**A dollar today is worth more than a dollar tomorrow
**$100 today is worth more than $100 tomorrow. This is because you could take the $100 today and invest it to earn interest.

The value of the money tomorrow is the **future value** and the value of the money today is the **present value**.

**Calculating Future Values**

Let’s say you invest $100 for 2 years at rate, *r = 5%*

Value of the investment after 1 year = *100 X (1+ r) = 100 X 1.05 = $105*

Value of the investment after 2 years = *105 X 1.05 = $ 110.25*

The value of the investment grows at compounded rate of interest and earns interest on interest too.

Future value of $100 = *$100 X (1+r) ^{t}*

**The present value of the same amount PV = $110.25/1.05^{2 }= $100**

**Calculating present values**** **

The present value of the same amount PV = *$110.25/1.05 ^{2 }= $100*

If C_{t }= Cash flow at time t. The Present Value, PV = *C _{t}/(1+r)^{t}*

The expression 1/(1+r)^{t }is called the **discount factor**.

**Net Present Value**

Net present value is the difference between present value of cash inflow and present value of cash outflow.

NPV = PV- Investments

In other words, **any project that has negative present cash flow is not worth undertaking**.

For Example, let’s say you need to buy an office space which is worth $10,000 now. This is the investment and let’s say that the net return from the office after 2 years is $15000. Suppose that the return on U.S. government securities during this period is r = 5% (r is the discount rate or the opportunity cost of capital).

PV =$15,000/(1+0.05)^{2 }= $13605.44

Thus, NPV= 13605.44- 10000 = $3605.44

The present cash flow is positive and hence the project is worth undertaking. In other words, it is worth more than the investment a time t=0.

Cash flow at present (C_{0}) is usually negative as this is the investment at t=0.

Hence, **NPV = C _{0 }+ C_{t}/(1+r)^{ t}**

**Rate of return**

The rate of return is simply the ratio of profit on initial investment.

In the above example,

Rate of return = (15000 – 10000)/ 100000 = 0.5

**If the rate of return is more than the opportunity cost of capital, then the project is worth undertaking.**

**Present Values for multiple cash flows**

This is also known as the discounted cash flow (DCF). Can write the same as:

**Perpetuity**

An asset that offers a fixed income every year forever (perpetual income). The payment will start after a one unit of time period (in this case a year).

Present value of a perpetuity is **(cashflow/rate).**

In some cases, the perpetuity might start after ‘n’ years and in such cases, the value till n^{th} year can be calculated and then PV of the value till n^{th} year can be calculated to arrive at a present value.

In case of growing perpetuity with growth rate g,

Where, C_{1} is the cashflow at year 1.

**Annuity**

An asset that offers a fixed income for a specified number of years.

**Present value of an n year annuity = present value of perpetuity stating now – present value of perpetuity starting at n ^{th} year.**

In case of a growing annuity with growth rate g,

**This can be remembered as a growing annuity with growth g minus the growing perpetuity after n years (The growth rate for which is g and discount rate is r).**

**How interest is calculated?**

If $1 is invested at rate of interest = r compounded m times a year, then at the end of the year the investment is worth **(1+r/m) ^{m }**