Present value of continuous income stream

[Unikid778]

I truly have no idea of the time frame of this question, as such, I have no idea where to even begin to get the values of a,b and t - any help with this would be greatly appreciated:


The Nobel Peace Prize is awarded each year and in 2020 has the value$1 million. It is funded by
an investment earning interest at 5%p.a., compounded continuously. What is the present value
of this fund? Take into consideration that the prize will be awarded in perpetuity,i.e.every year
in future. Assume the value of the prize each year will remain unchanged for the foreseeable
future.

Hint: Use the present value of a continuous income stream formula:

Present value= PV = R(t) er^(a-t) dt where R(t)is the value of the prize awarded in year t,
and t is the number of years since 2020.

 

[tkhunny]

You can work your way to it...

PV Compounded Annually i = 0.05 v = 1/(1+i) Pmt: 1,000,000
It converges nicely by the time it lasts for 200 years. No need to play all the way to infinitt.

It's heading right for 20,000,000

Okay, that's a good first guess. Now, let's work on the "Continuously" part. Note that the compounding is continuous, NOT the annual payments.

But first, what would you guess the result to be if we increase the compounding to semi-annual? If we compound more often, we accrue more interest, so we should come in lower. We're still paying only 1,000,000 annually.

Semi-Annual i = 0.05, m = 2, r = (1 + i/m)^m, v = 1/r

Sure Enough: , only 19,753,086.

Let's try Quarterly: i = 0.05, m = 4, r = (1 + i/m)^m, v = 1/r
, a little less again.

So, your task is to answer, "How low will it go?" Can you establish the end of this limiting process? Keep increasing "m" and see where it goes.

A few more, just to help you experiment:
Monthly: m = 12, PV = 19,545,796
Weekly: m = 52, PV = 19,513,778
Daily: m = 365.2425, PV = 19,505,535
Hourly: m = 8675.82, PV = 19,504,224

Well, how low will it go?

Hint: The secret is in the limit of "r".