MAY 2005 EXAM FM SOA/CAS 2 SOLUTIONS

Size: px

Start display at page:

Download "MAY 2005 EXAM FM SOA/CAS 2 SOLUTIONS"

Bennett Hardy
6 years ago
Views:

1 MAY 2005 EXAM FM SOA/CAS 2 SOLUTIONS Prepared by Sam Broverman 2brove@rogerscom sam@utstattorontoedu = 8l 8 " " 1 E 8 " œ@ = œ@ + œð" 3Ñ + œ + Á + (if 3Á! ) Ð" 3Ñ Answer: E 8l 8l 8l 8l 8l 2 Annual interest paid to the lender is "!ß!!! Þ!* œ *!! "!ß!!! Annual sinking fund deposit is = œ '*!Þ* "!lþ!) Total paid over 10 years is "! Ð*!! '*!Þ*Ñ œ "&ß *!$ Answer: C $ $ 3 The bond price (PV) is "!!@ "!!@ "!!@ "!!!@ $ $ "!!@!!@ $!!@ $!!!@ "!!@ "!!@ "!!@ "!!!@ "$"Þ*% ()*Þ$& Macaulay duration is $ $ œ œ Þ(! Answer: B 8 8 8l 7l 8l 7l 4 Seth's PV is \+ and Susan's PV is \@ + The difference is + Ó Answer: A 5 Suppose the 6-month yield rate on the zero coupon bond is 4 % Then '%Þ'! œ "!!!@ from which we get 4 œ Þ!"*)!% 4! 4!l4 Price of coupon bond is "!!!@ $!+ œ ""'(Þ!$ Answer: B 6 Let TZ denote the resent value of the portfolio of bonds at annual effective yield rate 3 The duration of the portfolio is Ð3TZÑÎTZ At the time of purchase, T Z œ T Z+ T ZF T ZG œ **& (sum of the PV's pf the three bonds) The duration of bond A is Ð 3T ZEÑÎT Z E, which is given as "Þ%( Since T ZE œ *)!, we have Ð 3T ZEÑ œ *)! "Þ%' œ "ß!$!Þ)! In a similar way, we have Ð 3T ZFÑ œ "!"& "Þ$& œ "ß &$&Þ& and Ð T Z Ñ œ "!! ""'Þ'( œ "'ß '(! 3 G 2brove@rogerscom sam@utstattorontoedu

2 6 continued 3 3 E F G 3 E 3 F 3 G Then, Ð TZÑ œ ÐTZ TZ TZ Ñ œ Ð TZ TZ TZ Ñ œ "ß!$!Þ)! "ß &$&Þ& "'ß '(!Þ!! œ &!ß $'Þ!& The duration of the portfolio is &!ß$'Þ)& 3 **& Ð T Z ÑÎT Z œ œ "'Þ((% Answer: D 7 "!!!!@ 3 "!!@ 3 œ $'%Þ%' We can solve for 3 three four ways (i) Trial and error Try each interest rate until the PV is correct We see that 3œÞ"! (ii) Solve the equation '%Þ%' œ!!@ "!!@ using the calculator financial functions with T Z œ '%Þ%' ß T QX œ!! and J Z œ "!! (iii) Solve the equation '%Þ%' œ!!@ "!!@ using the BA-II PLUS calculator cashflow spreadsheet (iv) Solve the quadratic equation "!!@!!@ '%Þ%' œ!!! È!! %Ð"!!ÑÐ œ œ Þ*!*" or Þ*!*" (ignore the negative œ " " 3 œ Þ*!*" p 3 œ Þ"! Answer: A 8 Just after the 10-th regular payment there are 15 payments remaining, and the (prospective) outstanding balance is $!!+ œ ß &'(Þ)% "&lþ!) After the additional payment of 1000, the amount owed is 1,56784 The revised annual payment, payable for 10 more years is "ß &'(Þ)% œ \+ p \ œ $$Þ'& "!lþ!) Answer: C \, where 9 The series of payment amounts is described in the following time line Time! " $ Þ &! Payment "!! "!" "! "%* This series of payments can be separated into two series: ** ** ** Þ Þ Þ ** " $ Þ &! The present value of the original series is the combination of the present values of the two series: **+ ÐM+Ñ œ "ß!)&Þ" "&Þ* œ "ß "!Þ&! &!lþ!* &!lþ!* Answer: D 2brove@rogerscom sam@utstattorontoedu

3 10 The one-year forward rate for year two is 4, where Ð"Þ!)&ÑÐ" 4Ñ œ Ð"Þ!*&Ñ Solving for 4 results in 4 œ Þ"!& Answer: C 11 The bond will be bought at a discount For a callable bond bought at a discount, the latest redemption date is the redemption date that provides the lowest price for a given yield, and the lowest yield for a given price In order to achieve a minimum desired yield, the price that should be paid is the price paid at the latest redemption date (this is true for a callable bond bought at a discount) Therefore, the price paid is 897 If the bond is called at the end of 20 years, the annual! yield rate is 4, where )*( œ "!&!@ 4 )!+!l4 Using the calculator unknown interest function (or the BA-II PLUS bond worksheet), we get 4 œ Þ!*% Answer: C 12 I False This is true only for a level perpetuity-immediate II True That is the definition of perpetuity (perpetual payments) III False For an increasing perpetuity-immediate at rate 3, the PV is ÐM+Ñ œ " " 3 3 The interest at the end of the first year is ÐM+Ñ 3 œ ", but the payment amount is 1 Answer: B " 3 13 The accumulate value at the end of the 2nd year of the two payments is 3 % 3 "!!!Ð" Ñ "&!!Ð" Ñ, which we are told is equal to If we let " 4 œ Ð" Ñ, then this equation becomes "!!!Ð" 4Ñ "&!!Ð" 3Ñ œ '!! The cashflow worksheet on the BA-II PLUS can be used to solve for 4, 4 œ Þ!)$% Alternatively, we can solve the quadratic equation and ignore the negative root 3 Then "Þ!)$% œ Ð" Ñ, so that 3 œ Þ!)" Answer: D 14 The time line of payments is Time! " $ Þ "! "" " Þ ) * Payment!!! Þ! "* ") Þ " The series is a combination of a 10-year annuity-immediate of amount 20 per year, plus a 19- year decreasing annuity-immediate (deferred for 10 years) The PV ÐH+Ñ œ!ð(þ$'!"ñ ÐÞ&&)$*&ÑÐ Ñ œ!þ Answer: E "!lþ!' "! "*lþ!' "* + Þ!' "*lþ!' 2brove@rogerscom sam@utstattorontoedu

4 15 Bond (i) with face amount 1 pays 104 at the end of 1 year, and Bond (ii) with face amount 1 pays 06 at the end of 1 year and 106 at the end of two years Let J" and J represent the face amounts of Bonds 1 and 2 needed to replicate the cash flow At the end of the first year, the cash flow will be "Þ!%J" Þ!'J and at the end of the second year, the cash flow will be "Þ!'J We want both of these to be 10,000, so "Þ!%J" Þ!'J œ "!ß!!! and "Þ!'J œ "!ß!!! Solving these two equations results in J" œ *ß!("Þ" ß J œ *ß %$$Þ*' The total cost of the two bonds is Ò*ß!("Þ"Ð"Þ!%Ñ *ß %$$Þ*'ÐÞ!'ÑÓ@ Þ!& *ß %$$Þ*'Ð"Þ!'Ñ@ Þ!& œ ")ß &*% Note that since both bonds have a yield rate of 5%, and the bonds are used to replicate the two casflows of 10,000 at the end of years 1 and 2, the total PV must be "!ß!!!@ Þ!& "!ß!!!@ Þ!& œ ")ß &*% It was unnecessary to find J" and J Answer: D 16 The dollar-weighted return equation is " " '! $ $ " " "!!! "!!!Ð Ñ!!Ð Ñ &!!Ð Ñ "!!!Ð" 3Ñ "!!!Ð" 3Ñ!!Ð" 3Ñ &!!Ð" 3Ñ œ "&'! Þ 3 œ œ Þ")&( Answer: A $ $ 17 The timeline of payments for the perpetuity is: Time! " $ % Þ Payment!! &! $!! $&! Þ Þ Þ This series of payments can be separated into the following two series: "&! "&! "&! "&! Þ Þ Þ &! "!! "&!!! Þ Þ Þ The total PV is "&!+ &!ÐM+Ñ œ "&!Ð Ñ &!Ð Ñ " " " We are given that this PV is 46,350 We can solve the quadratic equation in 3: " " " &!Ð Ñ!!Ð Ñ %'ß $&! œ! Ignoring the negative root, we have œ )Þ&", so 3 œ Þ!$& We can also try each of the possible answers in the equation to see which one is correct Answer: B 2brove@rogerscom sam@utstattorontoedu

5 \ "!! 18 The customer can pay " now, and that is equivalent to paying Þ* two months from now Therefore, the PV of the payment is 9 to be made in two months is equal to a payment of \ \ "!! "!! \ "Î' "!! Answer: E " now This can be expressed as " œðþ*ñ@ Ð" ÑÐ"Þ!)Ñ œ Þ* "Î' Þ!), or equivalently, Þ")* 19 The monthly interest rate is " œ Þ!"&(& The monthly discount rate 4 is related to the Þ!"&(& monthly interest rate through the relationship 4 œ œ Þ!"&&!' "Þ!"&(& The nominal annual discount rate convertible monthly is " Þ!"&&!' œ Þ")'" Answer: C 20 Suppose that the level deposit amount is \Þ The timeline of deposits and reinvested interest is: Time! " $ Þ * "! Deposit \ \ \ Þ \ \ Interest Þ"\ ÐÞ"\Ñ $ÐÞ"\Ñ *ÐÞ"\Ñ The total accumulated value at the end of 10 years is the combination of the 10 deposits of \, plus the increasing interest payments reinvested at 8% The AV is = "!lþ!) "! "!\ ÐÞ"\ÑÐM=Ñ"!lÞ!) œ \ "! ÐÞ"Ñ Þ!) œ \Ò"! ÐÞ"ÑÐ(!Þ&'*ÑÓ We want the total AV to be 10,000, so that \ œ &%"Þ%( Answer: A 21 For a loan of amount \, the customer is paying Þ"\ at the start of each month for a year At monthly rate 4, this means that \œþ"\+, or equivalently, + œ"! Using the calculator "l4 unknown interest function, we have 4 œ Þ!$&!$ as the interest rate per month The equivalent annual effective interest rate is Ð"Þ!$&!$Ñ " "l4 " œ Þ&""' Answer: D 22 Karen must deposit ÐÞ)ÑÐ&!Ñ œ %! into the margin account; that is the amount she invests at the start of the year At the end of the year, the amount she has after all transactions are completed is %! % Ð&! \Ñœ* \ We are told that she made a 20% return Since her initial investment is 40, this means she must have a net amount of %!Ð"ÞÑ œ %) when all transactions are completed at the end of the year Therefore, * \œ%), so that \œ%% Answer: B 2brove@rogerscom sam@utstattorontoedu

6 23 The dividends form a perpetuity with payments that follow a geometric progression With first payment amount 6 in one year, and subsequent payments 3% larger than the previous payment, at annual effective interest rate 3, the PV one year before the first dividend payment is ' 3 Þ!$ We are told that the stock price is 75 The stock valuation method that is implied by the wording of this question is that the stock price is the present value of the perpetuity of dividends ' Therefore, (& œ, from which we get 3 œ Þ"" Answer: D 3 Þ!$ 8 Ð" 3Ñ " 8 8l We are given that = œ œ "$Þ((' Using the given value Ð" 3Ñ œ Þ%(', we get œ Þ"!("%$, and 3 œ Þ" Then from Ð"Þ"Ñ œ Þ%(', we use the calculator 68 Þ%(' 68 "Þ" financial function or 8œ to get 8œ) Answer: E 25 This is a level payment amortization with payment amount Oœ! per quarter, at an interest rate of Þ!% per payment period (quarter) If we knew how many payments there were on 8 > " 8 $ the loan, say 8, then the principal in the >œ% -th payment would be O@ œ!@ The original loan equation is &!! œ!+ If we use the calculator function to solve for 8 we get an error message This is because " 8lÞ!% 8œ ; a perpetuity-immediate of 20 per period at 4% per period has PV!Ð Þ!% Ñ œ &!! Therefore, the payments of 20 are all interest Ð&! Þ!% œ!), and there is never any principal repaid Answer: A Þ!% Þ!% 2brove@rogerscom sam@utstattorontoedu

EXCERPTS FROM S. BROVERMAN STUDY GUIDE FOR SOA EXAM FM/CAS EXAM 2 SPRING 2007

EXCERPTS FROM S. BROVERMAN STUDY GUIDE FOR SOA EXAM FM/CAS EXAM 2 SPRING 2007 Table of Contents Introductory Comments Section 12 - Bond Amortization, Callable Bonds Section 18 - Option Strategies (1) Problem