anonymous
  • anonymous
You have a profit making scheme that is projected to pay at a rate of p[t] = (100,000+t) E^(t/5) dollars per year t years from now. A) Assuming a projected interest rate of 6% compounded every instant, what is the present value of your scheme? B) Still assuming a projected rate of 6% compounded every instant, how much would you have to plunk down for a perpetual annuity that would pay you at the same rate? C) what is the projected take on this scheme? D) How many years would it take for this scheme to play out in the sense that the future take will be next to nothing?
Mathematics
schrodinger
  • schrodinger
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anonymous
  • anonymous
For A I have r=0.06 ​p(t) = (100000+t) e^(-0.2 t) \[Present Value = \int\limits_{0}^{\infty} e^{-0.06 t} p(t) dx\]
IrishBoy123
  • IrishBoy123
that looks good for the PV i can help you on this stuff, but i think we are in conflicting times zones.
anonymous
  • anonymous
I mean, I know I dont understand how question B should be applied, but is it worded badly or something?

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anonymous
  • anonymous
how about... B Future Value = Present Value * E^(r t)
anonymous
  • anonymous
ah well, any suggestions at all appreciated
IrishBoy123
  • IrishBoy123
i'll have a look at this in a few hours time. "plunk in" and "take" are a bit Gordon Gekko for me and i worked in banking for some years! lol will post something here later, it might even be helpful!!
anonymous
  • anonymous
awesome thank you irish
IrishBoy123
  • IrishBoy123
@hughfuve are you around, now?
IrishBoy123
  • IrishBoy123
well if not, first there is a typo in the question \(p[t] = (100,000+t) E^{(t/5)}\) i think you worked this out because in your answer you switched to a negative exponential p[t]=(100,000+t)E(-t/5) for the present values we have as you say \(\int_{0}^{\infty} (100000+t) e^{(-.2 - 0.06)t} dt\) http://www.wolframalpha.com/input/?i=%5Cint_%7B0%7D%5E%7B%5Cinfty%7D+%28100000%2Bt%29+e%5E%7B%28-.2+-+0.06%29t%7D+dt
IrishBoy123
  • IrishBoy123
for the second one, what do you think the income from this scheme to me is just the same as an annuity, ie the PV of th eincome source is the PV, if the incomes are the same the PV's are the same we are discounting the same income at the same discount rate in perpetuity does that ring any bells with you? for part C, "the take", that particular piece of jargon does not ring any bells with me so i am going to have a stab at it meaning just the total amount of cash raised by the scheme with no discounting for this particular income function that's \(\int_{0}^{\infty} (100000+t) e^{-.2t} dt\) http://www.wolframalpha.com/input/?i=%5Cint_%7B0%7D%5E%7B%5Cinfty%7D+%28100000%2Bt%29+e%5E%7B-.2t%7D+dt it could conceivably also refer to the idea that the retae of return on the investment is greater than the 6% base rate by some amount and that is what we have to figure out. if that is so we need toplay again with the integration formulae. do you have this term described in your course notes?
IrishBoy123
  • IrishBoy123
for the last bit, again there is practical fudge that i would use in the absence of a taught method you can say that the amount of cash [no discounting, there's no point] received between time T and infinity is this: \(\int_{T}^{\infty} (100000+t) e^{-.2t} dt\) from Wolfram that is : \(5e^{-T/5} (T + 100005)\) you can then find when that becomes small, say \(5e^{-T/5} (T + 100005) < 1000\) i get this from a little script i wrote T 30 USD 1239.80986996 T 31 USD 456.105121531 T 32 USD 456.109680941 T 33 USD 456.114240351 T 34 USD 456.118799761 T 35 USD 456.12335917 T 36 USD 167.80008379 T 37 USD 167.801761103 T 38 USD 167.803438416 T 39 USD 167.805115729 this could all be total bs but let me know if you can shed any further light. this is a practical approach....but maybe not what the books want :(
anonymous
  • anonymous
Irish thank you thank you for this, much appreciated. B) I thought so too that the deposit must be the present value, but I was unsure adn thought it a trick. C) Negative on the notes on this. There was nothing taught on it. But Im with you on this, it must be the sum of p[t] D) Its as good as any method I would cook up, I was thinking to maybe just plot it and see where it approaches zero,
anonymous
  • anonymous
could they mean future value, by total take?
IrishBoy123
  • IrishBoy123
hi @hughfuve i played with the notion of using FV but kept getting stuck on this \(FV(T) =\int_{0}^{T} (100000 + t) e^{-t/5} e^{0.06(T-t)} dt\) from Wolf http://www.wolframalpha.com/input/?i=%5Cint_%7B0%7D%5E%7BT%7D+%28100000+%2B+t%29+e%5E%7B-t%2F5%7D+e%5E%7B0.06%28T-t%29%7D+dt the definite integral breaks suggesting there is something wrong and if you let \(T \rightarrow \infty\) the dominant terms is \(\frac{50}{169} e^{-T/5}(1300050)e^{13T/50} = \\ e^{3T/50} \times \frac{50}{169} (1300050)\) which \(\rightarrow \infty\) i "think" the intuitive explanation is that the income stream never actually becomes zero, it just gets smaller and smaller, and doing that forever will sill bring in an infinitely large amount of money.
IrishBoy123
  • IrishBoy123
https://gyazo.com/20adf710966e2d03c94adcdfd7bcd6d4
anonymous
  • anonymous
I see what you mean, Future Value must implies that there is an actual date of maturation of some kind and infinity is a too far out there.

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