Introduction to Mathematical Economics, ORMS1030

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1 Uiversity of Vaasa, sprig 04 Itroductio to Mathematical Ecoomics, ORMS030 Exercise 6, week 0 Mar 3 7, 04 R ma 0 D5 R5 ti 4 6 C09 R ma 4 6 D5 R6 to 4 C09 R3 ti 08 0 D5 R7 pe 08 0 D5 R4 ti 4 C09 R8 pe 0 D5. Fid the itegrals a 3x + 4x + 7dx, b 5 4x dx a 3x + 4x + 7dx = 3 3 x3 + 4 x + 7x +C = x 3 + x + 7x +C b 5 4x dx = / 5 x x = 5 5 = 36 Figure for the exercise : The figure is here i order to get all the calculatios for exercise o the same page.

2 . The iitial ivestmet of the project is H = 800e. The resultig cotiuous cash flow is k = 00 e/moth. The cash flow starts at t = 0 year ad stops at t =,5 year. The residual value is JA = +500e. The discout rate is 6,0% per aum i.e. ρ = l,06 a. The Net Preset Value by cotiuous compoudig is t NPV = H + e ρt ktdt + e ρt JA = H + k t ρ e ρt e ρt + e ρt JA. a Fid NPV, whe k = 00e/a, ρ = l,06 a, t = 0a, t =,5a. b Fid NPV, whe k = 00e/moth, ρ = l,06 / moth, t = 0moth, t = 30moth. c What you ca say about the iteral rate of retur? a b NNA = H + k ρ e ρt e ρt + e ρt JA = 800e + 00 vuosi e e 0 e l,06 vuosi,5vuosi +... l,06 vuosi... + e l,06 vuosi,5vuosi 500e = 800e + 00e l,06,06, e = 43,95e,06,5 NNA = H + k ρ e ρt e ρt + e ρt JA = 800e + 00 kk e l,06 / kk e 0 e l,06/ kk 30kk e l,06/ kk 30kk 500e = 800e + 00e l,06, e = 43,95e,06 30 c Because the Net Preset Value is positive the Iteral Rate of Retur is more tha the used discoutig rate 6% per aum. If the iteral rate of retur is i tod ad r = + i tod, the NNA = 800e + 00e lr r, e r,5 The graph of this fuctio is draw by Excel i the figure o the previous page. Accordig to the picture the Iteral Rate is about 7,5% per aum.

3 3. Compare followig two projects. The iitial ivestmets of project A is 000 e, ad it creates a cash flow of 00eper moth for two years. The iitial ivestmets of project B is e, ad it creates a cash flow of 00eper moth for te years. The cash flows iclude oly the productio costs ad reveues. Fiacial costs are ot icluded. Fid the Net Preset Values of the projects whe the discout rate is 8,0% per aum. Are the projects profitable? Ratkaisu: NNA A = 000e 4 00e,08 / j = 000e 00e,08 / NNA B = 6000e 0 = 6000e 00e,08 /,08 / 4,08 / Both projets are profitable with the discout rate of 8%. = 7,9e > 0e ok 00e,08 / j,08 / 0,08 / = 686,48e > 0e ok The Net Preset Value of the project B is three times larger tha NPV of A. Still the ivestmets of B is eight times the ivestmets of A ad the returs of B take 0 years to come. I spite of NPV mosts ivestors would prefer the project A. 4. We wefie the Proportioal Preset Value PPV as: PPV = Preset value of Icome flow Preset Value of Cost flow Fid the PPV for the projects of the exercise 3. Which oe is the better oe? PPV A = 7,9e 000e PPV B = 6686,48e 6000e =,086 > ok =,049 > ok Both projects are profitable with discoutig rate 8%. The project A is better PPV A > PPV B.

4 5. Usig Excels IRR-fuctio fid the Iteral Rate of Returs of the projects i the exercise 3. Which oe is ow the better? Ratkaisu: The values of IRR-fuctio of Excel are: IRR moth,a =,53% ja IRR moth,b = 0,74%. The aual iteral rates of retur are the IRR a,a =,053 = 0,9747 i it,a = 9,75% IRR a,b =,0074 = 0,09044 i it,b = 9,04% The project A gives its retur faster ad it pays better iterest for the ivestmet. So A is clearly the better oe. 6. a Fid the values of Retur o Ivestmet ROI II for the projects of the exercise 3. The results are ot ecessarily reasoable. ROI is a soud measure oly for a log project. Which project is ow beter oe? b For the both previous project fid the values of the followig ratio: myrate = k a b a H/ 00%, where k a is the aual Net Cash Accumulatio A: 00e, B: 400e, b a is the aual pay back of capital A: 000e, B: 600e, ad H/ is average capital reserved A: 000e, B: 8000e. a ROI II,A = 00e 00% = 60%, 000e ROI II,B = 400e 00% = 5%. 6000e The values of ROI are ow too big to be substitutes for iteral rates. ROI is a measure for Fiacial Statemets. Now we must use it with care. b myrate A = myrate B = 00e 000e 000e 400e 600e 8000e These umbers are usefull. They are close the values of IRR. 00% = 0%, 00% = 0%. 7. Fid the pay back periods for the projects of the exercise 3. Which oe is ow better? Ratkaisu: The pay back period is = lk/k ih. l + i A = l00e/00e,08/ 000e l,08 / B = l00e/00e,08/ 6000e l,08 / =,5kk = vuosi9,5kk =,7kk = 9vuotta4,7kk The compariso of the projects by pay back periods is ot trivial whe the duratio of the projects are so differet. At least the both projects are profitable. They pay back the ivestmets durig the projects.

5 Kaavoja: Korkolasku yksikertaie korkolasku: K t = + itk 0 = + p 00 tk 0, ku 0 < t < korokorkolasku: K t = + i t K 0, ku t =,,3,... jatkuva korkolasku: K t = + i t K 0 = e ρt K 0, ku t > ja + i = e ρ Jaksolliset suoritukset prologoititekijä, diskottaustekijä, kuoletuskerroi s,i = + i, a,i = + i i + i i i + i, c,i = + i Tasaerälaia ja osamaksukauppa k = c,i K 0, k = c,i H h + m k= a + k d = a + a, a q k = a q k= q Kassavirra ettoykyarvo NPV = k 0 + k j + i j Projekti ettoykyarvo Pääoma tuottoaste ROI I = ROI II = Takaisimaksu-aika NPV = H + k j + i j ettovuositulos keskimääri sidottu pääoma 00% ettovuositulos alussa sidottu pääoma 00% = lk/k ih l + i