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Annuities
Annuities are essentially a series of fixed payments required from you or paid to you at a specified frequency over the course of a fixed time period. The most common payment frequencies are yearly, semi-annually (twice a year), quarterly and monthly. There are two basic types of annuities: ordinary annuities and annuities due.
Ordinary Annuity: Payments are required at the end of each period. For example, straight bonds usually pay coupon payments at the end of every six months until the bond's maturity date.
Annuity Due: Payments are required at the beginning of each period. Rent is an example of annuity due. You are usually required to pay rent when you first move in at the beginning of the month, and then on the first of each month thereafter.
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Present Value of an Annuity
The present value an annuity is the sum of the periodic payments each discounted at the given rate of interest to reflect the time value of money.
PV of an Ordinary Annuity = R (1 − (1 + i)^-n)/i
PV of an Annuity Due = R (1 − (1 + i)^-n)/i × (1 + i)
Where,
i is the interest rate per compounding period;
n are the number of compounding periods; and
R is the fixed periodic payment.
Example :
1. Calculate the present value on Jan 1, 2015 of an annuity of 5,000 paid at the end of each month of the calendar year 2015. The annual interest rate is 12%.
Solution
We have,
Periodic Payment R = 5,000
Number of Periods n = 12
Interest Rate i = 12%/12 = 1%
Present Value
PV = 5000 × (1-(1+1%)^(-12))/1%
= 5000 × (1-1.01^-12)/1%
= 5000 × (1-0.88745)/1%
= 5000 × 0.11255/1%
= 5000 × 11.255
= 56,275.40
2. A certain amount was invested on Jan 1, 2015 such that it generated a periodic payment of 10,000 at the beginning of each month of the calendar year 2015. The interest rate on the investment was 13.2%. Calculate the original investment and the interest earned.
Solution
Periodic Payment R = 10,000
Number of Periods n = 12
Interest Rate i = 13.2%/12 = 1.1%
Original Investment = PV of annuity due on Jan 1, 2015
= 10,000 × (1-(1+1.1%)^(-12))/1.1% × (1+1.1%)
= 10,000 × (1-1.011^-12)/0.011 × 1.011
= 10,000 × (1-0.876973)/0.011 × 1.011
= 10,000 × 0.123027/0.011 × 1.011
= 10,000 × 11.184289 × 1.011
= 1,13,073.20
Interest Earned = 10,000 × 12 − 1,13,073.20
= 1,20,000 – 1,13,073.20
= 6926.80
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Net Present Value
Net present value is the difference between the present value of cash inflows and the present value of cash outflows that occur as a result of undertaking an investment project. It may be positive, zero or negative. These three possibilities of net present value are briefly explained below:
Positive NPV:
If present value of cash inflows is greater than the present value of the cash outflows, the net present value is said to be positive and the investment proposal is considered to be acceptable.
Zero NPV:
If present value of cash inflow is equal to present value of cash outflow, the net present value is said to be zero and the investment proposal is considered to be acceptable.
Negative NPV:
If present value of cash inflow is less than present value of cash outflow, the net present value is said to be negative and the investment proposal is rejected.
Net present value method (also known as discounted cash flow method) is a popular capital budgeting technique that takes into account the time value of money. It uses net present value of the investment project as the base to accept or reject a proposed investment in projects like purchase of new equipment, purchase of inventory, expansion or addition of existing plant assets and the installation of new plants etc.
To be at Net Present Value you also need to subtract money that went out (the money you invested or spent):
Add the Present Values you receive
Subtract the Present Values you pay
1. Company A is considering a new piece of equipment. It will cost Rs. 6,000 and will produce a cash flow of Rs. 1,000 every year for the next 12 years (the first cash flow will be exactly one year from today).
(a) What is the NPV if the appropriate discount rate is 10%?
You can either discount each individual cash flow or recognise that the Rs. 1,000 cash flows are just a twelve year annuity. So,
PV = a/i[l -1/(1 +i)n]
PV= 1,000/0.1 [1 - 1/(1.1)12]
PV = Rs. 6,814
Adding this to the original investment gives an NPV of
NPV = Rs. 6,814 - Rs. 6,000
NPV =Rs. 814
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(b) What is the NPV if the appropriate discount rate is 12%?
PV= 1,000/0.12 [1 -1/(1.12)12]
PV = Rs. 6,194
Adding this to the original investment gives an NPV of
NPV = Rs. 6,194-Rs. 6,000
NPV=Rs. 194
(c) What is the NPV if the appropriate discount rate is 15%?
PV= 1,000/0.15 [1-1/(1.15)12]
PV = Rs. 5,421
Adding this to the original investment gives an NPV of
NPV = Rs. 5,421-Rs. 6,000
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