__Present Value__

Present valueÂ describes how much a future sum ofÂ moneyÂ is worth today.Â Three most influential components of present value are : time, expected rate of return, and the size of the future cash flow. The concept of*Â present value*Â is one of the most fundamental and pervasive in the world of finance. It is theÂ basisÂ forÂ stockÂ pricing,Â bondÂ pricing, financial modeling, banking, insurance, pensionÂ fund valuation. It accounts for the fact thatÂ moneyÂ we receive today can be invested today to earn a return. In other words, present value accounts for theÂ time valueÂ of money.

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The formula for present value is:

**PV = CF/(1+r)n**

Where:

CF =Â cash flowÂ in future period

r = the periodicÂ rate of returnÂ or interest (also called the discount rate or the required rate of return)

n = number of periods

**Example :**

Assume that you would like toÂ putÂ moneyÂ in an account today to make sure your child has enough money in 10 years to buy a car. If you would like to give your child 10,00,000 in 10 years, and you know you can get 5% interest perÂ yearÂ from aÂ savings accountÂ during that time, how much should you put in the account now?

**PV = 10,00,000/ (1 + .05)10Â = 6,13,913/-**

Thus, 6,13,913Â willÂ be worth 10,00,000 in 10 years if you can earn 5% each year. In other words, the present value of 10,00,000 in this scenario is 6,13,913.

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Future Value

The value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today. It refers to a method of calculating how much theÂ present valueÂ (PV) of anÂ assetÂ or cashÂ willÂ be worth at a specific time in the future. There are two ways to calculate FV:

1) For an asset with simple annual interest: = Original Investment x (1+(interest rate*number of years))

2) For an asset with interest compounded annually: = Original Investment x ((1+interest rate)^number of years)

Example:

1) 10,000 invested for 5 years with simple annual interest of 10% would have a future value of

FV = 10000(1+(0.10*5))

= 10000(1+0.50)

= 10000*1.5

= 15000

2) 10,000 invested for 5 years at 10%, compounded annually has a future value of :

FV = 10000(1+0.10)^5)

= 10000(1.10)^5

= 10000*1.61051

= 16105.10

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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:**Â 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 bondâ€™s maturity date.

**Annuity Due:**Â 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 âˆ’ (1 + i)^-n)/i

PV of an Annuity Due = R (1 âˆ’ (1 + i)^-n)/i Ã— (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 PaymentÂ Â Â Â Â Â RÂ = 5,000

Number of PeriodsÂ Â Â Â Â nÂ = 12

Interest RateÂ Â Â Â Â Â Â Â Â iÂ = 12%/12 = 1%

Present Value

PV = 5000 Ã— (1-(1+1%)^(-12))/1%

= 5000 Ã— (1-1.01^-12)/1%

= 5000 Ã— (1-0.88745)/1%

= 5000 Ã— 0.11255/1%

= 5000 Ã— 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 PaymentÂ Â Â Â Â Â RÂ = 10,000

Number of PeriodsÂ Â Â Â Â nÂ = 12

Interest RateÂ Â Â Â Â Â Â Â Â iÂ = 13.2%/12 = 1.1%

Original InvestmentÂ Â Â Â Â Â = PV of annuity due on Jan 1, 2015

= 10,000 Ã— (1-(1+1.1%)^(-12))/1.1% Ã— (1+1.1%)

= 10,000 Ã— (1-1.011^-12)/0.011 Ã— 1.011

= 10,000 Ã— (1-0.876973)/0.011 Ã— 1.011

= 10,000 Ã— 0.123027/0.011 Ã— 1.011

= 10,000 Ã— 11.184289 Ã— 1.011

= 1,13,073.20

Interest EarnedÂ = 10,000 Ã— 12 âˆ’ 1,13,073.20

= 1,20,000 â€“ 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**Â (also known asÂ **discounted cash flow method**) is a popular capital budgeting technique that takes into account the time value of money.Â 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 atÂ NetÂ 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 â€“ 1/(1.1)12]

PV = Rs. 6,814

Adding this to the original investment gives an NPV of

NPV = Rs. 6,814 â€“ 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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