDF is defied as: f(x) lx µ exp x > 0 σx π σ r 7

25 Formulae ad Tables iii. b. Mea exp µ + σ c. Variace exp (µ + σ ) [exp (σ ) ] Pareto Distributio: a. The PDF is defied as: f(x) α β β β + x α +, x > 0 b. Mea of a pareto distributio is give by, E(X) β α iv. c. Var (X) Gamma Distributio: αβ (α )( α ) β βx α a. The PDF is defied as: f(x) e (x) 0 x < τ(α) b. Mea α β c. Variace α β v. Idividual Risk Model: vi. a. Expected Aggregate Loss: E(S) E(Y ) j j j q µ j j α Y j The amout of claim from the -th policy q j The probability of a claim from the j-th policy µ The amout of beefit associated with the j-th policy j b. Variace of aggregate loss: Var (S) Var(Y j) j q ( q )µ j j j j Y j The amout of claim from the -th policy q j The probability of a claim from the j-th policy µ j The amout of beefit associated with the j-th policy Collective Risk Model: a. Mea: E(S) E[Y i ] E[] b. Variace Var (S) E[] Var [Y i ] + E[Y i ] Var [] Y i Amout of claim from the i-th policy Number of policies. 8

26 0. Policy Values i. For a Whole Life assurace policy, policy value is give by: Actuarial Priciples ad Practice ii. iii. tv x A x+t, P x A x+t Preset value of Assurace beefits P x Level aual premium The policy value uder prospective method for a Edowmet assurace policy is give by: V t x: x t: t A + x t: t P. x: && a x + t: t A + Preset value of Assurace beefits a age of x + t years P Level Aual Premium x: && Preset value of a immediate auity due for life of Re..00 a x + t : t to a auitat of age x + t years for a term of t years. Uder prospective method, the policy value for Temporary assurace policy is give by: V x: t A P.&& + + x t: t x t: t x t : t a x + t: t A + Preset value of Assurace beefits a age of x + t years P + Level Aual Premium x t : t && Preset value of a immediate auity due for life for Re..00 to a x + t : t. Surplus ad it s Distributio a auitat of age x + t years for a term of t years. i. Loadig profit that is profit due to lower expeses is expressed as: i ( P P E) x + P Total amout of office premium received P Total of premiums take credit for i the last valuatio E Actual expeses i Valuatio rate. 9

27 . Supply ad Demad Aalysis i. Price elasticity of demad ii. iii. iv. a. Poit Elasticity e p Q P P Q II. Ecoomics Q Ifiitisimal chage i quatity demaded P Ifiitisimal chage i price P Origial price of the good Q Origial quatity demaded of the good b. Arc Elasticity e p Q P0 + P P Q + Q 0 Q Chage i quatity demaded P Chage i price of the good P 0 Origial price of the good P New price of the good Q 0 Origial quatity demaded of the good Q New quatity demaded of the good Margial Reveue MR AR ep AR Average reveue e p Price elasticity of demad Icome elasticity of demad e y Q Y Y Q Q Chage i quatity demaded Y Chage i icome of the cosumer Y Icome of the cosumer Q Quatity demaded of the good Cross price elasticity of demad e cij Q P i j P Q j i Q i Chage i quatity demaded of the good i P j Chage i price of the good j P j Price of the good j Q i Quatity demaded of the good i

28 Ecoomics v. Promotioal elasticity of demad e A Q A A Q Q Chage i quatity demaded A Chage i uits of advertisemet expediture o the good A Uits of advertisemet expediture o the good Q Quatity demaded of the good vi. Price-elasticity of supply e s Qs P P Q P Price of the good Q s Quatity supplied of the good s Q s Chage i quatity supplied o the good P Chage i price of the good.. Cosumer Behavior ad Aalysis i. Margial Rate of Substitutio of good X for good Y ii. iii. MRS X,Y MU MU X Y MU X Margial Utility of good X MU Y Margial Utility of good Y Cosumer equilibrium MU P X X MU P Y Y MU X Margial Utility of good X MU Y Margial Utility of good Y P X Price of good X P Y Price of good Y Budget costrait I P X X + P Y Y I Icome of the cosumer X Number of uits of good X Y Number of uits of good Y P X Price of good X P Y Price of good Y.

29 Formulae ad Tables 3. Productio Aalysis i. Average product of labor ii. iii. iv. AP L TP L L TP L Total product of labor L Number of labor uits Margial product of labor MP L TP L L TP L Chage i total product of labor L Chage i the umber of labor uits Margial rate of techical substitutio betwee Labor (L) ad Capital (K) MRTS L,K MP MP L K MP L Margial product of labor MP K Margial product of capital Cost costrait of a firm C 0 wl + rk C 0 A give amout of moey that the firm speds L Number of labor uits K Number of capital uits w Wage rate r Iterest rate v. Efficiet iput combiatio MP MP L K w r MP L Margial product of labor MP K Margial product of capital w Wage rate r Iterest rate. 4. Aalysis of Costs i. TC TFC + TVC TC Total cost TFC Total fixed cost TVC Total variable cost

30 Ecoomics ii. AFC TFC Q TFC Total fixed cost Q Number of uits produced iii. MC TC Q TC Chage i total cost Q Chage i quatity produced iv. Break-eve output (Q) P Price FC P AVC FC Fixed cost AVC Average variable cost. 5. Market Structure: Perfect Competitio i. Profit of a firm (π) TR TC TR Total reveue TC Total cost ii. Tax burde o the buyer iii. e d e s + e s Tax e s Price elasticity of supply e d Price elasticity of demad Profit maximizatio a. First order coditio MC MR AR P b. Secod order coditio TR TC < Q Q TR Total reveue TC Total cost MR Margial reveue MC Margial cost AR Average reveue P Price Q Quatity. 3

31 Formulae ad Tables 6. Market Structure: Moopoly i. Profit maximizatio ii. iii. Margial Reveue (MR) Margial Cost (MC) Lerer Idex of moopoly power a. L b. P MC P P MC P e P Price p MC Margial Cost e p Elasticity of demad The Herfidahl s Idex (H) 3 H S + S + S +...S S % share of the largest firm i the market S % share of the secod largest firm i the market S % share of the th firm i the market. 7. Market Structure: Oligopoly i. Output determiatio Q QP + Q p Output if the market would be a competitive oe Number of firms i Oligopoly. 8. Measuremet of Macro Ecoomic Aggregates i. Gross Net + Depreciatio ii. iii. iv. Market Price Factor Cost + [Idirect Tax Subsidy] Natioal Domestic + Net Factor Icome from Abroad The Laspeyre Price Idex I t t 0 Pi qi i 0 0 Pi qi i 00 0 q i Quatity of ith good purchased i the base year 0 p i Price of the ith good i the base year t q i Quatity of ith good purchased i the curret year t p i Price of the ith good i the curret year v. GNP Deflator Nomial GNP Real GNP. 4

32 Ecoomics 9. The Simple Keyesia Model of Icome Determiatio i. Y C + I + G + E M Y Equilibrium icome C Cosumptio expediture I Ivestmet expediture G Govermet expediture E Exports M Imports ii. Average Propesity to Cosume (APC) C Cosumptio expediture Y Icome C Y iii. Margial Propesity to Cosume (MPC) C Y C Chage i cosumptio expediture Y Chage i icome iv. Average Propesity to Save (APS) S Savigs Y Icome v. Margial Propesity to Save (MPS) S Chage i savigs Y Chage i icome vi. Multiplier (m) [ β( t) π + µ ] S Y S Y β Margial propesity to cosume t Tax coefficiet π Iduced ivestmet coefficiet µ Margial propesity to import. 0. Icome Determiatio Model icludig Moey ad Iterest i. Goods market equilibrium Y C + I + G + X M C Cosumptio expediture Y Icome 5

33 Formulae ad Tables I Ivestmet expediture G Govermet expediture E Exports M Imports ii. Moey market equilibrium M s M d M s Supply of moey M d Demad for moey.. Moey Supply ad Bakig System i. High powered moey (H) Moetary liabilities of the Cetral bak + Govermet moey ii. Multiplier (M) iii. + C C u u + r C u Currecy deposit ratio r Cash reserve ratio Moey supply (M s ) H m H High powered moey m Moey multiplier iv. Fiace Ratio Total Issues Natioal Icome v. Fiacial Iterrelatio Ratio (FIR) vi. New Issue Ratio (NIR) vii. Itermediatio Ratio (IR) viii. Velocity of moey (v) Total Issues Net Capital Formatio Primary Issues Net Capital Formatio Y M s Secodary Issues Primary Issues Y Icome M s Moey supply.. The Ope Ecoomy ad Balace of Paymets i. Trade balace Exports Imports ii. iii. Curret accout balace Credit (Curret accout) Debit (Curret accout) Capital accout balace Credit (Capital accout) Debit (Capital accout). 6

34 Ecoomics 3. Moder Macro Ecoomics: Fiscal Policy, Budget Deficits ad Govermet Debt i. Fiscal Deficit Borrowigs ad other liabilities ii. Primary Deficit Fiscal deficit Iterest paymets iii. Reveue Deficit Reveue expediture Reveue receipts. 7

35 III. Fiacial Maagemet. Time Value of Moey i. Future Value of a Lump Sum (Sigle Flow) ii. iii. FV PV( + k) FV Future value of the iitial flow years hece PV Iitial cash flow k Aual rate of iterest Life of ivestmet Effective rate of iterest r k + m m r Effective rate of iterest k Nomial rate of iterest m Frequecy of compoudig per year Future Value Iterest Factor of Auity FVIFA(k,) ( + k) k k Rate of iterest Time horizo iv. Sikig Fud Factor FVIFA(k,) FVIFA(k, ) v. Preset Value Iterest Factor of Auity PVIFA(k,) Future value iterest factor for auity at k% for years ( + k) k( + k) k Rate of iterest Time horizo vi. Capital Recovery Factor vii. k Rate of iterest Time horizo PVIFA(k, ) Preset Value Iterest Factor of a Perpetuity P /k k Rate of iterest.

36 Fiacial Maagemet. Risk ad Retur i. Rate of retur k D + (P P ) t t t P t k Rate of retur P t Price of security at time t, i.e., at the ed of the holdig period P t Price of the security at time t i.e., at the begiig of the holdig period or purchase price D t Icome or cash flows receivable from the security at time t ii. Expected rate of retur ( ) k i p k k i Rate of retur from the ith outcome p i Probability of the ith outcome Number of possible outcomes i Outcome i iii. Variace of a asset s rate of retur, VAR(k) VAR (k) Variace of returs i i i p (k k) p i Probability associated with ith possible outcome k i Rate of retur from the ith possible outcome k Expected rate of retur Number of years i Outcome i iv. Stadard deviatio, σ VAR(k) v. CAPM model: k j r f + β j (k m r f ) k j Expected or required rate of retur o security j r f Risk-free rate of retur β j Beta coefficiet of security j k m Retur o market portfolio Covim vi. Beta of security i, β i σ Cov im m Covariace of security i with true market σ m Variace of returs o the market idex i i 9

37 Formulae ad Tables vii. Alpha of security i (α) E(r i ) R(r i ) E(r i ) r + β ( E ( r ) r ) f im m f α The differece betwee expected retur ad required retur r f Risk-free rate β im Beta coefficiet of security i E(r i ) Expected retur of security i R(r i ) Required retur from security i E(r m ) Retur o market portfolio viii. Systematic risk of security, i sice R im ρ im β im Beta coefficiet of security i m σ Market variace i σ Variace of security i im im m β σ ρ ρ im σi σm σm im σ i ρ The correlatio coefficiet, ad im R im σ i R The coefficiet of determiatio betwee the security i ad the market portfolio ix. Usystematic Risk, ( σ ) ei Or i im m σ β σ i im σ i σ ρ σi ( ρ im ) σi ( Rim ) 30 σ i Variace of Security i β im Beta coefficiet of security i σ m Market variace ρ im The correlatio coefficiet, ad im R The coefficiet of determiatio betwee the security i ad the market portfolio.

38 Fiacial Maagemet 3. Valuatio of Securities i. Equity Valuatio: a. The itrisic value or preset value of equity share ii. (P 0 ) D t t ( + k ) e t P + ( + k ) e P 0 Curret market price of the equity share or itrisic value of the share D t Expected equity divided at time t P Expected price of the equity share at time k e Expected rate of retur or required rate of retur Ivestmet period t Time t b. The value of equity share whe there is costat growth P 0 D ( + g) 0 k e g D 0 Curret divided per share g Expected costat growth rate i divideds k e Expected rate of retur or required rate of retur Bod Valuatio: a. The itrisic value or the preset value of a bod V 0 or P 0 I(PVIFA kd, ) + F(PVIF kd, ) V 0 Itrisic value of the bod P 0 Preset value of the bod I Aual iterest payable o the bod F Pricipal amout (par value) repayable at the maturity time Maturity period of the bod k d Cost of Capital or Required rate of retur b. Curret yield Coupo Iterest Prevailig Market Price c. Yield to maturity r i the equatio P 0 t I F + t ( + r) ( + r) Maturity period of the bod I Aual iterest payable o the bod F Pricipal amout (par value) repayable at the maturity time 3

39 Formulae ad Tables iii. Valuatio of a Covertible: The value of covertible t C (P ) Coversio ratio + t ( + r) ( + r) C Coupo amout r Required rate of retur P Expected price of equity share o coversio Number of years to maturity. 4. Fiacial Statemet Aalysis i. Liquidity Ratios: ii. iii. a. Curret Ratio Curret Assets/Curret Liabilities b. Quick Ratio Curret Assets Ivetories Curret Liabilities c. Bak fiace to workig capital ratio Leverage Ratios: a. Log -term Debt-Equity Ratio b. Total Debt-Equity Ratio c. Debt-Asset Ratio Coverage Ratios: a. Iterest coverage ratio Short-term bak borrowigs Workig capital gap Log -term debt Net worth Total debt Net worth Total debt Total assets EBIT Iterest EBIT Earig before iterest ad tax b. Cash flow coverage ratio EBILT + D I L LR P ( t) ( t) EBILT Earigs before iterest, lease paymets ad taxes D Depreciatio I Iterest charges L Lease paymets t Margial Tax Rate LR Loa Repaymet P Preferece divided 3

40 Fiacial Maagemet iv. c. Debt Service Coverage Ratio PAT PAT + Depreciatio + Other o-cash charges + Iterest o term loa Re paymet of the term loa Iterest o term loa + t Profit after tax t Margial tax rate Turover Ratios: a. Ivetory turover b. Accouts receivables turover Cost of goodssold Averageivetory Net credit sales Average accouts receivable c. Total assets turover v. Profitability Ratios: a. Gross profit margi b. Net profit margi Net sales Average total assets Gross profit Net sales Profit after tax Net sales c. Retur o ivestmet (Earig Power) EBIT Average total assets EBIT Earig before iterest ad tax Profit after tax d. Retur o Net Worth Average et worth 5. Fiacial Forecastig i. Exteral fiacig requiremet A L EFR ( S) ( S) ms ( d) S S EFR Exteral fiacig requiremet A/S Curret assets ad fixed assets as a proportio of sales S Expected icrease i sales L/S Spotaeous liabilities as a proportio of sales m Net profit margi S Projected sales for ext year d Divided pay-out ratio 33

41 Formulae ad Tables ii. Sustaiable growth rate (g) m Net profit margi d Divided pay-out ratio m( d)a / E A / S m( d)a / E g Sustaiable growth rate with iteral equity A/E Total Assets Equity 0 Curret ad fixed assets as proportio of equity 6. Leverages A/S 0 Curret ad fixed assets as proportio of sales at time 0. i. Degree of Operatig Leverage (DOL) [Q(S V)] / [Q(S V) F] Q Quatity sold S Sellig price per uit V Variable cost per uit F Total fixed deposit EBIT ii. Degree of Fiacial Leverage (DFL) Dp EBIT I ( T) I Iterest amout D p Preferece divided T Tax rate EBIT Earigs Before Iterest ad Tax iii. Degree of Total Leverage (DTL) DOL DFL Q(S V) Dp Q(S V) F I ( T) DOL Degree of operatig leverage DFL Degree of fiacial leverage Q Quatity sold S Sellig price per uit V Variable cost per uit F Total fixed deposit I Iterest amout D p Preferece divided T Tax rate 34

42 Fiacial Maagemet iv. Overall break-eve poit (Q) S Sellig price per uit V Variable cost per uit F Total fixed deposit I Iterest amout D p Preferece divided T Tax rate v. Operatig break-eve poit (Q) S Sellig price per uit V Variable cost per uit F Total fixed deposit D p F + I + ( T) (S V) F (S V) vi. Fiacial break-eve poit (EBIT) I + ( T) I Iterest amout D p Preferece divided T Tax rate. 7. Cost of Capital i. Cost of Term Loas I ( T) ii. iii. I Iterest rate T Tax rate D P I( t) F Cost of Debetures, P + t ( + k ) ( + k ) t d d k d Post-tax cost of debeture capital I Aual iterest paymet per debeture capital t Corporate tax rate F Redemptio price per debeture P Net amout realized per debeture Maturity period D F Cost of Preferece Capital, P + t ( + k ) ( + k ) k p Cost of preferece capital t p p D Preferece divided per share payable aually F Redemptio price 35

43 Formulae ad Tables P Net amout realized per share Maturity period iv. Cost of Equity Capital a. Divided forecast approach, P e D k g P e Price per equity share D Expected divided per share at the ed of oe year k e Rate of retur required by the equity shareholders g Growth rate of divideds b. Cost of Exteral Equity, e k k e D e + P o ( f ) k g e e (Method ) ( f ) k Cost of exteral equity k e Cost of equity D Divided expected at the ed of year (Method ) P o Curret market price per share g Costat growth rate applicable to divideds f Floatatio costs as a percetage of the curret market price v. Weighted Average Cost of Capital E P D k + k + k ( T) E + P + D E + P + D E + P + D e p d E Market value of equity P Market value of preferece capital D Market value of debt k e Cost of equity k p Cost of preferece capital k d Cost of debt T Tax rate. 8. Capital Structure i. Overall capitalizatio rate of the firm B k o k d + k e B + S S B + S k d The cost of debt B The market value of the outstadig debt S The market value of equity k e The cost of equity k o The weighted average cost of capital 36

44 Fiacial Maagemet ii. Preset value of a tax shield of iterest paymets: a. Whe debt is perpetual t c B t c The tax rate o corporate icome B The market value of the debt b. Whe corporate taxes are cosidered the value of the levered firm V O( t c ) + tc B k O Operatig icome t c The tax rate o corporate icome B The market value of the debt k Iterest rate o debt c. If the persoal tax rate is t p, the tax advatage of debt t c B ( t p ) t c The tax rate o corporate icome B The market value of the debt d. Whe the tax rate o stock icome (t ps ) differs from the tax rate o debt icome (t pd ), the tax advatage of debt capital t c The tax rate o corporate icome ( t c )( t ps ) B ( t ) B The market value of the debt. 9. Divided Policy i. Traditioal Model (Graham-Dodd Model), P m (D + E/3) P The market price per share m The multiplier D The divided per share E The earigs per share ii. Walter Model, P D + (E D)r / k P The market price per share D The divided per share E The earigs per share r The iteral rate of retur k e The cost of equity capital iii. Gordo Model, P 0 0 e k e Y ( b) k br P 0 The market price per share at the begiig of period 0 Y 0 The earigs per share for period 0 e pd 37

45 Formulae ad Tables b The retetio ratio (retaied earigs/total earigs) r The retur o ivestmets k e The cost of equity capital or (Cost of capital of firm) iv. MM Approach, P 0 D + P + k e P 0 The market price per share at the begiig of period 0 D The expected divided per share for period P The market price per share at the ed of period k e The cost of equity capital v. Corporate Divided Behavior (Liter Model) D t cr EPS t + ( c) D t D t The divided per share for the time period t c The weightage give to curret earigs by the firm r The target pay-out rate EPS t The earigs per share for the time period t D t The divided per share for the time period (t ). 0. Estimatio of Workig Capital Needs i. Duratios at various stages of productio a. Raw Material Storage Period AverageStock of Raw Material ad Stores Average Raw Materials ad Stores cosumed per day b. Work-i-process period c. Fiished goods storage period d. Average collectio period Average Work-i-process ivetory Average daily cost of productio Average fiished good ivetory Average daily cos t of sales Average accouts receivable Average daily credit sales e. Average paymet period Average accouts payable Average credit purchases per day ii. Net operatig cycle period a + b + c + d e iii. Weighted Operatig Cycle D woc W rm D rm + W wip D wip + W fg D fg + W ar D ar W ap D ap D woc Duratio of weighted operatig cycle W rm Weight of raw material expressed as a percetage of raw material cost to sales D rm Duratio of raw material 38

46 Fiacial Maagemet W wip Weight of work-i-progress expressed as a percetage of work-i-progress cost to sales D wip Duratio of work-i-progress W fg Weight of fiished goods expressed as a percetage of cost of goods sold to sales D fg Duratio of fiished goods W ar Weight of accouts receivables expressed as a percetage of sales to sales D ar Duratio of accouts receivables W ap Weight of accouts payables expressed as a percetage of raw material cost to sales D ap Duratio of accouts payables.. Ivetory Maagemet i. Ecoomic Order Quatity EOQ UF PC uits U Aual usage rate F Orderig cost C Carryig cost P Price per uit ii. Reorder poit S L + F (S R L) S Usage i uits L Lead time i days R Average umber of uits per order F Stock out acceptace factor.. Receivables Maagemet i. Effect of relaxig the credit stadards o profit P S( V) k I b S P Chage i profit S Icrease i sales V Variable costs to sales ratio k Cost of capital I Icrease i ivestmet i receivables S Average collectio period V 360 b Bad debts loss ratio o ew sales V Cotributio to sales ratio 39

47 Formulae ad Tables ii. iii. iv. Effect of icreasig the credit period o profit P S ( V) k I b S The compoets of the formula are same exceptig I (ACP N ACP O ) S V(ACP S N) 360 I Icrease i ivestmet i receivables ACP N New ACP (after icreasig credit period) ACP O Old ACP V Ratio of variable cost to sales S Icrease i sales k Cost of capital S 0 Sales before icreasig the credit period The effect o profit for a chage i cash discout rate P S ( V) + k I DIS S Icrease i sales V Ratio of variable cost to sales k Cost of capital I Savigs i ivestmet i receivables DIS S (ACP S O ACP N ) V ACPN 360 Icrease i discout cost p (S 0 + S)d p o S 0 d o p Proportio of discout sales after liberalizig S o Sales before liberalizig S Icrease i sales d New discout percetage p 0 Proportio of discout sales before liberalizig d 0 Old discout percetage ACP O Average collectio period before icreasig cash discout ACP N Average collectio period after icreasig cash discout Effect of decreasig the rigor of collectio program o profit: P S( V) k I BD P Chage i profits S Icrease i sales V Variable costs to sales ratio k Cost of capital 40

48 Fiacial Maagemet I Icrease i ivestmet i receivables S o 360 (ACP N ACP O ) + S 360 ACP N V BD Icrease i bad debts cost ACP O ACP N b (S o + S) b o S o Average collectio period before relaxig collectio effort Average collectio period after relaxig collectio effort b o Proportio of bad debts to sales before relaxig collectio effort b Proportio of bad debts to sales after relaxig collectio effort. 3. Cash Maagemet i. Baumol Model, TC I (C/) + b (T/C) TC Total costs (total coversio costs + total holdig costs) I Iterest rate o marketable securities per plaig period C Amout of securities liquidated per batch T Estimated cash requiremet over the plaig period b Fixed coversio cost The poit where total costs are miimum: C bt I ii. Miller ad Orr Model, RP LL Lower cotrol limit RP Retur poit UL Upper cotrol limit b Fixed coversio cost 3 3b σ 4I UL 3 RP LL + LL ad, I Iterest rate per day o marketable securities. 4. Capital Expediture Decisios i. Accoutig Rate of Retur ii. (ARR) Net Preset Value (NPV) NPV t Average profit after tax Average book value of theivestmet CFt I t 0 ( + k) k Cost of fuds CF t Cash flows at the ed of the period t I 0 Iitial ivestmet Life of the ivestmet 4

49 Formulae ad Tables iii. Beefit-Cost Ratio (BCR) BCR PV I iv. BCR Beefit-Cost Ratio PV Preset Value of future cash flows I Iitial ivestmet Net-beefit-cost Ratio NBCR NPV I NPV Net preset value I Iitial ivestmet v. Iteral Rate of Retur (IRR) I 0 CFt t t ( + k) k IRR, which is that rate of retur where CF Cash flow I 0 Iitial ivestmet Life of ivestmet. t CFt I t 0 0 ( + k) 4

50 IV. Fiacial Risk Maagemet. Corporate Risk Maagemet i. Historical (ex-post) ii. iii. iv.. Futures a. Arithmetic mea retur, ri r it t b. Variace (risk), σ ( r r ) i it i t c. Stadard deviatio, σ i Variace Expected (ex-ate) a. Expected retur, E(r i ) r P is s i ris E (r i ).Ps s b. Variace (Risk), σ [ ] s r it Historical (ex post) retur geerated by the ith stock i time period t r is Expected (ex ate) retur for the ith stock assumig that S state of the world occurs P s Probability that the S state of the world will occur r i Retur o a security i Estimated retur o a stock (R s ) α + β r m r m Retur o market β Measure of stock s sesitivity to the market idex α Estimated retur whe the market retur is zero Accordig to the CAPM, the required retur o a security R s R f + β(r m R f ) R f Retur o risk-free ivestmet R m Retur o market β Measure of stock s sesitivity to the market idex. i. Effective price Sp + (Ft Ft ) If bases remais the same Effective price Sp Sp Spot price at time t Sp Spot price at time t Ft Futures price at time t Ft Futures price at time t Basis Curret cash price Futures price

51 Formulae ad Tables ii. Margi Iitial margi µ + 3σ µ Mea σ Stadard Deviatio iii. Relatioship betwee the cash price ad the futures price of ay commodity: F t,t C t + C t S t,t T t Gt,T C t Cash price at time t S t,t Aualized iterest rate o borrowigs G t,t Storage costs T t Time period F t,t The futures price at time t, which is to be delivered at time period T iv. Futures positio Hedge Ratio (HR) Uderlyig asset positio v. σsp Miimum variace hedge ratio, h F p σ vi. vii. viii. h Hedge ratio F p Coefficiet of correlatio betwee S p ad F t σ Ft Stadard deviatio of F t σ Sp Stadard deviatio of S p Ft F t Chage of futures price durig hedgig S p Chage i spot price durig hedgig % discout Days to maturity T-bill purchase price Face value IRR (Implied Repo Rate) IRR (FP t,t CP t,t )/(CP t,t ) 360/T t FP t,t Price of futures T-bill CP t,t Cash price of T-bill T t Time period Trasactio price or cash price of the bod, P Quoted price + Accrued iterest Ivoice price (Futures settlemet price Coversio factor) + Accrued iterest Cash market pricipal ix. HR Coversio factor Futures market pricipal 44