# Discount

In finance and economics, discounting is the process of finding the present value of an amount of cash at some future date, and along with compounding cash forms the basis of time value of money calculations. The discounted value of a cash flow is determined by reducing its value by the appropriate discount rate for each unit of time between the time when the cashflow is to be valued to the time of the cash flow. Most often the discount rate is expressed as an annual rate.

To calculate the net present value of a single cash flow, it is divided by one plus the interest rate for each period of time that will pass. This is expressed mathematically as raising the divisor to the power of the number of units of time.

As an example, suppose an individual wants to find the net present value of $100 that will be received in five years time. There is a question of how much is it worth presently, and what amount of money, if one lets it grow at the discount rate, would equal$100 in five years.

Let one assume a 12% per year discount rate.

NPV = 100 dollars divided by 1 plus 12% (0.12) divided by 1 plus 12% (0.12), etc.

$\rm NPV}=\frac{100}{(1+0.12)^5$

Since 1.125 is about 1.762, the net present value is about \$56.74.

## Discount rate

The discount rate which is used in financial calculations is usually chosen to be equal to the cost of capital. Some adjustment may be made to the discount rate to take account of risks associated with uncertain cashflows, with other developments.

The discount rates typically applied to different types of companies show significant differences:

• Startups seeking money: 50 – 100 %
• Early Startups: 40 – 60 %
• Late Startups: 30 – 50%
• Mature Companies: 10 – 25%

Reason for high discount rates for startups:

• Reduced marketability of ownerships because stocks are not traded publicly
• Limited number of investors willing to invest
• Startups face high risks
• Over optimistic forecasts by enthusiastic founders.

One method that looks into a correct discount rate is the capital asset pricing model. This model takes in account three variables that make up the discount rate:

1. Risk Free Rate: The percentage of return generated by investing in risk free securities such as government bonds.

2. Beta: The measurement of how a company’s stock price reacts to a change in the market. A beta higher than 1 means that a change in share price is more exaggerated then rest of shares in the same market. A beta less than 1 means that the share is stable and not very responsive to changes in the market. Less than 0 means that a share is moving in the opposite of the market change.

3. Equity Market Risk Premium: The return on investment that investors require above the risk free rate.

Discount rate= risk free rate + beta*(equity market risk premium)

## Discount factor

The discount factor, P(T), is the number by which a future cash flow to be received at time T must be multiplied in order to obtain the current present value. Thus for a fixed annually compounded discount rate r we have

$1}{(1+r)^T$

For fixed continuously compounded discount rate we have

$-rT$