This is going to be a boring but very important lesson. When investing, knowing the actual current value of a financial instrument is important. This current value, known as the present value, is less than the value of the cash returned at a future date because of the time value of money. Money in your hand right now is worth more than the same amount in the future because you can invest and grow the value of the money; this is the time value of money. Anyways, to calculate the present value of any future cash inflow, you need to know the relationship between the present value; the future value, which is the monetary value you will receive; and the periodic interest rate.

The Equation

To derive the equation for present value, first we must look at the equation for the total yield of an investment with a known fixed interest for a certain number of years, which is shown below. This equation is one of the most important; it shows how much money you can expect to make from a simple investment like money stored in fixed rate bank account. In this case, we can substitute the principal investment (P) for the present value because the principal investment is the actual money you put down when you make the investment. Then, we can substitute the total return (A) for future value because that is the value of the investment after the investment has reached maturation, or has been cashed out. Then we can solve for present value to get the final equation.

A = P * (1 + r)n

A = total return
P = principal investment
r = interest rate
n = number of years

future value = present value * (1 + r)n

present value = future value / (1 + r)n

Some Calculations

Imagine if your friend proposes to pay you $1,200 in five years if you loan him $1,000. If you want to earn a 3% interest rate, should you accept this deal? Well, let’s plug it in to the equation; your friend is willing to pay you $1,200 in five years, so the future value is $1,200 and the n is equal to 5. Then if you plug 0.03 (3% interest) in for r and solve, you find the present value of the investment is $1,035.13, or more than you have to pay. Therefore this is a good deal for you.

present value = future value / (1 + r)n = $1,200 * (1 + 0.03)5 = $1,035.13

However, say the economy is in great shape and other investments will net you a 5% interest rate. Then is this still a good deal? If you plug in 0.05 for r instead of 0.03, the present value comes out to be $940.23, making the deal unfavorable to you. Thus as interest rates rise and future value stays constant, the present value falls. The present value and interest rates are indirectly proportional. What if your friend offers to pay you back the same amount in 10 years, instead of five, and you wish for 3% interest on your investment? Plugging those numbers into the equation spits out a present value of $892.91. Thus the number of years until maturation and the present value are also indirectly related.

present value = future value * (1 + r)n = $1,200 * (1 + 0.05)5 = $940.23

present value = future value * (1 + r)n = $1,200 * (1 + 0.03)10 = $892.91

Conclusion

This might not have been the most flashy or interesting of lessons, but it introduces an important equation in calculating the actual value of future cash. This is not only important when deciding what investments to make but also demonstrates opportunity costs. The reason that money in future is not worth as much as money now is because if we had the money, we could use it to make more. Therefore the lost opportunity of not having the money and not being able to use it is a real and calculable cost.

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