Question

Discounting/Compounding . What is an annuity? What is the difference between an ordinary annuity & annuity due? .How does the FV and PV increase/decrease as the time and interest rates increase/decrease? TVM problems (lump sum problems only) that ask you to solve for the following: o Number of periods o Interest rate o Present Value o Future Value

Definition/Explanation

Answer 1

Part 1:

Discounting vs Compounding

Discounting:
Discounting method is used to determine the present value of future cash flows.
The concept here is: to get a certain amount in future how much we should invest today.
In the method of discounting, we use discount rate.
Here, in the method of discounting, future value of cash flow will be known.
Formula used: PV = FV / (1 + r)^n
PV is the present value of an investment.
FV is the future value of an investment or a cash flow.
r is the rate of interest
n is the number of period. Example, to discount a future cash flow in year 5, we will use n as 5.

Compounding:
Compounding method is used to to calculate the future value of an investment done in present.
The concept here is: how much amount we will get in future if we invest a certain amount of money today.
In the method of compounding, we use compound interest rate.
Present value must be known to use compounding method.
Formula used: FV = PV*(1 + r)^n
Where FV is the future value of an investment.
PV is the present value of an investment.
r is the rate of interest
n is the number of periods in the investment horizon.

Part 2:

Annuity refers to a series of payments made at regular intervals.
Example: When an insurance company makes retirement income payments at regular intervals to the people who brought retirement plan from the insurance company.

Ordinary annuity:
When a payment is made after the due date of payment we call it as ordinary annuity.
In simple terms, payments are made at the end of the period.

Annuity due:
When a payment is made before the due date of payment we call it as annuity due.
Here, payments are made at the beginning of the period.

Annuity Due = Ordinary annuityx(1 + i)
i is the interest rate associated with the payments.
This shows that the value of an annuity due will be more than ordinary annuity as the payment incase of an annuity due is made at the beginning of a period.

Part 3:

FV = PV*(1 + r)^n (this is called as the time value of money formula)
and PV = FV / (1 + r)^n

Where FV is the future value of an investment.
PV is the present value of an investment.
r is the rate of interest
n is the number of time periods in the investment horizon.

Effect of increase in time period on FV ad PV:
FV: From the equation FV = PV*(1 + r)^n we can see that if "n" increases FV will also increase and viceversa.
PV: In the equation PV = FV / (1 + r)^n, we can see that if "n" increases PV will decrease and viceversa.

Effect of increase in interest rates on FV ad PV:
FV: From the equation of FV we can see that if rate of interest increases, the value of FV will increase and vice versa.
PV: From the equation of PV we can see that if rate of interest increases, the value of PV will decrease and vice versa.

Part 4:

The formula used for time value of money is FV = PV*(1 + r)^n.
We can rearrange the formula to solve for number of periods, interest rate, present value and future value.

Number of periods (n): Here we need to use log functions in the formula FV = PV*(1 + r)^n = FV/PV=(1+r)^n
We will take log function on both sides and then we will get the following:
ln(FV/PV)=n*ln(1+r)
=>n= ln(FV/PV)/ln(1+r)

Interest rate (r) : FV = PV*(1 + r)^n = FV/PV=(1+r)^n
=>[FV/PV]^1/n=1+r
=>r=[FV/PV]^1/n-1

Present value: PV = FV / (1 + r)^n
Future value: FV = PV*(1 + r)^n