Fnance 402: Problem Set 1 Solutons Note: Where approprate, the fnal answer for each problem s gven n bold talcs for those not nterested n the dscusson of the soluton. 1. The annual coupon rate s 6%. A sem-annual coupon payment mples we get 6%/2 = 3% of the face value every sx months, or $3. Gven the APR s 12% per annum compounded sem-annually, the perodc nterest rate s 12%/2 = 6%. The cash flow stream conssts of 12 years of semannual coupons n the amount of $3 and 1 balloon payment at maturty for the face value of $100. The cash flows are presented graphcally n Fgure 1. Fgure 1: Cash Flow 0 $3 $3 $3 $3 $3 $3 + $100 Tme Perod 0 1 2 3 22 23 24 We can recognze these cash flows as correspondng to two other types of bonds. The stream of coupons can be thought of as an annuty and the prncpal payment can be thought of as a zero coupon bond. Thus, we can value the coupon payments usng our annuty formula, the prncpal payment usng our zero coupon formula and add the results to get the value of the coupon bond. 1
Valung the annuty porton yelds The value of the zero porton s 1 (1 + ) 24 c 1 (1 + 0.06) 24 = $3 0.06 = $37.65 $100 24 = $24.70 (1.06) Addng these two fgures gves the current value of the bond: $62.35. 2. The queston s askng us to fnd the payment that would make us ndfferent between recevng that lump-sum amount at the end of the year and monthly payments of $10,000 throughout the year. The most drect way to do ths s to smply compute the value of the monthly payments at the end of one year, whch amounts to computng the future value of an annuty. ( ) 1 + 0.12 12 1 $10, 000 = $126, 825.03 12 0.12 12 Thus, we are ndfferent between recevng a lump-sum payment of $126,825.03 at the end of the year or $10,000 each month. 3. 3.a Usng the same logc as n problem 1., we can value ths bond as the sum of an annuty and a zero. For the annuty porton, the perodc payment, a, s $25, whch follows from a 5% coupon rate, $1,000 face value and sem-annual coupons. The perodc nterest rate = 3% follows from the market rate (R) beng 6% and a sem-annual compoundng perod (m = 2). Fnally, the total number of payments (or perods n the annuty) s 7 years (T ) tmes 2 2
payments per year so that N = T m = 14 perods. The present value of the annuty, usng our annuty formula s c 1 (1 + ) N $25 1 (1.03) 14 = $282.40 0.03 The value of the zero porton of the bond s smply the present value of the par amount, or $1, 000 = $661.12 (1.03) 14 The current prce s the sum of the annuty and zero values, or $943.52. Thus, the bond s currently sellng at a dscount relatve to par (.e. face value). 3.b The effectve annual rate may be found from the followng relatonshp: (1 + r) = (1 + ) 2 Substtutng for, whch s just the perodc nterest rate of 3%, computed above, yelds an effectve annual rate r equal to 6.09%. Note that ths s not equal to the annual market rate of 6% and, n fact, s greater than 6%. Ths a consequence of nterest earnng nterest. That s, the nterest you earn on your money after the frst compoundng perod earns nterest durng the second compoundng perod. By ths logc, we would lke our nvestments to be compounded as frequently as possble. 3.c Ths prce was calculated n part 3.a: B 0 = $1, 000 = $661.12 (1.03) 14 3
3.d The bond from part 3.a s now worth $25 1 (1.035) 14 0.035 (the value of the annuty porton) plus = $273.01 $1, 000 = $617.78 (1.035) 14 (the value of the zero porton). The current prce s thus $890.79. The bond from part 3.c was calculated n valung the coupon bond and s worth $617.78. 4. 4.a The queston s askng to fnd the perodc payment of a 180-perod annuty wth a present value of $400,000. A summary of the nformaton usng our notaton s: T = 15 m = 12 N = 180 A 0 = $400, 000 R = 0.095 = 0.095 = 0.0079167 12 The present value formula for an N-perod annuty s c A 0 = (1 + ) + c (1 + ) +... + c 2 (1 + ) + c N 1 (1 + ) N = c ( ) 1 1 (1 + ) N 1 = c (1 + ) N Multplyng both sdes of above equaton by the perodc payment, c: c = A 0 1 (1 + ) N 1 (1+) N yelds an equaton for 4
Substtutng produces the answer: c = 0.0079167 $400, 000 1 (1.0079167) 180 = $4,176.9084. 4.b Ths s most easly done n Excel but here s an explanaton of what s happenng. Consder the end of the frst month when the frst payment s due. The prncpal s $400,000 and one month s worth of nterest has accrued 400, 000 0.0079167 = 3, 166.70. You pay 4,176.90 whch means the prncpal s reduced by 4, 176.90 3, 166.70 = 1, 010. 20. Contnue ths process 179 more tmes or let Excel do the rest of the work. 4.c After you make your 12th payment you are left wth an annuty that has 168 payments of $4,176.9084 each. Thus, the prncpal left on your loan can be found by usng the annuty formula: $4, 176.9084 1 (1.0079167) 168 0.0079167 = $387, 335.19. You apply your $200,000 bonus to ths amount so that the new prncpal remanng s $187,335.19. You can contnue wth the same amortzaton dea presented n 4.b n your Excel spreadsheet. You should fnd that you wll need another 56 months for a total payback perod of 68 months. The last month wll not requre the full $4,176.90, only $2,580.76. 5. Let s just compare the monthly payments snce the terms (lengths) of the two contracts are the same. Ths requres fndng the fxed payment of an annuty. Usng formula A 0 = = c c (1 + ) + c (1 + ) +... + c 2 (1 + ) + c N 1 (1 + ) ( ) N 1 = c 1 (1 + ) N 5 1 (1 + ) N
.and rearrangng above equaton yelds: c = A 0 1 (1 + ) N The present value of the annuty s smply the sze of the loan, $58,000, snce that s what you receve today. The perodc nterest rate,, s R/m or 5.5%/12. The maturty s 5 years wth monthly payments mplyng 60 perods. Thus, the monthly payments for the frst deal are 0.055/12 c = $58, 000 60 = $1, 107.87 1 (1 + 0.055/12) The monthly payments for the second deal, wth the $4,000 dscount subtracted from the loan prncpal and the hgher nterest rate, s 0.06/12 c = $54, 000 60 = $1, 043.97 1 (1 + 0.06/12) Therefore, snce the monthly payments are lower, you should take the $4,000 dscount and hgher rate. 6. Begn by fndng the monthly payment requred on an annuty wth present value $155,000. 0.11/12 c = $155, 000 360 = $1, 476.10 1 (1 + 0.11/12) (Ths s just algebrac manpulaton of the annuty formula) Next we need to fnd the amount of remanng prncpal after 10 years of repayment, whch s the value of an annuty wth 240 perods remanng and a perodc payment of $1, 476.10 or $1, 476.10 1 (1 +.11/12) 240.11/12 = $143, 006.84 The prepayment penaltes and closng costs equal 5% $143, 006.84 = $7, 150. 34. Ths mples that ther new prncpal amount, nclusve of penaltes and closng costs, s $143, 006.84 + $7, 150.34 = $150, 157.18. 6
To see f ths opton s better than the orgnal mortgage, we just need to see f the monthly payment s bgger or smaller than the orgnal. 0.09/12 c = $150, 157.18 240 = $1, 351.00 1 (1 + 0.09/12) The cost s less so they should refnance. 7. Let s value ths by creatng a replcatng portfolo. That s, let s buy some amount (possbly 0) of each bond so that the cash flows of our portfolo exactly match the cash flows of the bond we are tryng to value. By the prncple of no-arbtrage, the value of our replcatng portfolo should exactly equal the value of the bond. See Table 1. Table 1: Perodc Cash Flows Securty 1 2 3 Cost Coupon Bond 5,000 5,000 105,000? Replcatng Portfolo 50 unts of bond A 5,000 0 0 50 $95.24 = $4, 762.00 50 unts of bond B 0 5,000 0 50 $89.85 = $4, 492.50 1,050 unts of bond C 0 0 105,000 1050 $83.96 = $88, 158.00 Total Cash Flows 5,000 5,000 105,000 $97,412.50 Snce the portfolo perfectly replcates the cash flows of the coupon bond, the values of the two must be equal by arbtrage arguments. Therefore, the bond prce s $97,412.50. 8. Fgure 2 presents a tme lne of the cash flows: One way to value ths bond s to fnd the present value of the cash flows as of 3 months ago and then fnd the 3-month future value of ths number (ths mplctly assumes that the nterest rate dd not change between 3 months ago and today). Three months ago, our bond was maturng n sx and a half years, mplyng a total of 13 semannual payments of $0.12 mllon. 7
Fgure 2: Cash Flow 0 $0.12 $0.12 $0.12 $0.12 + $2 (Mllons) Tme Perod (Months) 0 3 6 9 12 69 72 75 Recognzng that the value of the coupon bond s the sum of the annuty porton and zero porton we get: $0.12 1 (1 + 0.03) 13 0.03 + 2 = $2.64 mllon (1.03) 13 Ths was the value of our bond, 3 months ago. To get the value today, we must fnd the future value of ths number, 3 months hence. $2.64 (1.03) 0.5 = $2.68 mllon An alternatve approach s to value the bond as of 3 months from today and then dscount ths value back to today. In 3 months we wll receve $0.12 mllon. Includng that ntal payment, whch need not be dscounted, the value of our bond n 3 months s: $0.12 + $0.12 1 (1 + 0.03) 12 0.03 + $2 = $2.72 mllon 1.0312 Now we need to dscount ths value back half a perod. $2.72 = $2.68 mllon 1.031/2 The value of the bond s $2.68 mllon 9. The tme lne of our cash outlays for the college expense are presented n Fgure 3. 8
Fgure 3: Cash Flow 0 $20,000 $20,000 Tme Perod 0 1 2 9.a The present value of these cash flows, and thus the amount we must nvest today, s: $20, 000 $20, 000 + = $35,587.21 1.04 2 1.04 4 The dscount rate follows from the 8% yeld-to-maturty (R = 0.08) and sem-annual compoundng (m = 2), mplyng = 4%. 9.b Scenaro 1: Unchanged Interest Rates. If we spend $35,587.21 on 4-year zeros today, ths means we own bonds wth a total face value (or par amount) of: $35, 587.21 (1.04) 8 = $48, 703.55 At the end of year 1, we need $20,000 for the frst payment. Thus we need to sell some fracton of our bond holdngs; that fracton s determned by the market n the followng manner: par = $20, 000 (1) (1.04) 6 We are just solvng the zero formula n reverse to determne the par value we must sell n the market to receve $20,000 n year 1. The soluton s par = $25, 306.38. After the sale, we are left wth bonds wth a total face value of $48, 703.55 $25, 306.38 = $23, 397.17 At the end of year 2, we need to make another payment of $20,000. Solvng a smlar equaton to (1): par = $20, 000 (2) (1.04) 4 9
yelds a par value of $23,397.17, exactly equal to the par amount of our remanng bonds. So we smply sell all of the remanng bonds, collect the $20,000 and make the payment. Scenaro 2: Instantaneous Decrease n Interest Rates to 7%. Before solvng ths part, here s some ntuton. We have locked n (.e. guaranteed) a return of 8% whle the nterest rate has decreased to 7%. Ths s good for us because we are gettng a hgher return on our nvestment relatve to the new rate. Therefore, we should expect to have excess cash after we pay off our expenses, gven that the above dervaton shows we exactly meet our expenses when the nterest rate does not change. From above, we own bonds wth a par amount of $48,703.55. At the end of year one, we must go to the market and redeem some fracton of our holdngs to make the $20,000 payment. The par amount of bonds we need to sell s gven by: par = $20, 000 (1.035) 6 whch yelds $24,585.11 (note the use of the new dscount rate). Ths leave us wth bonds wth a total face value of $48, 703.55 $24, 585.11 = $24, 118.44. In year two, we must go to the market agan and sell some fracton of our bonds to make the second payment of $20,000. Solvng par = $20, 000 (1.035) 4 yelds a par amount of $22,950.46 whch mples we wll have bonds wth a par value of $24, 118.44 $22, 950.46 = $1, 167.98 left over. Note that ths s not the same as havng $1,167.98 n cash. To get cash we have to go to the market and sell these bonds. If we dd, we would receve: $1, 167.98, 44 (1.035) 4 = $1, 017.83 Thus, we have an excess of $1,017.83. Scenaro 3: Instantaneous Increase n Interest Rates to 9% Before solvng ths part, here s some ntuton. We have locked n at 8% whle the nterest rate has ncreased to 9%. Ths s not good for us because we are gettng a lower return on our nvestment. Thus, we should expect to have a shortage of cash for our expenses. 10
Approachng the problem as before, we must sell: par = $20, 000 (1.045) 6 $26,045.20, par value, of our bonds. Ths transacton leaves us wth bonds totalng $48, 703.55 $26, 045.20 = $22, 658.34 n par value. In the second year, f we sell all of our remanng bonds, we wll receve: mplyng a shortfall of $999.59. $22, 658.34 (1.045) 4 = $19, 000.41 9.c Scenaro 1: Unchanged Interest Rates. From part 9.b, the present value of our expense s $35,587.21. Spendng ths amount on 1-year zeros mples a face value of $35, 587.21 (1.04) 2 = $38, 491.13. At the end of year one, when the bonds mature and pay ther par value, we must sell $20,000 and renvest the remanng $18,491.13 n one-year zeros agan. At then end of the second year, our bonds wll be worth $18, 491.13 (1.04) 2 = $20, 000, exactly equal to the second payment. Scenaro 2: Instantaneous Decrease n Interest Rates to 7%. Before solvng ths part, here s some ntuton. We have locked n (.e. guaranteed) a return of 8% only for the frst year, whle the nterest rate has decreased to 7%. Thus, our nvestment n the second year wll suffer from ths lower rate of return and we should have a shortage of funds. We know from above that we are left wth $18,491.13 after the frst payment. Investng ths n one-year zeros at the new nterest rate of 7% ensures a payoff at the end of year two of equal to $18, 491.13 (1.035) 2 = $19, 808.16. Thus, we wll have nsuffcent funds n the amount of $20, 000 $19, 88.16 = $191.84. Scenaro 3: Instantaneous Increase n Interest Rates to 9%. Before solvng ths part, here s some ntuton. We have locked n (.e. guaranteed) a return of 8% only for the frst year, whle the nterest rate has ncreased to 11
9%. Thus, our nvestment n the second year wll beneft from ths hgher rate of return and we should have a surplus of funds. We know from above that we are left wth $18,491.13 after the frst payment. Investng ths n one-year zeros at the new nterest rate of 9% ensures a payoff at the end of year two equal to $18, 491.13 (1.045) 2 = $20, 192.78. Thus, we wll have excess funds n the amount of $20, 192.78 $20, 000 = $192.78. 10. 10.a Recall that the yeld (.e. yeld-to-maturty, YTM) s the one rate that dscounts all of the bonds cash flows to gve the present value of the bond. The yeld may be found by solvng the followng equaton. 100 = 8 1 + y + 8 (1 + y) 2 + 108 (1 + y) 3 You may solve ths equaton by tral and error or usng Excel s solver. 1 The yeld s 8%. In fact, we ddn t really need to solve ths equaton to fnd the yeld. Snce the bond s tradng at par and we are valung t on a payment date, the yeld has to equal the coupon rate. 10.b The Duraton 2 s computed accordng to the followng duraton formula 1 [T 1 P V (c 1 ) + T 2 P V (c 2 ) +... + T N P C(c N )] B 0 ( 1 1 8 100 1.08 + 2 8 1.08 + 3 108 ) = 2.78 2 1.08 3 The modfed duraton measure adjusts ths fgure by dvdng by (1 + ) = 1.08, whch equals 2.57. DV01, or dollar value of one bass pont, s computed usng equaton DV 01 = B(R 0.01%) B(R) 1 A techncal note: In fact, there s an analytc soluton to the cubc equaton. You wll get three answers correspondng to the order of the polynomal, but only one of the solutons wll be real. The other two wll be a complex conjugate par. 2 Duraton s also referred to as Macaulay Duraton. 12
The bond prce at R = 8% s gven as $100. The prce at R = 7.99% s: B(7.99%) = 8 (1.0799) + 8 (1.0799) + 108 2 (1.0799) 3 = 100.02578 Therefore, DV01 s 100.02578 100 = 0.02578. 10.c We know from above that a one bass pont declne results n a prce ncrease of $0.02578. Therefore, a 10 bass pont declne results n an approxmate prce ncrease of 10 $0.02578 = $0.2578. 13