Learning outcomes, after completing this video you should be able to calculate the future value of a lump-sum amount given a present value. Calculate the present value of a lump-sum amount given the future value. Time value of money, in this video we introduce the concept of time value of money and present some related formulas. Time value of money is based on the idea that having $100 today is worth more than having $100 a year from today. This is easy to understand if you think in terms of having a bank account and are promised a certain amount of money in a year's time. Let's say that the bank promises you an interest rate of 5% over one year. If you deposit the $100 today, you will earn 100 times 0.05 which equals $5 in interest over the year. Add that to the $100 you invested, you will have $105 in your bank account after a year. Clearly having $100 today is better than having $100 in a years time. As $100 invested today becomes $105 in a year's time. The $105 is referred to as the future value of the $100 after one year. Alternatively, we can say that the $100 is the present value of the $105. We can ask the investment question a little differently. If you want $100 in your bank account after one year and the bank is paying an interest of 5% per year, how much should you invest today? For now I'll give you the answer, it is $95.24. We'll get to the actual calculation later. If you invest $95.24 today and the bank pays you 5% interest, you will earn 95.24 x 0.05, which equals $4.76 in interest. That combined with your investment of $94.24 will give you $95.24 plus $4.76, which equals $100. Here again, we say that the present value of $100 is $95.24. Or conversely, the future value or $95.24 after one year is $100. Time value of money is one of the important concepts in finance. Almost all calculations in finance are based on it. It is also referred to as the discounted cash flow, in short, DCF methodology. Because we are discounting a future cash flow to the present using an interest rate. In our example, we discounted the $100 at 5% a year back to today which yielded a present value of $95.24. What if the bank offers you a higher interest rate of 10% a year? How much must you invest today to have $100 after a year? The answer is $90.91, an interest rate of 10% will yield you $90.91 x 0.1 which equals $9.09 in interest. Which when added to your initial investment of $90.91 will give you $90.91 plus $9.09, which equals $100. As you can see, the present value is lower when the interest rate is higher. This makes sense, because you're earning more through interest, and hence you can invest a smaller amount today. Next, we need a formula to calculate the present value, given a future value, or vice versa. Notice, I said that the present value of $100, when interest rate is 10% is $90.91, but I didn't tell you how I arrived at $90.91. Let's revisit the equation we wrote out earlier, $90.91 plus $9.09 equals $100. The $90.91 is the present value $9.09 is the interest earned, and $100 is the future value. Let's denote the present value as PV, future value as FV, and interest rate as r. Interest earned, 9.09, is nothing but 90.91 times 0.1 which is nothing but PV times r. So we can rewrite the first equation as follows, 90.91 + 9.09 = 100. PV + PV times r = FV. And then we can simplify that mathematically by writing PV times one plus r within parenthesis equals FV, which is for a one year investment. What if you make a two year investment? You invest $90.91 for two years, at 10% a year. After one year you will now $100 as we've already seen. That $100 will continue to be in the bank account and earn an additional 10% in the second year. The interest earned in the second year is now 100 times 0.1, which equals $10. Add that to the investment of $100 at the start of the second year, and your bank account will have $110 after two years. Let's distinguish the future value after year one and after year two, by introducing a subscript to FV. FV subscript one is the future value after the first year and FV subscript two is the future value after the second year. If you want to generalize this, we write 100 plus 10 is equal to 110. 100 is nothing but FV subscript 1 plus ten is nothing but FV subscript 1 times r which is equal to FV subscript 2. FV subscript one times (1 + r) = FV subscript 2. What we already know that FV subscript 1 = PV x (1 + r). So now, we have PV x (1 + r) which is substituted for FV sub 1 x (1 + r) = FV subscript 2. Or, alternatively, PV times (1 + r) the whole squared is equal to FV subscript 2. This can further be generalized for an annual investment as FV subscript n = PV x (1 + r) the whole thing raised to the power of n. On the other hand, if you want to calculate the present value given a future value, the formula is PV = FV subscript n / (1 + r) the whole thing raised to the power of n. In this video we looked at calculating the future value given a present value and vice versa for a single amount or lump sum. What if we have periodic identical cash flows? How do we calculate the present and future values of this stream of cash flows? We will talk about this next time