[SOUND] Hello, today we're going to consider the time value of money. And in this video we will discover the basic notions the present value, future value and interest rate. Our first idea is that value of money is different. If you choose between two options, $10 in your pocket or $10 in the bank vault in Switzerland. Most people prefer $10 in cash. Well it is more liquid. Now situation changed and you have $10 million. Would your answer change? Now, bank looks better, at least more safe. So conclusion, the value of money is different and depends on the location of that money. Here value is not some mathematical thing but something expressing preferences. Preferences can also depend on time. Most people prefer $1 today to $1 next year, but $10 today are usually better than $1 today and there exists some amount X Which is equivalent to $1 today in terms of preferences. Moreover, the value of money could depend on the person. For Alex, $1 today is the same as $6 next year. For Boris $100, today is equivalent to $105 next year. And for Christian $95 today is equivalent to $100 next year. Now let's be a little bit more formal. Considerate timeline and the time moment 0, which is today amount of money at the time moment 0 is called present value. For Alex for example present value was $1. Now consider time moment 1, in our example it was next year. Amount of money that gives us the same happiness is called future value. So in short present value dollars today is equivalent to future value dollars in the future for example next year in terms of preferences. In our example for Alex present value is 1$ and future value is $6. For Boris, Present value is $100 and future value is $105. And for Christian, present value is $95 and future value is $100. Connection between present value and future value can be expressed with a concept called interest rate. Here is the formula interest rate is the ratio of future value of a present value minus 1. In our example for Alex interest rate was 500%, you can check yourself how this was found and we are going to find the interest rates for Boris and Christian. Let's start with Boris at time moment 0, $100 have the same board value for Boris as $105 a time moment 1. So here we have the data between shortly and we need to find the interest rate. We have a nice formula for the interest rate future value over present value -1. And after a substitution numbers we have $105 over $100 minus 1 which is 1.05 minus 1 and equals to 5%. So interest rate for Boris is 5%. And now let's go to Christian. So now for Christian $95 a time of 0 have the same value as $100 in time moment 1. So here we are again with a short data and we will use the same formula for interest rate. Future value of a present value minus 1. So we can simplify it a little bit by getting to the common denominator and here it is future value minus present value. So it is the difference in the values and here it is the initial time value of money, present value. So when we substitute the numbers, we will have $100 minus $95 over $95, which is 5 over 95, which is approximately 0.0526. So 5.26% this is interest rate for Christian. If we took the same route for Boris it would be 100 here in the denominator. And now let's go back to the slides. We can see that for Alex, Boris and Christian future value is $5 higher than present value, but returns are all different. This happens because for Alex return is based on $1 present value for Boris, on $100 present value, and for Christian on $95 present value. So for Alex it is 500%. For Boris it is 5% and for Christians 5.26%. Again return is measured in persons not in dollars, it is a relative thing not absolutely. Now let's look at a useful expressions of future value and present value. Future value could be found as present value times 1 plus interest rate and present value couldn't be found as future value over 1 plus interest rate. Let's consider a couple examples, in our first example here we need to find 1 period future value of $250 if interest rate is 7%. So here we are with the data 250 is present value 7% interest rates 0.07. We need to find the future value. So let's do it, we have the formula for the interest rate, the definition of interest rate, future value of the present Value minus 1. And we can express future value from this formula so how to do this. First of all we can find the value of future value to present value as interest rate plus 1. And then we can multiply the whole fraction by present value. Okay so future value, it's going to be equal to Present value 1 plus interest rate. And we already saw this formula in the slides. So let's substitute the numbers. The future value is $250 times 1 plus 0.07. Yeah which is equal to $267.50. So thanks to the formulas we know that $250 today equivalent to $267 next year, for the person with interest rates 7%. In our next example here we need to find the present value of money. So here we have a plan, we plan to pay $900 for the CFA Level 1 exam for example next month. And we need to find the present value of this payment if monthly interest rate is 2%. So the data future value is $900, interest rate 0.02 and present value is unknown. From the previous example, we know that future value could be found as present value, times 1 plus interest rate. And if we divide this equation by 1 plus interest rate, we can find the value of the present value. So present value is future value over 1 plus interest rate. So once again we already saw this formula in the slides. So let's substitute the numbers. Present value is future value, $900 over 1 plus interest rate 1.02. So it is going to be approximately $882.35. So this is the present value for $900, we need to pay 1 month from now. If interest rate is 2%,, we considered formal issues concerning interest rate. Now let's discover what we really mean when we use this concept. Interest rates can be interpreted in three different ways. The first 1 is discount rate, discount rate is about equivalent in terms of preferences. The second 1 required rate of return. It is about equivalent in terms of investing goals and the third 1 is opportunity cost, opportunity cost is about equivalent. In terms of alternative strategies, interest rate can be decomposed into five parts. Each part explains a certain reason why future value some investment might be higher all over and the first part is real risk free rate. It is theoretical return with no risk. In practice of, the closest thing is US Treasury bonds. Inflation premium is another part of interest rate. It shows the fact that with higher inflation the same present value is equivalent to higher future value. Another part of interest rate is default risk premium. It captures the fact that if a project has higher risk of default, the future value of investment should be also higher. Liquidity premium shows that if assets is hard to convert into cash, then it's future value should be quite high to be equivalent to cash today. And the last part, this maturity premium majority premium shows that if we need to wait longer or for our investment return future value of this investment should be quite high.