[MUSIC] Learning outcomes. After watching this video, you will be able to calculate the future value of a lump-sum amount give a present value, calculate the present value of a lump-sum amount given the future value. 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 say that the bank promises you an interest 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 year's time, as the $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 a present value of the $105. We can ask the investment question a little differently. If you want $100 in your bank account after 1 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 times 0.05 which is equal $4.76 in interest. That combine with their initial investment of $95.24, 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 of $95.24 after one year is $100. Time value of money is one of the most important concepts of finance. Almost all calculations in finance are based on it. It is also referred to as the discounted cash flow methodology, in short DCF, because we are discounting a future cash flow to the present using a discount 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 give you $90.91 times 0.01, which equals $90.91 in interest. Which when added to your initial investment of $90.91 will give you 90.91 plus 9.09 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 that 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 that $90.91. Let's be with the equation we wrote earlier, 90.91 plus 9.09 equals $100 $90.91 is the present value. $9.09 is the interest earned, and $100 is the future value. Let's denote present value as PV, future value as FV, and interest rate as r. Interest earned, 9.09, is equal to 90.91 times 0.10, which is equal to PV times r. So we can rewrite a first equation as as follows, 90.91 + 9.09 = 100 in other words PV + PV*r= FV, and we can simplify that to PV ( l + r ) = FV which is borrow one 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 have $100. The $100 will continue to be in the bank account and own an additional 10% in the second year. The interest on in the second year is 100 times 0.10 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 sub 1 is the future value after the first year, and FV sub 2 is the future value after the second year. If you want to generalize this, we can write 100 + 10 =110. FV sub 1 + FV sub 1 time r = FV sub 2. FV sub 1 times 1 plus r equals FV sub 2. But we already know FV sub 1 equals PV times 1 plus r. So we now have PV times 1 plus r times an additional 1 plus r, which equals FV sub 2. Or, we can simplify that to PV times 1 plus R, the whole square equals FV sub 2. This can further be generalized for a n year investment as FV sub n equals PV times 1 plus r to the whole power of n. On the other hand if you want to calculate the present value given a future value, the formula is PV equals FV sub n divided by 1 plus r the whole raise to the power of n. In this video, we look at calculating the future value given the 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. [MUSIC]