Institute of Actuaries of India

Transcription

1 Insttute of Actuares of Inda Subject CT1 Fnancal Mathematcs May 2010 Examnatons INDICATIVE SOLUTIONS Introducton The ndcatve soluton has been wrtten by the Examners wth the am of helpng canddates. The solutons gven are only ndcatve. It s realzed that there could be other ponts as vald answers and examner have gven credt for any alternatve approach or nterpretaton whch they consder to be reasonable.

2 1 () (a) Cost:- Weather Scenaro& Strke Scenaro & Combned PV of Cost Expected Probablty Probablty 10% PV of cost PV of Cost* Probablty Good Weather(2/3) No Strke (1/2) 2/ Good Weather(2/3) Strke (1/2) 2/6 72* Bad Weather(1/3) No Strke(3/4) 3/ v Bad Weather(1/3) Strke(1/4) 1/12 (72+36v)* Total Check : Total probablty should be =1 (2/6 + 2/6 + 3/12 + 1/12 =1) Expected PV of cost = crores Revenue: Expected PV(Revenue) = Prob(Success).PV(Revenue/Success) + Prob(Falure).PV(Revenue/Falure) = 4/5 * 60 * ä 3 v 2 + 1/5 * 30 * ä 3 v 10% = (48+6) * ä 3 v 10% = 54 * = crores NPV = EPV(Revenue) EPV(Cost) = = crores b) Prob(Mn Cost) = 2/6 Prob(Max Revenue) = 4/5 Prob(Best Scenaro) = (2/6)*(4/5)=8/30=4/15 [10] 2. () P = 75v+75*1.03v 2 +75* v * v v 8.25% = 75v( v + (1.03v) 2 + (1.03v) (1.03v) 19 ) v 8.25% =(75/1.0825) ä % v 8.25% = = P = () To fnd n: = 20 a n + 100v 8.5% = v n +100 v n v n = / = n = ln( )/ln(v) where ln = natural logarthm n = ~ 15 years When there s a possblty of default, we have to multply each payment wth the probablty Page 2 of 8

3 of payment. Dscountng factor of t th payment = ( ) t v 10% = (0.8547) t = (1/1.17) t Hence, calculate 17% whch wll be equvalent to multplyng by probablty of default. PV of payment streams = 20 a v 17% = [10] 3. () Let PV(Annuty 1)=A & PV(Annuty 2)=B Then, A = 10v + 9v 2 + 8v v 9 + 1v A(1+) = v +8v v 8 + 1v 9 Subtractng, we get A = 10 (v+v 2 +v v 9 +v 10 ) = 10 a 10 A = 10 a 10. (1) B = v + 2v 2 + 3v v (v 11 +v 12 +v 13 +v 14 + = (Ia) v 10 a = (Ia) v 10. (2) (1)+(2) gves A+B = 10 a 10 + (Ia) v 10 Gven 2A = B. Thus A+B = A+2A = 3A Hence, A+B = 3A = 10 a 10 + (Ia) v 10 => 3A = 10 a 10 + ä 10-10v v 10 = 10 v v 10 =11 Hence A = (11/3 ). (3) From (1) and (3), 10 a 10 = 11/3 => a 10 = 10-11/3 = 19/3 = From tables, at 10% a 10 = At 9% a 10 = By nterpolaton = Substtutng ths value n (3), we get A = 11/(3* )= ~ 39.4 PV(Annuty 1) = 39.4 [10] 4. ) a) (12) =7.75% (12) /12 = % Captal content of t th nstallment = (Installment amount)* v % Page 3 of 8

4 Captal content of 16 th nstallment = v (12n-15) = v (12n-15) = n-15=ln(0.2349)/ln(v) => 12n= ~ 240 months or 20 years n=20 years Y = a % = * =17,80,000 Hence, Y = Rs.17,80,000 b) Loan o/s after 36 nstallments = a % = * = Loan o/s after lump-sum payment of 300,000= Interest rate applcable = 7.5% p.a. effectve Let New EMI=Q Then 12Qa (12) % Q = /(12* ) = /12 Q= Revsed EMI = ) Intt content f I repay n one lumpsum = X[ ] Intt content f I repay through 10 level nstallments = [Installment*10-X] Where Installment =X/a 10 Gven, X[ ]= [10*X/a 10 - X ] X[ ] = X = / = 1000 X= ) Force of Interest:- The force of nterest can be defned as the nomnal rate of nterest per unt tme at tme t convertble momently..e., for each value of t there s a number δ (t) such that lm h (t ) = δ(t), where δ(t) s called the force of nterest per unt tme at tme t. h->0+ We may also defne δ (t) drectly n terms of the accumulaton factor as δ (t) = lm A(t,t h) 1 h->0+ h ) Gven (2) = =(1+ (2) /2)^2-1 = ( /2)^2-1= =7.9002% δ = ln(1+)=ln(1.079)=7.6036% [14] Page 4 of 8

5 d (12) = 12*(1-v^(1/12)) = 12*(1-(1/1.079)^(1/12)) = % (1/2).=(1/2)*((1+)^2-1) = 0.5*(1.079^2-1) = % ) (1+ 1/ä t -d ) = d + v = (d+v-v t+1 ) = 1-v t+1 = a t+1 (1-v t ) (1-v t ) (1-v t ) a t LHS = Σ log 10 (1/ä t -d + 1 ) = Σ log 10 (a t+1 / a t ) = Σ (log 10 a t+1 - log 10 a t ) = log 10 a 31 - log 10 a 1 = log 10 (a 31 / a 1 ) = log 10 ä 31 = ä 31 = = From Tables, At 6% a 31 = Hence, ä 31 = 1.06* = Hence =6% v) From frst prncples, (Iä) n = 1+2v+3v 2 +4v 3 +5v nv Multplyng by v on both sdes, v(iä) n = v+2v 2 +3v 3 +4v 4 +5v nv n Subtractng, (1-v)(Iä) n = 1+v+v 2 +v 3 +v 4 +v v n-1 - nv n (1-v)(Iä) n = (1- v n )/(1-v) - nv n (Sum of a GP formula) d(iä) n = ä n - nv n (Iä) n = ä n - nv n d Hence Proved. v) If s the money rate of nterest and e s the nflaton rate then, Real rate of nterest ' = (-e)/(1+e) = ( )/1.012 = % Lumpsum = PV of payments receved = v 6.3% * * a % = * * = = v) Accumulated fund after 10 years = 3,00,000 S 10 Gven that : (1+ t )~ logn(μ, σ 2 ) S 10 ~ logn(10μ, 10σ 2 ) log S 10 ~ N(10μ, 10σ 2 ) Page 5 of 8

6 (logs 10 10μ)/ (10σ 2 ) ~ N(0,1) So, the probablty that the amount wll become at least Rs. 5,00,000 s: P(3,00,000 S 10 5,00,000) = 1 P(S 10 5/3) = 1 P(S ) = 1 Ф(log µ) σ 10 = 1 Ф( ) = Ф(0.1879) = [19] 6 () Opton 1 : If Rs.100 s nvested n bank then maturty value = 100*1.08^10 Hence IRR= 8% Opton 2 : Purchase prce per = 6[a (v 2 a 2 ).08 +(v 4 a 2.09 )+ (v 6 a 2.10 )+ (v 8 a 2.11 )] v Rs.100 nomnal = 6[ ] = 6* = IRR s gven by the equaton: a v 10 =0 At 11% LHS = At 10% LHS = By nterpolaton, IRR = % Opton 3: Redempton proceeds of 105 at tme 5 s renvested at 10% for a further perod of 5 years. So, IRR s gven by the equaton: 90 = 5a (1.1) 5 v 10 At 9% RHS = At 10% RHS = By nterpolaton, IRR = 9.125% Snce the IRR obtaned s maxmum under opton 2, the nvestor should select Opton 2. () The other crtera that can be used to decde between alternatve nvestment projects are: (Any two) 1. Net present value and accumulated proft 2. Payback perod 3. Dscounted payback perod [11] Page 6 of 8

7 7 a) b) Money weghted rate of return s gven by the equaton: 8.6(1+) (1+) 7/4 1.0(1+) = For = 20%, LHS = So, MWRR = 20% p.a. Tme weghted rate of return s gven by: (1+) 2 = 8.4 x ( ) x ( ) 10.5 (1+) 2 = = = TWRR = 19.48% p.a. c) The effectve rate of return for year 2008 s gven by solvng the equaton of value: 8.6(1+) + 0.6(1+) 3/4 = 11.5 = The effectve rate of return for 2009 s gven by: (1+) = = So, the lnked annual rate of return s gven by: (1+) 2 = x = LIRR = 19.79% p.a. d) () The money weghted rate of return wll ncrease as the new money receved (0.6) whch wll accumulate for a smaller perod but gves the same Fund Value ( ) at the end. () The tme weghted rate of return wll decrease as the second growth factor n the equaton gven n b) above wll decrease. [10] 8. () Any four of the followng : Corporate Bonds are part of loan captal of companes. They are more rsky (less secure) than the Government bonds They are usually less marketable than Government Bonds The lower securty and marketablty means that nvestors requre a yeld greater than on the correspondng government bonds. The nvestors lend a lump sum of money to the company. In return the company pays regular nterest payments and a fnal payment representng a return of captal at the end of the term of the contract. The level of securty depends on the type of bond, the company whch has ssued t, whether they are secured on some assets of the ssung companes and the term. Page 7 of 8

8 () We frst check f there s captal gan on redempton. g(1-t 1 ) = 0.10 x 0.8 = 0.08 (2) 0.08 = As g(1-t 1 ) > (2), there s captal loss. The stock s redeemable at the opton of the nvestor. The nvestor wll wsh to defer the captal loss as long as possble, so we assume that the nvestor wll choose the latest possble redempton date and the bond wll be redeemed after 20 years. Let P be the maxmum prce that Investor X can pay n order to acheve a net yeld of 8% p.a. P = 0.8 x 10 a (2) v 8% P = 8 x x x P = Rs ) Agan to check f there s any captal gan g(1-t 1 ) = 0.10 x 0.75 = (2) 0.09 = As g(1-t 1 ) < (2), there s captal gan. Ths means Investor Y wll want to make the captal gan as soon as possble, so we assume that the bond wll be redeemed at the earlest possble date.e. after 12 years and 10 months. Let P 1 be the maxmum prce that Investor Y can pay n order to acheve a net yeld of 9% p.a. Equaton of value s P 1 = (1.09) 2/12 [0.75 x 10 a (2) v (100-P 1 )v 13 9% P 1 = ( )x[7.5x7.4869x x100x P 1 x P 1 = Rs v) Actual net yeld,, obtaned by Investor X s gven by solvng the equaton = 0.8 x 10 a (2) v % At 3% RHS = At 2% RHS = By nterpolaton, = % ********************** [16] [Total 100] Page 8 of 8