Question-89385

Question Number 89385 by pete last updated on 17/Apr/20

Commented by pete last updated on 17/Apr/20

I need help on the above question, please.

$$\:\mathrm{I}\:\mathrm{need}\:\mathrm{help}\:\mathrm{on}\:\mathrm{the}\:\mathrm{above}\:\mathrm{question},\:\mathrm{please}. \\ $$

Answered by Joel578 last updated on 17/Apr/20

First option PV_1 = 48000v^6 + 11300v^(10) = $ 48607.51 Second option PV_2 = X + 12100v^8 = $ (X + 9551.85) Now, PV_1 = PV_2 ⇒ $ 48607.51 = $ (X + 9551.85) ⇒ X = $ 39055.65

$$\mathrm{First}\:\mathrm{option} \\ $$$${PV}_{\mathrm{1}} \:=\:\mathrm{48000}{v}^{\mathrm{6}} \:+\:\mathrm{11300}{v}^{\mathrm{10}} \:=\:\$\:\mathrm{48607}.\mathrm{51} \\ $$$$ \\ $$$$\mathrm{Second}\:\mathrm{option} \\ $$$${PV}_{\mathrm{2}} \:=\:{X}\:+\:\mathrm{12100}{v}^{\mathrm{8}} \:=\:\$\:\left({X}\:+\:\mathrm{9551}.\mathrm{85}\right) \\ $$$$ \\ $$$$\mathrm{Now}, \\ $$$${PV}_{\mathrm{1}} \:=\:{PV}_{\mathrm{2}} \\ $$$$\Rightarrow\:\$\:\mathrm{48607}.\mathrm{51}\:=\:\$\:\left({X}\:+\:\mathrm{9551}.\mathrm{85}\right) \\ $$$$\Rightarrow\:{X}\:=\:\$\:\mathrm{39055}.\mathrm{65} \\ $$

Commented by Joel578 last updated on 17/Apr/20

i^((2)) = 0.06 v = (1 + (i^((2)) /2))^(−1) = (1 + ((0.06)/2))^(−1) = (1.03)^(−1)

$${i}^{\left(\mathrm{2}\right)} \:=\:\mathrm{0}.\mathrm{06} \\ $$$${v}\:=\:\left(\mathrm{1}\:+\:\frac{{i}^{\left(\mathrm{2}\right)} }{\mathrm{2}}\right)^{−\mathrm{1}} \:=\:\left(\mathrm{1}\:+\:\frac{\mathrm{0}.\mathrm{06}}{\mathrm{2}}\right)^{−\mathrm{1}} \:=\:\left(\mathrm{1}.\mathrm{03}\right)^{−\mathrm{1}} \\ $$

Commented by pete last updated on 18/Apr/20

thanks very much sir Joel. But please what does P, V_1 , V_2 and v stand for?

$$\mathrm{thanks}\:\mathrm{very}\:\mathrm{much}\:\mathrm{sir}\:\mathrm{Joel}. \\ $$$$\mathrm{But}\:\mathrm{please}\:\mathrm{what}\:\mathrm{does}\:\mathrm{P},\:\mathrm{V}_{\mathrm{1}} ,\:\mathrm{V}_{\mathrm{2}} \:\mathrm{and}\:\mathrm{v}\:\mathrm{stand} \\ $$$$\mathrm{for}? \\ $$

Commented by Joel578 last updated on 19/Apr/20

PV is present value (the value of money right now) In order to determine the upfront amnout now, we should know what is the value of $ 48000 and $ 11300 if they are given right now. v is discount factor = (1 + i)^(−1) Let′s say you deposits $K on a bank, which give you effective annual interest rate i. Then, after n years from now, your deposit is $K(1+i)^n → Future Value (FV) If you know the Future Value, then you can find its Present Value (= $K) PV = FV v^n = $K(1+i)^n . (1+i)^(−n) = $K

$${PV}\:\mathrm{is}\:\mathrm{present}\:\mathrm{value}\:\left(\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{money}\:\mathrm{right}\:\mathrm{now}\right) \\ $$$$\mathrm{In}\:\mathrm{order}\:\mathrm{to}\:\mathrm{determine}\:\mathrm{the}\:\mathrm{upfront}\:\mathrm{amnout}\:{now}, \\ $$$$\mathrm{we}\:\mathrm{should}\:\mathrm{know}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\$\:\mathrm{48000}\:\mathrm{and}\: \\ $$$$\$\:\mathrm{11300}\:\mathrm{if}\:\mathrm{they}\:\mathrm{are}\:\mathrm{given}\:{right}\:{now}. \\ $$$$ \\ $$$${v}\:\mathrm{is}\:\mathrm{discount}\:\mathrm{factor}\:=\:\left(\mathrm{1}\:+\:{i}\right)^{−\mathrm{1}} \\ $$$$\mathrm{Let}'\mathrm{s}\:\mathrm{say}\:\mathrm{you}\:\mathrm{deposits}\:\${K}\:\mathrm{on}\:\mathrm{a}\:\mathrm{bank},\:\mathrm{which}\:\mathrm{give}\:\mathrm{you} \\ $$$$\mathrm{effective}\:\mathrm{annual}\:\mathrm{interest}\:\mathrm{rate}\:{i}.\:\mathrm{Then},\:\mathrm{after}\:{n}\:\mathrm{years}\:\mathrm{from}\:{now}, \\ $$$$\mathrm{your}\:\mathrm{deposit}\:\mathrm{is}\:\${K}\left(\mathrm{1}+{i}\right)^{{n}} \:\rightarrow\:{Future}\:{Value}\:\left({FV}\right) \\ $$$$ \\ $$$$\mathrm{If}\:\mathrm{you}\:\mathrm{know}\:\mathrm{the}\:\mathrm{Future}\:\mathrm{Value},\:\mathrm{then}\:\mathrm{you}\:\mathrm{can}\:\mathrm{find} \\ $$$$\mathrm{its}\:\mathrm{Present}\:\mathrm{Value}\:\left(=\:\${K}\right) \\ $$$$\mathrm{P}{V}\:=\:{FV}\:\:{v}^{{n}} \:=\:\${K}\left(\mathrm{1}+{i}\right)^{{n}} \:.\:\left(\mathrm{1}+{i}\right)^{−{n}} \:=\:\${K} \\ $$

Commented by pete last updated on 25/Apr/20

may you continually increase in knowledge as you continue to give help, Sir

$$\mathrm{may}\:\mathrm{you}\:\mathrm{continually}\:\mathrm{increase}\:\mathrm{in}\:\mathrm{knowledge} \\ $$$$\mathrm{as}\:\mathrm{you}\:\mathrm{continue}\:\mathrm{to}\:\mathrm{give}\:\mathrm{help},\:\mathrm{Sir} \\ $$

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