# Derivatives Basics – Time Value of Money

## Derivatives Basics – Time Value of Money

### posted By : Nauman Mir

This blog is the first in a series that introduces the theory and some of the fundamental concepts behind the valuation and risk management of financial derivatives. The concepts presented in these introductory blogs form the basis for the understanding of the material that will be present in the subsequent blogs on derivatives.

This blogs explains the time value of money and the concept of discounting future cash flows derived from the time value of money. But before we can look at time value of money and discounting, let us look into e, the universal growth constant.

# e – The universal growth constant

The universal growth constant e is derived from continuously compounding interest rates. With continuous compounding, any interest earned over a period of time is ‘re-invested’ at the given rate, allowing greater yield compared to non-compounding. For example, consider that we invest $1 at an annual rate of 100%. Without compounding, it would be$2 at the end of the year. Now lets consider the scenario with 6-month compounding period. After 6 months, $1 becomes$1.50. We then invest $1.50 at a rate of 100% for the next 6 months that gives us a total of$2.25 (original $1 plus the interest earned in green boxes below) which is 25 cents higher than the non-compounding scenario. In the above example, we had two compounding periods of 6 months each. The growth in the capital increases with the number of compounding periods. The formula for growth is: $$Growth = (OriginalAmount + \frac{InterestRate}{Periods})^{Periods}$$ As the compounding periods approach $$\infty$$, with an original amount of 1 and interest rate of 1 (100%), the growth is: $$(1 + \frac{1}{\infty})^{\infty} = 2.7182 = e$$. e is the universal constant that represents the infinitely compounding growth rate. The formula for growth using e is: $$Growth = Principle * e^{rate * time}$$ For example, if we have starting principle of$1000, and an annual compounding rate of 5%, then the growth after 6 months would be:

$$Growth = \1000 * e^{0.05 * 0.5} = \1025.32$$

Time value of money is a concept that is fundamental to the valuation of financial derivatives and is linked to the interest rates. For the purpose of explaining this concept, we will assume that the interest rates are always positive (although they can be negative as well and the theory presented here can easily be adjusted to cater for negative rates). Assuming that interest rates are positive, time value of money dictates that a dollar today is worth more than a dollar say, in a years time. Why? Because I can deposit the one dollar in a bank, that will earn interest over a year. If the interest rate is continuously compounding 5%, then the dollar will become $1.05127 (using the formula for growth from the previous section) in a year’s time. Alternatively, we can also say that$1.05127 in a year’s time is same as $1 of today. Why do we need to use the concept of time value of money? This is due to the fact that all financial derivatives involve payments (also known as cash flows) in the future and the dollar value of these payments drive the price of the derivatives. Since the payments are in the future, they need to be converted into today’s value (present value – PV) due to the time value of money. If we do not do that, the price of the derivative will not be the fair price. For example, a future contract stipulates the transfer of some asset from one party to another for a price in one year’s time. The fair price of the future contract is the spot price (the current price) of the asset plus the interest earned over a period of one year for the cash price of the asset. If the price of future contract was not fair, it would result in an arbitrage opportunity. For the purpose of this discussion, lets assume that the spot price of the asset is$100 and the current interest rate is 3%. The fair value of the future contract would be $1.03. Lets assume that it is$1.10. If this is the case, a party would get into a contract to sell the asset for $1.10 in a year’s time. The party would borrow$100 from the market at the rate of 3%, costing them $1.03 over the year. The party will then use the borrowed$100 to buy the asset at its spot price, sell it in a year’s time for $1.10 and make a risk free profit of 7 cents (sell price minus the cost). When such arbitrage situations exist, the players in the market exploit them until the contract prices return to their fair value. # Discounting Discounting is the mechanism that allows us to determine the present value for future cash flows. It is essentially the reverse of the growth formula introduced in the first section. To convert a future cash flow into today’s present value, we simply add a minus (-) sign in the growth formula. $$Growth = Principle * e^{-rate * time}$$ For example, if I have$1000 in a year’s time with a growth rate of 5%, then the present value (P.V.) of $1000 dollars from one year on would be: $$Growth = \1000 * e^{-0.05 * 1} = \951.23$$ This basically means that$1000 one year from now is equal to $951.23. This is because I can invest$951.23 at 5% annual rate to get to \$1000 in a year’s time. These simple formulas to covert future cash flows to today’s present value and vice-versa are fundamental to the pricing of derivatives.

## Summary

In this blog we have learnt:

I. The time value of money

II. Formula for continuously compounding growth ($$Principle * e^{rate * time}$$)

III. Formula for finding present values of future cash flows ($$Principle * e^{-rate * time}$$)