Present Value (PV) vs. Future Value (FV)

Single sum of money
A sum of money in a specific point of time

PV = FV / (1 + r)ⁿ
FV = PV x (1 + r)ⁿ

Where:
r – periodic interest rate
n – no. of compounding periods

Ordinary annuity
A finite set of sequential equal sum of payments at the end of each compounding period.

PV_ordinary = PMT / (1 + r)¹ + PMT / (1 + r)² + PMT / (1 + r)³ + …..+ PMT / (1 + r)ⁿ
= PMT{[1-1/(1+r)ⁿ]/r}

FV_ordinary = PMT(1 + r)^n-1+ PMT (1 + r)^n-2 + PMT (1 + r)^n-3 + …..+ PMT(1 + r)⁰
= PMT{[(1+r)ⁿ-1]/r}

Annuity due
A finite set of sequential equal sum of payments at the beginning of each compounding period.

PV_Due = PMT + PMT / (1 + r)¹ + PMT / (1 + r)² + PMT / (1 + r)³ + …..+ PMT / (1 + r)^n-1
= PMT(1+r){[1-1/(1+r)ⁿ]/r}

FVDue = PMT(1 + r)ⁿ+ PMT (1 + r)^n-1 + PMT (1 + r)^n-2 + …..+ PMT / (1 + r)¹
= PMT(1+r){[(1+r)ⁿ-1]/r}


Ordinary annuity and annuity due conversion

PV_Due = (1+r) PV_ordinary
FV_Due = (1+r) FV_ordinary

Perpetuity
A set of never-ending sequential equal sum of payments.

PV_Perpetuity =PMT/(1+r)¹ + PMT/(1+r)²+ ….. = PMT / r

Continuous compounding
Earning interest on top of interest constantly and continuously.

PV=FVe^-rt
FV=PVe^rt

Where,
r – continuously compounded rate
r = ln(FV/PV)/t.

A series of unequal cash flows
PV = CF1 / (1 + r)¹ + CF2 / (1 + r)² + CF3 / (1 + r)³ + …..+ CFn / (1 + r)ⁿ
FV = CF1 (1 + r)^n-1 + CF2 (1 + r)^{n- 2} + CF3 / (1 + r)^n-3 + …..+ CFn

My tips:

• The cash outflows for mortgage repayment are always at the end of each period and that for education fees for students are at the beginning of each school year unless specified otherwise.
• Be aware that the compounding periods may not necessarily be annual.

PV = FV / (1 + r/m)^mxn

where: m-periods per year; n- years in the future

• Use financial calculator to solve the unknown variable. Be remembered to change the BEG/END mode.