Time value of money

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 = $110.25/1.052 = $100

Calculating present values 

The present value of the same amount PV = $110.25/1.052 = $100

If Ct = Cash flow at time t. The Present Value, PV =  Ct/(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)= $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 (C0) is usually negative as this is the investment at t=0.

Hence, NPV = C0 + Ct/(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 nth year can be calculated and then PV of the value till nth year can be calculated to arrive at a present value.

In case of growing perpetuity with growth rate g,

Where, C1 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 nth year.

Hence,

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

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