In this first part I will introduce the various notions of interest rates and related contracts. Starting from the basic definition of interest paid on a loan we derive simple and continuously compounded rates, also called LIBOR, and yield. They depend on the length of the lending period and at the short hand coincide and are equal to the so called short rate. Investing and continuously reinvesting at the short rate gives us the money market account and we will see that this is a risk free asset. In turn, the zero coupon bond is the securitized form of a loan. And they can be used to discount future cash flows which is why they are also called discount bonds. Interest refers to the rent paid by a borrower of money to a lender of money over a period of time. Here we have a period of time. It starts at calendar date t, and it ends at the maturity date T. The lender will pay a loan of 1 unit of currency to the borrower. And the borrower in turn will pay back this principal amount plus interest at the maturity T. Interest is expressed in annualized interest rates. With simple compounding, we arrive at a simple rate that we call L. Which is defined by this equation here which simply states that L is the interest R divided by the length of the lending period. There's also a continuously compounded version which is also called the yield and we label it by the letter y. And is defined by this equation here. Here is a market example, the LIBOR. LIBOR stands for London InterBank Offered Rate. This is the rate at which high credit financial institutions can borrow on the London InterBank money market. This rate however has become an important reference rate for many interest rate contracts beyond the London InterBank money market. It is quoted daily as a simple rate and this is why I choose the letter L to denote the simple rate. And it is quoted for various maturities ranging from 1 day to 1 year, and it is quoted for various currencies. Interest rates depend on the lending period this is why we also use the argument t and T when we notate for simple and continuously compounded rates. If we freeze the current calendar date t and we vary the maturity T. We obtain what is called the term structure of interest rates prevailing at t. We see the curve here in black the simple interest rate. And in red, we see the term structure of yields which we also call the yield curve. It is a mathematical fact that for the same interest, the simple rate is larger than the yield. While at the short end both curves coincide and we will see later this is what we call the short rate. If we vary the calendar date t we obtain a time series of terms structures that is plotted in this graph overhear. So what we saw on the previous slide was a snapshot for a given calendar date t, we saw the term structure of interest rates. It is clear as calendar time proceeds, economic conditions change and this term structure will fluctuate. As you can see from this plot, which is taken from empirical data. These fluctuations of the term structures are manifold. Modeling this time series, is the aim of this course on interest rate models. Such models are useful then for financial risk management and pricing interest rate derivatives in a consistent and arbitrage free way. Here I come back to what we saw on the previous slide with term structure of interest rate and the yield curve. They both coincide at the short end. This common short end is called the short rate. The short rate is the rate earned on a loan over the short period [ t, t + dt ]. Investing 1 unit of currency at the short rate, and continuously reinvesting it at the short rate gives the money market account. The return on the money market account over the period B(t + dt) is given by 1 + r(t) dt. with the initial normalization, this equation is equivalent to this expression here. So what we see on the plot on the right hand side is, for a given trajectory of the short rate evolving over time. We have it on that scale here. The red line shows us how the money market evolves. At each time the rate of return on this money market account is given by the respective short rate. The money market account is a risk free asset. Risk free in the sense that, it's return over a time interval is known at the beginning of the time interval. This is very much in difference to risky assets where the return is only known at the end of the time period. And where mathematically we would have to add a noise term here. A zero coupon bond with maturity T is a contract that pays its holder 1 unit of currency at the maturity. This zero coupon bond with maturity T for short is also called a T-bond. Its price at time t is denoted by P(T, t). A zero coupon bond is the securitized that is, tradable form of a loan. It is related to a loan in the following sense. At t, the lender buys a zero coupon bond from the borrower. and pays the price P. At the maturity T, the lender receives 1 from the borrower. Buying a T-bond at time t and holding it until maturity generates a simple rate of return which is given by the simple rate L. Formally speaking, the zero coupon bond price is related to the simple rate by this equation. Buying a T-bond and holding it until maturity generates a logarithmic rate of return which is given by the yield. Again this is expressed by this relation down here. A zero coupon bond. is also called a discount bond. P is the price at calendar time t. over cash flow of 1 at maturity T. Now for varying maturities T we obtain the term structure of zero coupon bond prices prevailing at t. This is also called the discount curve. This discount curve is typically decreasing and smooth. It reflects a positive time value of money. At the short end. The discount curve equals 1 because the cash flow of 1 at calendar date t has value 1 at T. In reality, interest rates depend on various factors. They can depend on the credit worthiness of the borrower. The less credit worthy the borrower, the more the lender will charge for a loan. They also depend on the liquidity needs of the lender. The more liquidity the lender needs, the more the lender will charge for a loan. Interest rates also depend on regulatory requirements and the monetary policy by the central banks. And interest rates also depend on the market micro structure. Various interest rate contracts are traded on various markets. And these markets are not always perfectly integrated. In theory however, for this course we will assume that there is no credit risk; which means buying a bond will give the holder of the bond the face value at the end of the maturity. And we also assume that there exists a frictionless market. where all T-bonds for all maturities are traded.
