Dept of Mathematcs and Statstcs Kng Fahd Unversty of Petroleum & Mnerals AS201: Fnancal Mathematcs Dr. Mohammad H. Omar Major Exam 2 FORM B Soluton November 27 2012 6.30pm-8.00pm Name ID#: Seral #: Instructons. 1. Please turn o your cell phones and place them under your char. Any student caught wth moble phones on durng the exam wll be consdered under the cheatng rules of the Unversty. 2. If you need to leave the room, please do so quetly so not to dsturb others takng the test. No two persons can leave the room at the same tme. No extra tme wll be provded for the tme mssed outsde the classroom. 3. Only materals provded by the nstructor can be present on the table durng the exam. 4. Do not spend too much tme on any one queston. If a queston seems too d cult, leave t and go on. Return to t after you attempted other questons. 5. Use the blank portons of each page for your work. Extra blank pages can be provded f necessary. If you use an extra page, ndcate clearly what problem you are workng on. 6. Only answers supported by work wll be consdered. Unsupported guesses wll not be graded. 7. Whle every attempt s made to avod defectve questons, sometmes they do occur. In the extremely rare event that you beleve a queston s defectve, the nstructor cannot gve you any gudance beyond these nstructons. 8. Moble calculators, or communcable devces are dsallowed. Use scent c calculator wth mathematcal equaton solvng capablty or SOA approved nancal calculators only. No other materals such as lecture notes, assgnments, soluton, etc are allowed. 9. Wrte mportant steps to arrve at the soluton of the followng problems. The test s 90 mnutes, GOOD LUCK, and you may begn now! Queston Marks Comments 1 10 2 10 3 10 4 5 5 5 Total 40 1
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1. (3+3+2+2 =10 marks) Today you borrow a loan of $125,000 n order to repay t n 20 years at an nterest rate of (12) = 12%, compounded monthly. You amortze ths loan wth monthly level payments (meanng payng every month the same amount) startng one month from today. a) Determne your monthly level payment. b) What s your outstandng balance just after your payment 8 years from now? c) How much prncpal do you repay n ths payment 8 years from now? d) What are the total nterest payments over the 20 years? 3
2. (4+3+3=10 marks) Today you borrow a loan of $125,000 n order to repay t n 20 years at an nterest rate of (12) = 12%, compounded monthly. For ths loan, you delver perodc payments only for nterest, and a sngle payment of the prncpal of $125,000 n 20 years. You amortze the loan wth level payments nto a separate snkng-fund, whch earns an nterest rate of j (12) = 9%, compounded monthly. a) Fnd your monthly outlay (nterest on orgnal loan + depost nto snkng fund). b) What s your outstandng balance just after your payment 8 years from now? c) How much prncpal do you repay n ths payment 8 years from now? 3. (5+5=10 marks) A bond wth face value of $2,500 matures on September 1, 2018. The sem-annual coupon rate s quoted at a nomnal rate of r (2) = 7% a) Determne the purchase prce on March 1, 2007, whch guarantees the buyer a yeld of j (2) = 6:8%: b) What s the market quotaton or clean prce, on Nov 18, 2010, whch guarantees the buyer a yeld of j (2) = 7:2%: 4
4. (5 marks) On January 1, 2009, an nvestment account s worth 100,000. On Aprl 1, 2009, the value has ncreased to 103,000 and 9,000 s wthdrawn. On January 1, 2011, the account s worth 104,000. Assumng a dollar-weghted method for 2009 and a tme-weghted method for 2010, and the e ectve annual nterest rate was equal to x for both 2009 and 2010. Calculate x. 5. (5 marks) A 30-year bond wth a face value of 1000 and 12% coupons payable quarterly s sellng at 800. Calculate the annual nomnal yeld rate convertble quarterly. 5
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IRR: Solve for j n P n k=0 C kv t k j = 0 Pro tablty Index I = Payback Perod = rst k n Present value of cash n ows Present value of cash out ows P t s=0 C s P k Dscounted Payback Perod = rst k n Dollar-weghted I = B Tme-weghted = [ F1 A FORMULA SHEET r=t+1 C r P t s=0 C sv s P k r=t+1 C rv r [A + P n k=1 C k] = I F2 F 1+C 1 F3 F 2+C 2 ::: F k F k 1 +C k 1 B A+ P n k=1 C k(1 t k ) F k +C k ] 1 Trapezodal rule to approx R b a f(x)dx u b a 2 [f(a) + f(b)] Descartes rule of sgns of Polynomal P (x) for countng types of roots: () n +ve roots n sgn changes n (Cn;C n 1;:::;C 1;C 0) () n -ve roots n sgn changes n (( 1) n C n;( 1) n 1 C n 1;:::;( 1)C 1;C 0) P = Cvj n + F r a nej = C + (F r Cj) a nej = K + g j (C K) where K = Cvn j P = F vj n + F r a nej = F + F (r j) a nej = K + r j (F K) where F = C () P = F Bought at Par () P > F Bought at a Premum () P < F Bought at a Dscount () P = F $ r = j () P > F $ r > j () P < F $ r < j # of days snce last coupon pad t = # of days n the coupon perod P t = P 0 (1 + j) t prce t = P t tf r: BV t+1 = BV t (1 + j) F r I t+1 = BV t j P R t+1 = F r I t+1 BV t = F [1 + (r j):a n te ] I t = F [j + (r j)(1 v n k+1 j )] P R t = F (r j)v n k+1 j L = K 1 v + K 2 v 2 + :::: + K n v n OB t+1 = OB t (1 + ) K t+1 I t+1 = OB t P R t+1 = K t=1 I t+1 Retrospectve: OB t = OB 0 (1 + ) t K 1 (1 + ) t 1 K 2 (1 + ) t 2 ::::K t 1 (1 + ) K t Prospectve: OB t = K t+1 v + K t+2 v 2 + :::: + K n v n t Level payments: OB t = L(1 + ) t Ks te = K(a ne (1 + ) t s te ) = K(a n te ) I t = Ka n t+1e = K(1 v n t+1 h ) P R t = Kv n t+1 P R t = P R t 1 (1 + ) = P R 1 (1 + ) t 1 + 1 = I t + P R t Snkng Fund perodc Outlay: L Snkng Fund h perodc Amortzaton schedule: OB t = L 1 s te j P R t = OB t 1 OB t = L (1+j)t 1 I t = L L s t 1e j j = L Makeham s sngle loan: A = Lvj n + L sa nej = K + j (L K) Makeham s m loans wth scheduled repayments: A s = L s v ts j +L sa tsej = K s + j (L s K s ) A = P m s=1 A s = P h m s=1 K s + j (L s K s ) Accumulated value of n-payment annuty-mmedate of 1: s ne = (1+)n 1 Present value of n-payment annuty-mmedate of 1: a ne = 1 vn Present value of a perpetuty-mmedate: a 1e = 1 Annuty-due: a ne = 1 sne = (1+)n 1 d ; vn d h (1+j) t 1 1 = K + j (L K) Contnuous annutes: s ne = R n 0 (1 + )n t dt = (1+)n 1 ; a ne = R n 0 vt dt = 1 vn = 1 e n Present value of n-term m thly payable annuty-mmedate of 1=m: a (m) ne = 1 vn (m) = a ne (m) Present value of annuty wth non-level payments: K 1 v + K 2 v 2 + + K n 1 v n 1 + K n v n Present value of annuty wth payments followng geometrc seres: v + (1 + r)v 2 + + (1 + r) n 2 v n 1 + (1 + r) n 1 v n = 1 ( 1+r 1+ )n r a) f = r, then v + (1 + r)v 2 + + (1 + r) n 2 v n 1 + (1 + r) n 1 v n = v + v + + v = nv 7
Accumulated value of annuty wth payments followng geometrc seres: 1 ( 1+r 1+ )n r (1+) n = (1+)n (1+r) n r K Dvdend dscount model for present value of a stock: n-payment ncreasng annuty-mmedate: (Is) ne = r sne n ; (Ia) ne = a1e = 1 n-payment ncreasng perpetuty-mmedate: (Ia) 1e = d = 1 + 1 2 n-payment decreasng annuty-mmedate: (Ds) ne = n(1+)n s ne ; (Da) ne = n a ne Captalzed Cost: C = P + P S S ne + M Perodc Charge = (Captalzed Cost) = P + P S = P + P S + M S ne + M Deprecaton Methods 1) Declnng Balance (compound dscount) Method: P t = P 0 (1 d) t D t = P 0 (1 d) t 1 d 2) Deprecaton - Straght Lne Method: P t = P 0 t 1 n (P 0 P n ) D t = 1 n (P 0 P n ) 3) Deprecaton - Sum of Year Dgts Method: S k = 1 + 2 + + k = k(k+1) P t = P n + Sn t S n (P 0 P n ) D t = n t+1 S n (P 0 P n ) S 4) Deprecaton - Compound nterest Method: P t = P te 0 S ne (P 0 P n ) D t = (1+)t 1 S ne (P 0 P n ) t+1 = A(t+1) A(t) A(t), A(t) = A(0)a(t) a) a(t) = (1 + ) t Compound nterest accumulaton factor 2 S ne ane nv n b) a(t) = 1 + t Smple nterest accumulaton factor h v = 1 1+ ; 1 + = d = A(1) A(0) A(1) = 1 + (m) 1+ = d m m h (m) = m (1 + ) 1=m 1 1 d (1) = ln(1 + ) a) (1 d) t Compound dscount factor b) 1 dt Smple dscount factor h m 1 d = d (1) = ln(1 d) t = A0 (t) A(t) 1 d (m) m A(n) = A(0)e R n 0 tdt ; real = r 1+r 1 + x + x 2 + x 3 + + x k = 1 xk+1 1 x = xk+1 1 x 1 8