Now that we are familiar with the fundamentals of financial calculations and the valuations of such cash flows as perpetuaties and annuities, let's move on and talk about bond pricing. Bond pricing Is all about interest rates. That's why let's first consider the difference between spot and forward rates, which will be essential for our future calculations first. Let's get back to the timeline we considered in the previous lecture. Suppose I'm an investor who would like to invest his cash equal to $1,000 for two years. I have at least two possible investment opportunities. The first one is to invest my money for the full two year period with annual compounding at the interest rate 10% per annum. According to our previous calculations, we can declare that the future value of such an investment will be equal to $1,210. I have a different opportunity though. I can invest my money first for one year at the spot rate of 8%. And then reinvest them for one more consecutive year. This situation creates some uncertainty though because we cannot forecast how much the one year rate will be one year from now. So this is what makes the difference between those two types of investments. And this is the example I would like to use to make the difference between spot rates and forward rates. So let's move on to the definitions. A spot rate is the interest rate which is applicable to an immediate investment transaction. In our example, we have the two years spot rate of 10% per annum, and the one year spot rate of 8% per annum. This is the interest rate which is guaranteed and that you will be obtaining for sure. However, the rate which we don't know is called the forward rate which is the interest rate that is applicable to the financial transaction that will take place in the future, in our example, it's one year from now. So how should we determine a forward rate? In this case, we should refer to the non average principal we considered in the previous part. So let's question ourselves what makes us equally off between choosing those two investment opportunities. In this case, we can write down the equation you can see in front of you. And this is the only unknown which is the forward rate for one year, which starts one year from now. This equation implies that will be indeed indifferent between investing for the full two years or investing in one year. And then rolling over our investment for one more consecutive year if the forward rate will equal 12.04%. The problem is that it's not always possible to fix this rate for future. However, once we have a chance to, then, indeed, will be indifferent between those two investment opportunities. Once again, I would like to get back to the definition of a present value of a cash flow, which we also discussed last time, and make a very important common here. We'll be mostly focusing on the spot rates in this topic. And as we value cash flow, we can consider each part of this cash flow separately as a single payment, which implies that we should use appropriate spot rates as the discounting rate for each and every payment in this cash flow will be getting. Namely, we should use the one years pot rate as the discount rate for the cash flow expected one year from now. Then we should use two years pot rate as the discount rate for the cash flow expected two years from now and so on. In other words, we should use different discounting rates which correspond to different time periods. This is a very important thing that there is one more thing I would like to emphasize. We can call an n-year spot rate as the opportunity costs of investing your money for n-years, which can be also called the opportunity cost of capital. That's why we use it as the discounting rate. However, a very important thing is that the opportunity cost of capital is based on the economic approach to cash flow valuation. That's why it's important to note that this should be the return offered by an alternative investment with the same level of risk. In other words, we should use a risk-free interest rate in order to work out the present value of risk for investment. However, we should use the interest rates offered by alternative risky investments if we value some risky cash flows. Now that we know it, we can formulate the second underlying principle of asset valuation. As you remember, we have previously discussed that $1 received today is always more than $1 receive tomorrow because of investment opportunities. Here we should argue that a safe dollar is always worth more than the risky dollar. And the reason is the opportunity cost of capital. Safer investments have lower opportunity cost of capital. And as it's used as the discount rate, they are likely to have higher present values. On the contrary, risky investments should be offered a higher opportunity cost of capital to persuade investors to buy them. That's why they are expected to have the higher interest rates for discounting cash flows and consequently lower present value. At the end of this section, I would like to illustrate this idea with a numerical example. Suppose you have two investment opportunities. The first one is to receive $1,000 in one year for sure, which can be called a risk-free investment opportunity. Another one is getting $1,000 in one year with 95% certainty, which makes it a risky investment. Assuming the interest rate used for discounting our risk-free cash flows is 5%. We can use it as the opportunity cost of capital for our first investment. According to the present value formula, we can easily work out that it's present value should be $952.4. Then we can use the probability and the interest rate for the risk-free investment to work out the present value of our second investment opportunity. It can be worked out just in the same way, and it will be equal to $904.8. And now that we have identified the present value of the risky investment, we can use these formula backwards once again in order to identify the appropriate discount rate for the risky investment. According to this calculation, it will be equal to 10.53%, which is the appropriate opportunity cost of capital for an investment with such level of risk.