Discounting

Discounting means converting money in the future to its equivalent value today, which helps translate money from different dates to the same "today" basis. It's an important idea in finance because it helps compare money fairly.

DCA model uses discounting to compare cash flows occurring at different times, so decisions about value can be made by having a single present value number.

Options pricing uses discounting to convert an option's expected payoff at maturity into today's price, so traders can determine whether the option is priced fairly. Because whatever you receive in the future must be "brought back" to today at an appropriate date.

Intuitive example

Because a dollar today can earn interest, having $1,000 now is worth more than receiving$ 1,000 a year from now. Discounting captures the difference by applying an interest rate to translate future amounts into today's dollars.

If you invest $PV$ today at rate $r$ , it grows to $K$ by time $T$ . So discounting is just the reverse of that growth.

Compounding = $PV \rightarrow PV e^{rT}$
Discounting = $K \rightarrow Ke^{-rT}$

Say the interest rate ( $r$ ) is 5%. The $1,000 ( $PV$ ) we have now will grow and become $1,051.27 ( $K$ ) in a year ( $T$ ). And the $1,000 we will receive a year from now is actually$ 950.48.

The reason we're using $e$ is because in finance, we treat growth over time smooth and compounding continuously.

Pure discount bond

A pure discount bond is also known as a zero-coupon bond.

Pays no periodic coupons, ie. no interest payments along the way
Pays the face value at maturity Example:
Pay $950 today to buy a bond with face value at$ 1,000.
That $50 is the interest instead of paying off periodically