MS-E24 Ivestmet Sciece Exercise 2/206, Solutios 26.2.205 Perpetual auity pays a xed sum periodically forever. Suppose a amout A is paid at the ed of each period, ad suppose the per-period iterest rate is r. The the preset value of the perpetual auity is A P = ( + r) k = A + r + A ( + r) k = A + r + A + r ( + r) k = A + r + P + r k= P = A r. k=2 k= A ite-lifetime costat paymet is termed auity. Assume a amout A be paid periodically times, ad assume a per-period iterest rate r. The preset value of the auity is A P = ( + r) k = A ( + r) k A ( + r) k = A r A ( + r) ( + r) k = A r A ( + r) r k= k= k=+ k= P = A r ( + r) The amout A of the paymet ca be derived from the above formula to be P = A r ( + r) A = r( + r) P ( + r) The issuer of a bod has a obligatio to pay the bod holder (accordig to the rules specied at the time the bod is issued) a) the face value of the bod at the date of maturity. b) the possible coupo paymets C/m periodically, for example a percetage (coupo rate c) of the face value, i.e., c, where c = C/(m ) The preset value of a bod is P = yield to maturity). ( + λ m ) + k= C m ( + λ, where λ = the yield of the bod (i.e., the m ) Similarly to the case of the auity formula, the collapsed form of the preset value formula of a bod is C P = ( + λ + m m ) ( + λ = m ) ( + λ + C m ) λ ( + λ. m ) λ m
MS-E24 Ivestmet Sciece Exercise 2/206, Solutios 26.2.205 The duratio of a xed-icome security is a weighted average of the times that paymets (cash ows) are made. or a zero-coupo bod, the duratio equals the maturity, otherwise the duratio is always less tha the maturity. As the coupo rate icreases, the duratio decreases, because the cash ow has a greater weight i the earlier periods. The duratio is formally deed as D = P V (t 0)t 0 + P V (t )t + + P V (t )t, where P V = P V or every o-egative cash ow the, it is clear that t 0 D t. P V (t k ). t=0 Macalay duratio is deed as a duratio of which the preset values are calculated usig the yield of the bod (yield to maturity). Specically, suppose a acial istrumet makes m paymets i a year, with a paymet k beig c k (both coupo paymet ad possibly the face value), ad there are periods remaiig. The, the paymet times are t k = k/m ad the Macalay duratio ca be calculated as D = k= k m c k (+ λ m )k, where P V = P V k= c k ( + λ m )k. If the coupo paymets are idetical (c k = C/m k < ad c = C/m +, where is the face value of the bod ad C the aual coupo paymet), otig the coupo rate as c = C/(m ), the explicit formula for the Macalay duratio is D = + y my + y + (c y) mc( + y) + my, where y = λ m. This formula ca be derived as follows. Assume coupo rate c = C/(m ), whe the periodical coupo paymet is c, ad yield y = λ/m per period. c k c The preset value of the bod is P = ( + λ = k= m )k ( + y) + c ( + y) 2 + + c ( + y) + ( + y). Dieretiatig the preset value yields dp dy = c + y ( + y) + 2c ( + y) 2 + + c ( + y) + ( + y) = P m c + y P m ( + y) + 2 c m ( + y) 2 + + (c + ) m ( + y) () = P m + y D The preset value ca also be writte with the auity formula as P = c y Dieretiatig this yields dp dy = c y 2 ( + y) + c y ( + y) + = c ( + y) + y 2 ( + y) + ( + y). ( c/y) ( + y) ( + y) + (2)
MS-E24 Ivestmet Sciece Exercise 2/206, Solutios 26.2.205 Settig the two formulas () ad (2) for dp/dy equal the gives P m + y D = c y 2 ( + y) ( c/y) ( + y) + (3) The, we multiply both sides of (3) with the deomiators y 2 ad ( + y) + to get y 2 ( + y) P md = c( + y) + + y 2 c( + y + y). (4) We see that the auity formula form of the value P of the bod ca be modied ito + P = c y ( + y) ad substitutig P from (5) ito (4) yields ( + y) = c ( + y) + y y( + y), (5) y c ( + y) + y md = c( + y) + + y 2 c( + y + y) (6) Now, we elimiate ad divide the factor of md ito the right side of the above equatio to get md = c( + y)+ + y 2 c( + y + y) y {c ( + y) + y} (7) Last step of the proof is to take the partial fractio decompositio of the right side of (7). We solve A(y) ad B(y) from c( + y) + + y 2 c( + y + y) y {c ( + y) + y} = A(y) y + B(y) c ( + y) + y, (8) ad get (detailed steps of solvig these skipped here) A(y) = + y ad B(y) = + y + (c y). Substitutig (8) with the previous formulas for A(y) ad B(y) ito (7) ad dividig m ito the right side yields which completes the proof. D = + y my + y + (c y) mc( + y) + my, (9)
MS-E24 Ivestmet Sciece Exercise 2/206, Solutios 26.2.205. (L3.6) (The biweekly mortgage 0) Here is a proposal that has bee advaced as a way for homeowers to save thousads of dollars o mortgage paymets: pay biweekly istead of mothly. Specically, if mothly paymets are x, it is suggested that oe istead pay x/2 every two weeks (for a total of 26 paymets per year). This will pay dow the mortgage faster, savig iterest. The savigs are surprisigly dramatic for this seemigly mior modicatio - ofte cuttig the total iterest paymet by over oe-third. Assume a loa amout of 50 000 e for 20 years at 4% iterest, compouded mothly. a) Uder a mothly paymet program, what are the mothly paymets ad the total iterest paid over the course of the 20 years? b) Usig the biweekly program, whe will the loa be completely repaid, ad what are the savigs i total iterest paid over the mothly program? (You may assume biweekly compoudig for this part.) c) Which factors do the savigs cosist of? Is this a ewly iveted moey-makig-machie? Compare the aual paymets. Solutio: a) P = 50 000 e, r = 4%, r m = 4%/2 0.33% Loa time is 20 years. Hece the total amout of mothly periods is: = 20 2 = 240 The mothly paymets are: P = A r A = 0.0033(.0033)240 50000 (.0033) 240 909 e. ( + r) A = r( + r) P ( + r) The total paymet is A 28 53 e. The total iterest paid is the total paymets - the preset value P of the loa (50 000 e): The total iterest paid is: A P 68 53 e. b) or bi-weekly paymet there are 26 paymets i a year. The bi-weekly paymet is A = A/2 = 454 e. The per-period iterest i r 26 = r/26 = 4%/26 0.5% To solve whe the loa will be completely paid, we solve the for which A = r( + r) P ( + r) as follows: A rp = ( + r) A rp A = (+r) l ( A rp ) l A = l(+r) = l A l(a rp ) l( + r) By isertig the values of A, r ad P, we get 46, which is i years 46/26 7.7. Hece, the total paymets are A 209 548 e, from which the iterest paymets ca be solved similarly to part a) as A P 59 548 euroa. Savigs i total iterest over mothly program is 28 53 209 548 =8 605 e (2.6%). (We would get the same result whe comparig the iterest paymets.) c) The yearly paymets are part a): 2 909 = 0 908 e part b): 26 454 = 804 e Thus, the aual paymets are greater for the biweekly paymet. or this reaso it is atural that the iterest paymets are smaller. If the total aual paymet of part b) 804 e would be made mothly, the mothly paymet would be 984 e, ad usig this paymet the loa time ad the iterest paymets would equal those of the bi-weekly program. Hece the savigs result from faster paymet program. Note that mo. 4 wk. = 2 2 wk.
MS-E24 Ivestmet Sciece Exercise 2/206, Solutios 26.2.205 2. (L3.8) (Variable-rate mortgage) The Lehtie family just took out a variable-rate mortgage o their ew home. The mortgage value is 00 000 e, the term is 30 years, ad iitially the iterest rate is 4%. The iterest rate is guarateed for 5 years, after which time the rate will be adjusted accordig lo prevailig rates. The ew rate ca be applied to their loa either by chagig the paymet amout or by chagig the legth of the mortgage. a) What is the origial yearly mortgage paymet? (Assume paymets are yearly.) b) What will be the mortgage balace after 5 years? c) If the iterest rate o the mortgage chages to 5% after 5 years, what will be the ew yearly paymet that keeps the termiatio time the same? d) Uder the iterest chage i c., what will be the ew term if the paymets remai the same? Solutio: a) P = 00 000 e, r = 4% Paymet time is 30 years, ad because the paymets are yearly, = 30. A = r( + r) P ( + r) 5 783 e. b) Accordig to part a), aual paymet of 5783 e suces payig a debt of 00 000 e i 30 years. We calculate the preset value of a loa paid with these paymets i 25 years. = 25, A = 5 783 e P = A r ( + r) 90 343 e. This is the balace of the mortgage after 5 years. This 25 years correspod the years 6-30 i the 30-year paymet program. Sice a amout of 90 343 e ca be amortized durig these years, i years -5 a amout of 00 000 90 343 = 9 657 e will be amortized. or sake of clarity, Table below presets the pricipal of the loa, the paymets, ad the portios of the paymets allocated to the iterest ad pricipal paymets. Table : Loa amortizatio durig the rst 5 years of the paymet program. Year Pricipal Paymet Iterest paymet Pricipal paymet P A r P A r P 00000 5783.0 4000 783.0 2 9826.99 5783.0 3928.68 854.33 3 96362.66 5783.0 3854.5 928.50 4 94434.6 5783.0 3777.37 2005.64 5 92428.5 5783.0 3697.4 2085.87 9657.36 c) Accordig to part b) there is 90 343 e loa remaiig after 5 years. Sice the paid iterest is greater, the yearly paymet has to be icreased i order to keep the termiatio time the same. P = 90 343 e, r = 5% Paymet time 25 years with yearly paymets = 25. The ew yearly paymet is A = r ( + r ) P ( + r ) 640 e.
MS-E24 Ivestmet Sciece Exercise 2/206, Solutios 26.2.205 Hece the yearly paymet icreases 6 40 5 783 = 627 e. d) Accordig to part b) there is 90 343 e loa remaiig after 5 years. Sice the paid iterest is greater, the paymet time has to be icreased to keep the yearly paymet the same. P = 90 343 e, r = 5%, A = 5 783 e Usig the formula from Exercise for the loa payo time we get = l A l(a r P ) l( + r 3. years. ) The total life of the mortgage is 5 + 3.=36. years.
MS-E24 Ivestmet Sciece Exercise 2/206, Solutios 26.2.205 3. A youg couple is curretly livig o ret i a apartmet with a mothly ret of 600 e. They have saved 0 000 e for buyig a ow apartmet, ad have cosulted a real estate broker about the ew apartmet. The broker is oerig them a two-room apartmet with a price of 00 000 e, ad with mothly costs of 20 e (cosistig of maiteace ad water paymets). The apartmet is i good coditio ad it requires o reovatio. However, after three years the codomiium will be reovatig the widows of the apartmet, causig a 2000 e paymet at the start of the third year. The couple has decided to take a 20-year auity debt of 90 000 e, with a aual iterest of 5% ad with a opeig paymet of 00 e. However, because oly 70% of the value of the apartmet (70 000 e) ca be used as collateral for the loa, the couple buys a partial collateral by the govermet of 5% of the value of the apartmet (5 000 e) with a price of 375 e (2.5% of the size of the collateral), ad the remaiig 5000 e will be covered with a free collateral from the parets of the husbad. a) Cosiderig oly the paymets durig the time of the loa, should the couple move ito the apartmet? Note that eectively oly 70% of the iterest have to be paid, because 30% of the iterests of the rst apartmet ca be reduced from taxes. b) What if oly the paymets to bak, govermet ad the codomiium are cosidered, ad o amortizatios are icluded i the calculatios (amortizatios icrease the wealth of the couple)? Use the 5% iterest rate for discoutig. c) What if the prevalet iterest rate is 2.3% (2-moth Euribor + 0.7 margial)? Solutio: Ret ow 600 e/moth Savigs 0 000 e Price of ew apartmet 00 000 e Maiteace charge of the ew apartmet 20 e/moth Widow reovatio at the start of the third year 2000 e. Loa paymet time 20 years Number of periods = 20 2 = 240 Loa pricipal 90 000 e Loa opeig paymet 00 e Govermet partial collateral 375 e Iterest rate r = 5% Mothly iterest r m = 5% 2 = 0.4% = 0.004 a) ad b) The preset value of the ret of the old apartmet for the 20 years is A = 600 e P = A r m ( + r m ) 90 95 e. The mothly paymets for the P = 90000 e loa are A = r m( + r m ) P ( + r m ) 594 e The partial collateral ad opeig paymets are paid immediately. The preset values are 375 e ad 00 e. The preset value of the maiteace charges (usig same formula as for the PV of the ret) is 8 83 e. The widow reovatio paid after three years has a preset value of 2000/( + r) 3 728 e. Let us cosider the paymets ad their preset values after the rst period.
MS-E24 Ivestmet Sciece Exercise 2/206, Solutios 26.2.205 The mothly paymet for the P = 90000 eloa is 593.96 e. The iterest paymets are 0.004 90000 = 375.00 e. The pricipal paymet is 593.96 375 = 28.96 e. Because the eective iterest paymets are 70% of the total iterest paymets, the eective iterest paymets are 0.7 375 = 262.50 e. The preset values of the paymets are calculated usig the discout factor d m = /( + r m ). These calculatios are cocluded i Table 2 below. Table 2: The paymets after the rst period ad their preset values Loa pricipal Total paymet Iterest paymet Pricipal paymet E. iterest paymet 90 000 594 375.0 28.96 262.5 Preset value - 59.5 373.4 28. 26.4 After the rst period (ad at the start of the secod period) the remaiig loa pricipal is 90 000 28.96 = 89 78.04 e. Usig this value, the correspodig iterest paymets, pricipal paymets ad eective iterest paymets ca be calculated for the secod period. This procedure is repeated for the 240 periods util the whole pricipal of the loa is paid (see Excel solutio). Summatio of the preset values of the paymets of partial collateral, maiteace, loa opeig ad loa pricipal ad the eective iterest rates yields the preset value of the ivestmet to be 99 085 e (see the solutios Excel le). This is greater tha the preset value of the ret 90 95 e. If the icreased wealth of the couple is ot take ito accout, the couple should ot move i to the ew apartmet (part a) ). Nevertheless, if the icreased wealth after payig o the loa are take ito accout, the pricipal paymets of the loa are ot icluded i the ivestmet value calculatio. This calculatio yields the preset value of the ivestmet to be 46 753 e, which is sigicatly less tha the preset value of the ret. Hece, would the icreased wealth be cosidered, the couple should move i to the ew apartmet (part b) ). c) The calculatios are idetical to the earlier part of this exercise apart from the dieret iterest rate r = 2.3%. Repeatig the calculatios with ew r yields preset values of the ret 5 337 e, the loa w.o. icreased wealth cosidered 09 660 e ad the loa with icreased wealth cosidered 38 829 e. Hece at this iterest, the couple should move i to the ew apartmet i both cases.
MS-E24 Ivestmet Sciece Exercise 2/206, Solutios 26.2.205 I exercises 4-6 the coupo paymets are made every 6 moths (m = 2). The face values of the bods are 00 scores ad the aual coupo paymet is C is the omial value coupo rate. I formula (3.3) of the course book otatios c = C/(m ) are y = λ/m are used. 4. (L3.9) (Bod price) A 8% bod with 8 years to maturity has a yield of 9%. What is the price of this bod? Solutio: Par value = 00, coupo rate 8%, yearly coupo paymets C = 8 e. The bod matures i 8 years ad there are m = 2 yearly coupo paymets, therefore the total umber of paymets is = 2 8 = 36. Straightforward use of the formula for a bod price gives P = ( + λ + C m ) λ ( + λ 9.7 e. Note that the bod price formula does ot use the per-period m ) coupo rate c = 8%/2 = 4%. 5. (L3.0) (Duratio) id the price ad duratio of a 0-year, 8% bod that is tradig at a yield of 0%. Solutio: Par value = 00, coupo rate 8%, yearly coupo paymets C = 8 e. There are m = 2 yearly paymets, hece the per-period coupo rate is c = 8%/2 = 4% The bod matures i 0 years, therefore = m 0 = 2 0 = 20. The yearly yield is λ = 0%, ad the per-period yield is y = λ m = 0%/2 = 5%. Use of the bod price formula gives P = ( + λ + C m ) λ ( + λ 87.54 e. m ) Moreover, use of the Macalay duratio formula gives D = + y my + y + (c y) mc( + y) + my 6.84. 6. (L3.4) (Duratio limit) Show that the limitig value of duratio as maturity is icreased to iity is D + λ m λ. or the bods i Table (3.6) of the course book (where λ = 0.05 ja m = 2) we obtai D 20.5. Note that for large λ this limitig value approaches /m, ad hece the duratio for large yields teds to be relatively short. Solutio: D = + y my + y + (c y) mc( + y) + my D = + y +y my Whe, + y 0, 0 ad my mc (+y) ( + y) 0, but + (c y) + my. Hece D + y my = + λ m λ.