May 2005 Exam Solutions

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1 May 005 Exam Soluto 1 E Chapter 6, Level Autes The preset value of a auty-mmedate s: a s (1 ) v s By specto, the expresso above s ot equal to the expresso Choce E. Soluto C Chapter 1, Skg Fud The terest paymet made at the ed of each year s the terest rate of 9% tmes the loa amout of 10,000: 10, The skg fud paymets accumulate to 10,000 at the ed of 10 years. The skg fud paymet s foud below: SFP s , SFP 10, , 000 SFP SFP The total of the paymets made over the 10-year perod s equal to 10 tmes the sum of the terest paymet ad the skg fud paymet: X 10 ( ) 15,90.95 The BA-II Plus calculator ca be used to solve ths questo as follows: 10 [N] 8 [I/Y] 10,000[FV] [CPT] [PMT] [+/-] [+] 10,000 [] 0.09 [=] [] 10 [=] Result s 15, ActuaralBrew.com 015 Page 1

2 Soluto 3 B Chapter 16, Durato The prce of the bod s: ,100 PV0 CF t t 0 The umerator of the formula for the Macaulay durato s: t ,100 3 t PV0 CFt, The Macaulay durato s: t PV0 CFt 0, MacDur t.70 PV0 CFt t 0 Soluto 4 A Chapter 6, Level Autes Sce Seth receves the frst paymets ad Susa receves the ext m paymets, the dfferece betwee the preset values of ther paymets s: Xa Xv a X a v a Ths matches Choce A. m m Soluto 5 B Chapter 15, Bods The 6-moth effectve yeld s: 1 1,000 4 y The prce of the bod s: 1,000 P Coup a Rv 30 a y , The BA-II Plus ca be used to aswer ths questo: 1,000 [] [=] [y x ] 4 [1/x] [=] [] 1 [=] [] 100 [=] [I/Y] 0 [N] 30 [PMT] 1,000 [FV] [CPT] [PV] Result s 1, Aswer s 1, Page ActuaralBrew.com 015

3 Soluto 6 D Chapter 16, Portfolo Durato The durato of a portfolo s the weghted average durato of ts compoets: k , , MacDurPort w j MacDurj 980 1,015 1,000 j1 50, ,995 Soluto 7 A Chapter 13, Iteral Rate of Retur The tme-0 equato of value to be solved s: v 100v v 100v The easest way to obta the aual effectve terest rate that satsfes the equato above s to use the BA-II Plus calculator: [CF] CF0 = [+/-] [ENTER] C01 = 00 [ENTER] C0 = 100 [ENTER] [IRR] [CPT] The result s The aswer s 10.00%. Soluto 8 C Chapter 1, Loas After the 10 th paymet, the remag balace s equal to the preset value of the remag 15 paymets: 300a 15 After subtractg out the extra 1,000, the remag balace s: 300a 1, Ths balace s the pad off wth 10 paymets, so the tme-10 equato of value s: 300a 1,000 Pmt a ActuaralBrew.com 015 Page 3

4 We ca ow solve for the revsed aual paymet: ,000 Pmt ,000 Pmt Pmt The BA-II Plus calculator ca be used to solve ths problem as follows: 15 [N] 8 [I/Y] 300 [PMT] [CPT] [PV] [+] 1,000 [=] [PV] 10 [N] [CPT] [PMT] Result s Soluto 9 D Chapter 8, Varyg Autes Usg the PI method, we have: P1 100 I 1 50 The preset value s: 50 I I 1 1 (1.09) 150 PV0 P1 a v 100 (1.09) ,10.49 Usg the BA II Plus, we have: 50 [N] 9 [I/Y] 100 [+] 0.09 [1/x] [=] [PMT] 50 [] 0.09 [=] [+/-] [FV] [CPT] [PV] Result s 1, Aswer s 1, Page 4 ActuaralBrew.com 015

5 Soluto 10 C Chapter 18, Forward Rates Although the wordg of the questo s a bt ambguous, t s meat to refer to the 1-year forward rate, startg 1 year: 1 f 1 f t (1 s t ) t 1 t 1 (1 st 1) (1 s ) 1 1 s f f Soluto 11 C Chapter 15, Callable Bods Sce ths bod s a dscout bod, ts prce-to-worst s foud by assumg that the bod s held utl maturty. Therefore, the hghest prce that the vestor ca pay ad be assured of her desred yeld s 897. The bod s called after 0 years for 1,050, ad therefore the tme-0 equato of value s: 1, a 0 0 (1 y) The BA-II Plus ca be used to fd the yeld that satsfes the equato of value: 0 [N] 897 [+/-] [PV] 80 [PMT] 1,050 [FV] [CPT] [I/Y] Result s Aswer s 9.4%. Soluto 1 B Chapter 6, Perpetutes I s oly true for level autes-mmedate that pay aually. II s true. III s oly true for level autes-mmedate. ActuaralBrew.com 015 Page 5

6 Soluto 13 D Chapter 13, Iteral Rate of Retur The tme-0 equato of value to be solved s: 1,000 1,500v,600v 1,000 1,500v,600v 0 The easest way to obta the aual effectve terest rate that satsfes the equato above s to use the BA-II Plus calculator: [CF] CF0 = 1,000 [ENTER] C01 = 1,500 [ENTER] C0 =,600 [+/-] [ENTER] [IRR] [CPT] The result s.834%, whch s a aual effectve terest rate. We must covert t to a terest rate that s covertble semaually. [] 100 [+] 1 [=] [y x ] 0.5 [] 1 [=] [] [=] Aswer s Soluto 14 E Chapter 8, Varyg Autes At tme 10, the preset value of the remag paymets ca be foud usg the PI method: Usg the PI method, we have: P1 19 I 1 19 The preset value s: 19 I I 1 1 (1.06) PV10 P1 a v 19 (1.06) Dscoutg the preset value foud above for 10 years ad addg the preset value of the frst 10 paymets gves us the preset value of the autymmedate: v 0a We ca use the BA II Plus to aswer ths questo: 19 [N] 6 [I/Y] 19 [] 0.06 [1/x] [=] [PMT] 19 [] 0.06 [=][FV] [CPT] [PV] (Result s ) Page 6 ActuaralBrew.com 015

7 [+/-] [FV] 10 [N] 0 [PMT] [CPT] [PV] Result s Aswer s Soluto 15 D Chapter 17, Dedcato The preset value of the asset cash flows s equal to the preset value of the lablty cash flows, so the cost of the bods s equal to the preset value of the lablty cash flows: 10,000 10,000 18, Soluto 16 A Chapter 13, Dollar-Weghted Rate of Retur To fd the come we treat the fal balace as a wthdrawal ad we treat the tal balace as a depost: Icome Wthdrawals Deposts ,560 1,000 1, The exposure of the fud to terest s: Fud exposure (Net depost)(tme depost s the fud) ,000 1, , The smple terest approxmato to the dollar-weghted yeld s: Icome Fud exposure 1,400 Soluto 17 B Chapter 8, Varyg Autes We ca use the formula for the preset value of a creasg perpetutymmedate to solve for : P1 I PV ,530 46, , The quadratc formula gves us two solutos for, ad we use the postve oe: ActuaralBrew.com 015 Page 7

8 , , Soluto 18 E Chapter 3, Preset Value Let P be the purchase prce. The two offers have the same preset value: / P(1.08) P 1 X (1.08) 1/6 X 100 The fal equato above matches Choce E. Soluto 19 C Chapter 3, Iterest Rate Coversos The mothly effectve terest rate ca be coverted to the mothly effectve dscout rate: (1) d 1 (1) (1) The omal aual dscout rate s the mothly effectve rate tmes 1: (1) d Soluto 0 A Chapter 8, Varyg Autes There are two fuds, oe earg 1% ad oe earg 8%. The amout of each level depost s deoted by X. The deposts to the fuds are descrbed below: Tme 1% 8% 0 X 1 X X 0.1 X X X 9X X 0.1 Sce the terest from the 1% fud s pad to the 8% fud, the value of the 1% fud at the ed of 10 years s 10X. The accumulated value of the 8% fud ca be foud usg a creasg auty mmedate: Page 8 ActuaralBrew.com 015

9 10 X ( Is) 0.1X 10, s 10 10X X 10, / X 0.1X 10, X 0.1X 10, X X 10,000 X We ca use the BA II Plus to obta the value of ( Is ) : [N] 8 [I/Y] 1 [PMT] [CPT] [FV] [] 1.08 [+/-] [] 10 [=] [] 0.08 [=] Result s Soluto 1 D Chapter 7, Level Autes Let P be the purchase prce. The equato of value at tme 0 s: P P a 10 1 j 1 0.1a 1 j We ca use the BA-II Plus calculator to solve for the mothly effectve terest rate. Oce we have the mothly effectve terest rate, we covert t to the aual effectve terest rate: [ d ] [BGN] [ d ] [SET] [ d ] [QUIT] 1 [N] 1 [PV] 0.1 [+/-] [PMT] [CPT] [I/Y] (Result s 3.503) [] 100 [+] 1 [=] [y x ] 1 [=] [] 1 [=] Aswer s Soluto B Short Sellg The marg s the marg requremet tmes the value of the stock: ActuaralBrew.com 015 Page 9

10 O Jauary 1, 005, the proft to Kare s the prce at whch she sold the stock mus the prce at whch she covers the short sale plus the marg terest mus the dvded o the stock: 50 X 4 5 X The percetage retur s equal to the Jauary 1, 005 proft over the marg requremet: 5 X X 44 Soluto 3 D Chapter 14, Commo Stock We ca use the dvded dscout growth model to fd the retur o the stock: Dv PV 1 0 g Soluto 4 E Chapter 6, Level Autes The accumulated value of the auty-mmedate at tme ( 1) s equal to the accumulated value of a auty-due: s s (1 ) (1 ) d d d We ca use d to fd the value of : (1 ).476 (1 d).476 ( ).476 l(0.899) l(.476) 8.00 Page 10 ActuaralBrew.com 015

11 Soluto 5 A Chapter 1, Loas The paymet of 0 s exactly equal to the terest o the loa each quarter: Sce the paymet does ot exceed the terest, the amout of prcpal pad dow s zero for each paymet: Prt Pmtt Itt ActuaralBrew.com 015 Page 11

0.07 (12) i 1 1 (12) 12n. *Note that N is always the number of payments, not necessarily the number of years. Also, for

0.07 (12) i 1 1 (12) 12n. *Note that N is always the number of payments, not necessarily the number of years. Also, for Chapter 3, Secto 2 1. (S13HW) Calculate the preset value for a auty that pays 500 at the ed of each year for 20 years. You are gve that the aual terest rate s 7%. 20 1 v 1 1.07 PV Qa Q 500 5297.01 0.07