209) Expanded Formula for Invesment

209) Expanded Formula for Investment, 

FV= PV(1+i)^n

If you are in the Unit Trust industry, you would most probably have seen the formula below.

FV= PV(1+i)^n

FV = Future Value or End Value
PV = Present Value or Begin Value
i = interest rate or rate of return
n = number of years or periods

Yes, it is the same formula you learned during your preparation for the Unit Trust Exam, CUTE.

Let us see the formula from different perspectives. Most probably it is your first time ever.

As you know about Unit Trust and PRS investments, your invested money is converted into units.
The amount you invested is divided by the unit price to get number of units.

Eg,

If you invested $1000 into a fund with the unit price of $0.25, you will get 4000 units. Assume service charges are paid separately.

PV = UP x NU
where,
UP = Unit Price
NU = Number of Units

$1000 = $0.25 x 4000

Let us expand the formula to separate the n.
We assume it is for 3 years to simplify the calculation. In actual practice, the n can be any number of years.


FV= PV(1+i)^n

FV= PV(1+i1)(1+i2)(1+i3)
where,
i1 = rate of return for year 1
i2 = rate of return for year 2
i3 = rate of return for year 3

Let us expand the FV and PV, indicating:
PV1 as the Begin Value in Year 1
FV3 as the End Value in Year 3

So, the expanded formula is:
FV3= PV1(1+i1)(1+i2)(1+i3)

Now, expand PV1 and FV3 in terms of Unit Price (UP) and Number of Units (NU)
PV1 = UP1 x NU1
FV3 = UP3 x NU3

The expanded formula is:
FV3= PV1(1+i1)(1+i2)(1+i3)
(UP3 x NU3) = (UP1x NU1)(1+i1)(1+i2)(1+i3)

This formula above can only be used if the Number of Units Remains Constant.

What if the number of units increased?

Let us combine another formula from Second perspective.
Total Return is the calculated by the compounded return over the different years.

So the Total Return for 3 years can be shown as formula below:

(1+TR3) = (1+i1)(1+i2)(1+i3)

The formula, 
FV3= PV1(1+i1)(1+i2)(1+i3) becomes
FV3= PV1(1+TR3)   


Let us combine another formula from Third perspective.

(1 + Total Return) = (1 + Number of Units Growth) x (1 + Unit Price Growth)

(1+TR3) = (1+NUG3)(1+UPG3)

TR3 = Total Return for 3 years
NUG3 = Number of Units Growth for 3 years
UPG3 = Unit Price Growth for 3 years

The formula, 
FV= PV(1+i)^n

FV= PV(1+i1)(1+i2)(1+i3)

FV3= PV1(1+i1)(1+i2)(1+i3)
FV3= PV1(1+TR3)  

FV3= PV1(1+NUG3)(1+UPG3)
(UP3 x NU3) = (UP1x NU1)(1+NUG3)(1+UPG3) 

In General Terms, for n years, the formula is
(UPn x NUn) = (UP1x NU1)(1+NUGn)(1+UPGn)

This is the new derived formula to calculate the end investment value from the growth of unit prices and growth of number of units.

Try this formula out and check the answer.

Using the above formula, the original investment PV of $1000 with 4000 units at $0.25 each has grown into end value FV.

Let's say the unit price had increased to $0.30 and number of units became 5000. The investment value has now become $1500.

So,
Number of Units Growth = (5000 - 4000)/4000 = 0.25
Units Price Growth =  ($0.30 - $0.25)/$0.25 = 0.20

FV3 = (UP1x NU1)(1+NUG3)(1+UPG3) 
FV3 = ($0.25 x 4000)(1+0.25)(1+0.20)
FV3 = ($1000)(1.25)(1.20)
FV3 = $1500

FV3 = UP3 x NU3
FV3 = $0.30 x 5000
FV3 = $1500


More details from these links:

251) How Total, Annual & Annualized Returns are Related

https://highlevelrules.blogspot.com/2020/05/251-how-total-annual-annualized-returns.html

179) Unit Price and Number of Units Growth Rates Effect on Investment Values.
https://highlevelrules.blogspot.com/2018/06/unit-price-and-number-of-units-effect.html