Matematyka ekonomiczna

Lista zadań 02

Zadanie 1.

Dane: F = 10000zl l $n = 9mcy = \frac{9}{12} = 0,75$ d = 8% D = ? P = ? r = ?	Obliczenia: Dla odsetek pobieranych z góry D = F • d • n = 600 P = F − D = 9400 – tyle dostanie do ręki Dla odsetek pobieranych z dołu F = P(1+r•n) $$P = \frac{F}{1 + r \bullet n}$$ $$I = F - P = F - \frac{F}{1 + r \bullet n}$$
D = I $$F \bullet d \bullet n = F\left( 1 - \frac{1}{1 + r \bullet n} \right)$$ $$d \bullet n = 1 - \frac{1}{1 + r \bullet n}$$ d • n(1+r•n) = 1 + r • n − 1 d • n(1+r•n) = r • n d(1+r•n) = r d + d • r • n = r r(1−d•n) = d $$r = \frac{d}{1 - d \bullet n}$$ $$r = \frac{0,08}{1 - 0,08 \bullet 0,75} \approx 8,5\%$$

Zadanie 2

Dane: P = 100 000zl d = 12% $$n = \frac{90}{365}$$ F = ?	Obliczenia: P = F − D P = F − F • d • n P = F(1−d•n) $$F = \frac{P}{1 - dn}$$
$$F = \frac{100\ 000}{1 - \frac{12}{100} \bullet \frac{90}{365}} = \frac{100\ 000}{0,970411} = 103049,12$$

Zadanie 3

Dane: Fn = 200zl ri = 6% Fr = ? Wzór Fishera: 1 + rn = (1+rr)(1+ri) $$\frac{1 + r_{n}}{1 + r_{r}} = 1 + r_{i}$$	Obliczenia: Fn = P − P • rn = P(1 + rn) Fr = P − P • rr = P(1 + rr) $$\frac{F_{n}}{F_{r}} = \frac{P\left( 1 + r_{n} \right)}{P\left( 1 + r_{r} \right)} = \frac{1 + r_{n}}{1 + r_{r}} = 1 + r_{i}$$ $$F_{r} = \frac{F_{n}}{1 + r_{i}}$$
$$F_{r} = \frac{F_{n}}{1 + r_{i}} = \frac{200}{1,06} = 188,68zl$$

Zadanie 4

Dane: P = 10 000zl rn = 1, 5% ri = 6% Fn = ? Fr = ? In = ? Ir = ?	Obliczenia: Fn = P − P • rn = P(1 + rn) Fn = 10000 • 1, 015 = 10150zl In = Fn − P = 150zl $$F_{r} = \frac{F_{n}}{1 + r_{i}} = \frac{10150}{1 + \frac{0,06}{4}} = 10000$$ Ir = Fr − P = 0zl

Zadanie 5

Inwestycje: 1000 600 Korzyści: 600 1250 F = P(1 + r)^n Wszystkie kwoty należy zdyskontować ponieważ, żeby je porównywać musimy rozpatrywać je dla tego samego punktu w czasie $$P = \frac{F}{{(1 + r)}^{n}}$$ $$1000 + \frac{600}{{(1 + 0,1)}^{2}} = \frac{600}{{(1 + 0,1)}^{1}} + \frac{1265}{{(1 + 0,1)}^{3}}$$ $$\begin{matrix} 1495,86 = 1495,86 \\ \Downarrow \\ r = 10\% \\ \end{matrix}$$

Zadanie 6

Oferta A: P = 3000 − 0, 02 • 3000 − 80 = 2860 F = F1 = F2 = 1750 $$P_{1} = \frac{F}{{(1 + r)}^{1}}$$ $$P_{2} = \frac{F}{{(1 + r)}^{2}}$$ $$P = \frac{F}{{(1 + r)}^{1}} + \frac{F}{{(1 + r)}^{2}}\ \ \ / \bullet {(1 + r)}^{2}$$ P(1 + r)^2 = F(1+r) + F P(1 + r)^2 − F(1+r) − F = 0 1 + r = x Px^2 − Fx − F = 0 =b^2 − 4ac = F^2 + 4PF $$\sqrt{} = 4804,425$$ $$x_{1} = \frac{- b - \sqrt{}}{2a} = \frac{F - \sqrt{}}{2P} < 0\ $$ $$x_{2} = \frac{- b + \sqrt{}}{2a} = \frac{F + \sqrt{}}{2P} \approx 1,146$$ 1, 146 = 1 + r r = 0, 146 = 14, 6%	Oferta B: P = 3000 F = F1 = F2 = 1893 $$P_{1} = \frac{F}{{(1 + r)}^{1}}$$ $$P_{2} = \frac{F}{{(1 + r)}^{2}}$$ $$P = \frac{F}{{(1 + r)}^{1}} + \frac{F}{{(1 + r)}^{2}}\ \ \ / \bullet {(1 + r)}^{2}$$ P(1 + r)^2 = F(1+r) + F P(1 + r)^2 − F(1+r) − F = 0 1 + r = x Px^2 − Fx − F = 0 =b^2 − 4ac = F^2 + 4PF $$\sqrt{} = 5128,2988$$ $$x_{1} = \frac{F - \sqrt{}}{2P} < 0\ $$ $$x_{2} = \frac{F + \sqrt{}}{2P} \approx 1,170$$ 1, 170 = 1 + r r = 0, 170 = 17%
Oferta C: P = 3000 − 50 = 2950 F = F1 = F2 = 1893 $$P_{1} = \frac{F}{{(1 + r)}^{1,5}}$$ $$P_{2} = \frac{F}{{(1 + r)}^{2}}$$ $$P = \frac{F}{{(1 + r)}^{1,5}} + \frac{F}{{(1 + r)}^{2}}\text{\ \ }\ $$ P(1 + r)^2 − F(1+r)^0, 5 − F = 0 1 + r = x^2 Px^4 − Fx − F = 0 W Excelu obliczamy, że r = 0, 1536 = 15, 36%	Oferta D: P = 3000 + 100 = 3100 F1 = 1500       F2 = 2500 $$P = \frac{F_{1}}{{(1 + r)}^{1}} + \frac{F_{2}}{{(1 + r)}^{2}}\ \ \ / \bullet {(1 + r)}^{2}$$ P(1 + r)^2 − F1(1+r) − F2 = 0 Px^2 − F1x − F2 = 0 =b^2 − 4ac = F1^2 + 4PF2 $$\sqrt{} = 5766,2812$$ $$x_{1} = \frac{F_{1} - \sqrt{}}{2P} < 0\ $$ $$x_{2} = \frac{F_{1} + \sqrt{}}{2P} \approx 1,172$$ r = 0, 172 = 17, 2%
Odpowiedź: rA < rc < rB < rD więc najlepsza jest oferta banku A