Question

What is the present value​ (PV) of an investment that pays $90,000 every year for four years if the interest rate is 9​% ​APR, compounded​ quarterly?

Answer 1

Effective Annual Rate = ( 1 + r ) ^ n - 1
r = Int Rate per period
n = No.of periods per anum

Particulars	Amount
Ret period	2.2500%
No. of periods	4.0000

EAR = [ ( 1 + r ) ^ n ] - 1
= [ ( 1 + 0.0225 ) ^ 4 ] - 1
= [ ( 1.0225 ) ^ 4 ] - 1
= [ 1.0931 ] - 1
= 0.093083
I.e EAR is 9.308332 %

As the CFs are yearly CFs, Effective rate per anum is calculated.

Present value of Investment:

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here cash flows are happened at the end of the period.PV of annuity is current value of cash flows to be received at regular intervals discounted at specified int rate or discount rate to current date.

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
r - Int rate per period
n - No. of periods

Particulars	Amount
Cash Flow	$          90,000.00
Int Rate	9.3083%
Periods	4

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
= $ 90000 * [ 1 - [(1+0.0931)^-4]] /0.0931
= $ 90000 * [ 1 - [(1.0931)^-4]] /0.0931
= $ 90000 * [ 1 - [0.7005]] /0.0931
= $ 90000 * [0.2995]] /0.0931
= $ 289612.35
PV of Investment is $ 289612.35