Final Examination MATH NOTE TO PRINTER

Transcription

1 Fnal Examnaton MATH NOTE TO PRINTER (These nstructons are for the prnter. They should not be duplcated.) Ths examnaton should be prnted on paper, and stapled wth 3 sde staples, so that t opens lke a long book.

2 McGILL UNIVERSITY FACULTY OF SCIENCE FINAL EXAMINATION MATH THEORY OF INTEREST EXAMINER: Professor W. G. Brown DATE: Thursday, Aprl 21st, 2005 ASSOCIATE EXAMINER: Prof. N. Sancho TIME: 14:00 17:00 hours SURNAME: MR, MISS, MS, MRS, &c.: GIVEN NAMES: STUDENT NUMBER: 1. Fll n the above clearly. Instructons 2. Do not tear pages from ths book; all your wrtng even rough work must be handed n. 3. Calculators. Whle you are permtted to use a calculator to perform arthmetc and/or exponental calculatons, you must not use the calculator to calculate such actuaral functons as a n, s n, (Ia) n, (Is) n, (Da) n, (Ds) n, etc. wthout frst statng a formula for the value of the functon n terms of exponentals and/or polynomals nvolvng n and the nterest rate. You must not use your calculator n any programmed calculatons. If your calculator has memores, you are expected to have cleared them before the examnaton. 4. Ths examnaton booklet conssts of ths cover, Pages 1 through 7 contanng questons; and Pages 8 and 9, whch are blank. For all problems you are expected to show all your work, and to smplfy algebrac and numercal answers as much as you can. All solutons are to be wrtten n the space provded on the page where the queston s prnted. When that space s exhausted, you may wrte on the facng page. Any soluton may be contnued on the last pages, or the back cover of the booklet, but you must ndcate any contnuaton clearly at the bottom of the page where the queston s prnted! You may do rough work anywhere n the booklet. 5. You are advsed to spend the frst few mnutes scannng the problems. (Please nform the nvglator f you fnd that your booklet s defectve.) 6. Several useful formulas are prnted on page 3. You should not assume that any of these formulas s/are requred n the soluton of any of the problems on ths examnaton. PLEASE DO NOT WRITE INSIDE THIS BOX 1(a) 1(b) 1(c) 1(d) 2(a) 2(b) 2(c) 3 /2 /2 /3 /3 /3 /3 /4 /5 4(a) 4(b) 4(c) 5(a) 5(b) 5(c) 6(a) /2 /4 /9 /4 /2 /4 /5 6(b) 7(a) 7(b) Total /5 /5 /5 /70

3 Fnal Examnaton MATH Showng your work n detal, determne each of the followng; the rates you determne should be accurate to 4 decmal places, or as a percentage accurate to 2 decmal places. (a) [2 MARKS] The nomnal annual nterest rate, compounded quarterly, correspondng to an effectve annual nterest rate of 8%. (b) [2 MARKS] The effectve annual nterest rate correspondng to a nomnal dscount rate, compounded monthly, of 6%. (c) [3 MARKS] The effectve sem-annual nterest rate correspondng to a force of nterest of δ = (d) [3 MARKS] d ( 1 2), correspondng to a nomnal annual nterest rate, compounded every 2 months, of 12%.

4 Fnal Examnaton MATH Showng all your work, determne for each of the followng sequences of payments, the value as of the gven tme and for the gven nterest or dscount rate. (a) [3 MARKS] The value as of one year ago of 30 payments of 1 at the end of every half-year, the frst to be pad 18 months from now, at a nomnal nterest rate of 9% compounded sem-annually. (b) [3 MARKS] The value now of 24 payments of 1 at the end of every 3 months, the frst to be pad one year from now, at an effectve annual nterest rate of 9%. (c) [4 MARKS] The value exactly 5 years from now of a perpetuty consstng of an unendng sequence of payments of 1 at ntervals of one year, the frst to be pad n 6 months from now, at a nomnal annual nterest rate of 4% compounded every 4 months.

5 Fnal Examnaton MATH [5 MARKS] Gve a formula n terms of alone, for the value of the followng, showng all your work:. An annuty at an effectve semannual nterest rate of, of 100 now, 97 sx months from now, 94 one year from now, the payments decreasng by a constant amount untl a fnal payment of 34. Table 1: Several Useful Formulæ that you were not expected to memorze You may wsh to make use of some of the followng formulæ, but remember that your fnal answers for ths problem must be expressed n terms of alone. (Ia) n = än nv n (Is) n = s n n (Is) n = s n+1 (n+1) (Ia) = ä (Da) n = n a n (Ds) n = n(1+)n s n

6 Fnal Examnaton MATH The purchase of a new condomnum s partally fnanced by a mortgage of 120,000 payable to the vendor; the mortgage s amortzed over 25 years, wth a level payment at the end of each month, at a nomnal annual rate of 6% compounded monthly. (a) [2 MARKS] Determne the level monthly payments under ths mortgage. (b) [4 MARKS] Dvde the 120th payment nto prncpal and nterest. (c) [9 MARKS] When the payments from the mortgagor start to arrve, the vendor deposts them nto an account earnng 4% per annum compounded monthly. He wll contnue to do so untl the account contans exactly 120,000, wth the last depost beng possbly smaller than the others. Determne the tme and amount of that last depost.

7 Fnal Examnaton MATH The Smths have sold ther house for 500,000. They wsh to purchase a 20-year annuty-certan wth the money, but are concerned that lvng costs wll be rsng, and should be planned for by havng the amount of the payment n any month tmes the amount of the payment for the precedng month. The frst payment s to be 1 month after the sale. The nterest rate s 4% per annum, compounded monthly. (a) [4 MARKS] Showng all your work, determne the amount of the frst payment. (b) [2 MARKS] Determne the amount of the last payment, to be made 20 years from now. (c) [4 MARKS] Suppose that, wth the same frst payment, the annuty payments wll ncrease by a constant amount and not by a constant factor. Showng all your work, determne the amount of the last payment. (You may wsh to use the formulæ n Table 3 on page 3 of ths examnaton.)

8 Fnal Examnaton MATH X sold Y hs home for 500,000. In addton to a down payment of 100,000 cash, Y mortgaged the property to X for 400,000. The mortgage provdes for level quarterly payments over 25 years, at a nomnal rate of 4% compounded every 3 months. (a) [5 MARKS] Construct an amortzaton table showng the frst 6 payments, under the followng headngs: Payment Payment Interest Prncpal Outstandng Number Amount Pad Repad Loan Balance (b) [5 MARKS] After 10 years just after recevng the 40th payment X sells the (remanng payments of the) mortgage to Z at a prce to yeld 6% convertble quarterly. Determne the prce P that Z pays, and construct the frst 3 and last 3 lnes of an amortzaton table for the 60 payments showng, under the followng headngs, how Z s recoverng her nvestment, under the followng headngs: Payment Payment Interest Prncpal Outstandng Number Amount at new rate Repad Loan Balance 0 P

9 Fnal Examnaton MATH Consder a 10,000 par-value 10-year bond wth sem-annual coupons payng nterest at 5% compounded semannually. Suppose that the bond can be redeemed for 11,000 at the tme of ether of the 16th or 17th coupons, at 10,500 at the tme of the 18th or 19th coupons, or can be held untl maturty, at the tme of the 20th coupon, where t matures wthout premum. (a) [5 MARKS] What prce should an nvestor pay n order to be certan of a yeld rate of 6% compounded semannually? (b) [5 MARKS] What prce should an nvestor pay n order to be certan of a yeld rate of 4% compounded semannually?

10 Fnal Examnaton MATH contnuaton page for problem number You must refer to ths contnuaton page on the page where the problem s prnted!

11 Fnal Examnaton MATH contnuaton page for problem number You must refer to ths contnuaton page on the page where the problem s prnted!