1 + r. k=1. (1 + r) k = A r 1

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1 Perpetual auity pays a fixed 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 P = A r. k=2 A fiite-lifetime costat paymet is termed auity. Assume a amout A to 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=+ P = A [ r ( + r) The amout A of the paymet correspodig a preset value P uder periodical iterest rate r ad umber of paymet periods 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 specified 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 ) + 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

2 The duratio of a fixed-icome security is a weighted average of the times that paymets (cash flows) 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 flow has a greater weight i the earlier periods. The duratio is formally defied 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 flow the, it is clear that t 0 D t. P V (t k ). t=0 Macalay duratio is defied as a duratio of which the preset values are calculated usig the yield of the bod (yield to maturity). Specifically, suppose a fiacial 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 m c k (+ λ m )k, where P V = P V 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. The derivatio of this formula ca be foud as a additioal material for this exercise.

3 2. (L3.8) (Variable-rate mortgage) The Lehtie family just took out a variable-rate mortgage o their ew home. The mortgage value is 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 to 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? a) P = e, r = 0.04 Paymet time is 30 years, ad because the paymets are yearly, = 30. A = r( + r) P ( + r) e. b) Accordig to part a), aual paymet of 5783 e suffices payig a debt of e i 30 years. We calculate the preset value of a loa paid with these paymets i 25 years. = 25, A = e P = A r [ ( + r) e. This is the balace of the mortgage after 5 years. These 25 years correspod the years 6-30 i the 30-year paymet program. Because a amout of e ca be amortized durig these years, i years -5 a amout of = 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 first 5 years of the paymet program. Year Pricipal Paymet Iterest paymet Pricipal paymet P A rp A rp

4 c) Accordig to part b) there is e loa remaiig after 5 years. Because the paid iterest is greater, the yearly paymet has to be icreased i order to keep the termiatio time the same. P = e, r = 0.05 Paymet time 25 years with yearly paymets = 25. The ew yearly paymet is A = r ( + r ) P ( + r ) 640 e. Hece the yearly paymet icreases by = 627 e. d) Accordig to part b) there are e loa remaiig after 5 years. Because the paid iterest is greater, the paymet time has to be icreased to keep the yearly paymet the same. P = e, r = 0.05, A = e To solve the loa payoff time, we solve the for which A = r ( + r ) P ( + r ) as follows: A = r P ( + r ) A r P A l ( A r P ) l A = l( + r ) = ( + r ) = l A l(a r P ) l( + r ) 3. years. The total life of the mortgage is =36. years.

5 I exercises 2.2 ad 2.3 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 ) ad y = λ/m are used. 2.2 (L3.0) (Duratio) id the price ad duratio of a 0-year, 8% bod that is tradig at a yield of 0%. 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 = 0m = 0 2 = 20. The yearly yield is λ = 0%, ad the per-period yield is y = λ/m = 0%/2 = 5%. irst let us calculate the price ad duratio of the bod as P = D = P ( + λ/m) + C/m ( + λ/m) k k c k m ( + λ/m) k, where c k = C/m, if k < ad c = C/m +. The calculatios ca be made with a spreadsheet software as preseted i Table 2. I Table 2, discout factors refer to factors /( + λ) k, ad duratio factors refer to factors (k/m)/( + λ/m) k. The duratio term refers to duratio factors multiplied by the cash flow of period k. Table 2: Spreadsheet calcaulatio of preset value ad duratio Σ Period k Cash flow c k Time k/m Discout factor Duratio factor PV of cash flow Duratio term Duratio 6.84 The calculatios ca also be made usig the collapsed formulas for the bod price ad Macalay duratio. Use of the bod price formula gives P = ( + λ + C [ m ) λ ( + λ e, m ) ad use of the Macalay duratio formula gives D = + y my + y + (c y) mc[( + y) + my 6.84, where y = λ/m

6 2.3 (L3.4) (Duratio limit) Show that the limitig value of duratio as maturity is icreased to ifiity 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 Note that for large λ this limitig value approaches /m, ad hece the duratio for large yields teds to be relatively short. 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.

7 2.4 (L3.6) (The biweekly mortgage) 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. Specifically, 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 modificatio - ofte cuttig the total iterest paymet by over oe-third. Assume a loa amout of 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. a) P = 50, 000 e, r = 0.04, r m = 0.04/ Loa time is 20 years. Hece the total amout of mothly periods is: = 20 2 = 240 The mothly paymets are: P = A [ r A = (.0033) (.0033) e. ( + r) A = r( + r) P ( + r) The total paymet is A e. The total iterest paid is the total paymets - the preset value P of the loa ( e): The total iterest paid is: A P 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 is r 26 = r/26 = 0.04/ Usig the formula from Exercise 2.. for the loa payoff time we get = l A l(a rp ) 46, l( + r) which is i years 46/ Hece, the total paymets are A e, from which the iterest paymets ca be solved similarly to part a) as A P e. Savigs i total paymets over mothly program is =8 605 e (2.6%). (We would get the same result whe comparig the iterest paymets.) c) The yearly paymets are part a): = e part b): = 804 e Hece 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.

8 2.5 A youg couple is curretly livig o ret i a apartmet with a mothly ret of 600 e. They have saved e for buyig a ow apartmet, ad have cosulted a real estate broker about the ew apartmet. The broker is offerig them a two-room apartmet with a price of 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 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 ( e) ca be used as collateral for the loa, the couple buys a partial collateral by the govermet for 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 effectively oly 70% of the iterest have to be paid, because 30% of the iterests of the first 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 margial)? Numerical values: Ret ow = 600 e/moth Savigs = e Price of ew apartmet = 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 = e Loa opeig paymet = 00 e Govermet partial collateral = 375 e Iterest rate r = 0.05 Mothly iterest r m = 0.05/2 = 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 ) e. The mothly paymets for the P = 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) e.

9 Let us cosider the paymets ad their preset values after the first period. The mothly paymet for the P = e loa is e. The iterest paymets are = e. The pricipal paymet is = e. Because the effective iterest paymets are 70% of the total iterest paymets, the effective iterest paymets are = 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 3: The paymets after the first period ad their preset values Loa pricipal Total paymet Iterest paymet Pricipal paymet Eff. iterest paymet Preset value After the first period (ad at the start of the secod period) the remaiig loa pricipal is = e. Usig this value, the correspodig iterest paymets, pricipal paymets ad effective 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 effective iterest rates yields the preset value of the ivestmet to be e (see the solutios Excel file). This is greater tha the preset value of the ret 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 off the loa is take ito accout, the pricipal paymets of the loa are ot icluded i the ivestmet value calculatio. This calculatio results i the preset value of the ivestmet to be e, which is sigificatly 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 differet iterest rate r = 2.3%. Repeatig the calculatios with ew r yields preset values of the ret e, the loa w.o. icreased wealth cosidered e ad the loa with icreased wealth cosidered e. Hece at this iterest, the couple should move i to the ew apartmet i both cases.

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