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Interpolation of the Yield Curve U.U.D.M. Project Report 2020:49 Interpolation of the yield curve Gustaf Dehlbom Examensarbete i matematik, 15 hp Handledare: Maciej Klimek Examinator: Veronica Crispin Quinonez September 2020 Department of Mathematics Uppsala University Abstract In this thesis we study the yield curve and the properties of it. The thesis gives an introduction to what the yield curve is, the properties of it and the important financial concepts related to it. Furthermore, the monotone convex method is presented, a method specifically developed for finance. In particular how it works, what mathematical properties it focuses on and how it is used to construct the yield curve. 2 Contents 1 Introduction 5 2 Background 6 2.1 Bonds . .6 2.2 Yield to maturity . .7 2.3 Yield curve . .7 2.3.1 Normal . .9 2.3.2 Inverted . 10 2.3.3 Flat . 11 2.4 Spot rate . 11 2.4.1 Zero-coupon rate . 12 2.5 Discount factors . 12 2.6 Forward rates . 13 2.6.1 Useful properties of forward rates . 14 2.7 Bootstrapping . 14 3 The Monotone Convex Method 17 3.1 Interpolation of forward rates . 17 3.1.1 Calculating f0 ....................... 18 3.1.2 Calculating fi ....................... 19 3.1.3 Calculating fn ....................... 20 3.2 Ensuring monotonicity . 22 3.2.1 Sector (i) . 25 3.2.2 Sector (ii) . 26 3.2.3 Sector (iii) . 27 3.2.4 Sector (iv) . 28 3.3 Modifying g . 28 3.3.1 Modifications of (ii) . 28 3.3.2 Modification of (iii) . 30 3.3.3 Modification of (iv) . 31 3.4 Ensuring positivity . 32 3.4.1 Collaring f ........................ 34 3.5 Summary . 35 3 4 Example 35 4.1 Bootstrapping our yield curve . 36 4.2 Enforcing monotonicity . 37 4.3 Defining the interpolation function . 39 4.4 Recovering the risk free rate function . 39 5 Conclusions 40 4 1 Introduction The yield curve is the term structure of interest rates and is defined as the relationship between the maturity of a zero-coupon bond and the yield to ma- turity. This plot has a wide range of uses within the financial market where the main purpose consists of valuating various bonds, something that will de- scribe later in this thesis. When constructing the yield curve we begin with a set of discrete data points or financial instruments which we are looking to interpolate. This can however be done in many different ways, especially in regard to what financial instrument we use [10]. It is therefore interesting to look at one of the popular methods and see what makes it beneficial and how. Most of the popular interpolation algorithms are not developed for finance and this can cause problems. If we for example plot a yield curve using a linear interpolation to compare different bonds it can be problematic. It does not necessarily mean that it is a bad method, it would however be a poor representation of the financial market because of how dynamic and stochastic it is [8]. In addition it could create an arbitrage opportunity. An arbitrage opportunity is a type of risk-free investment that can occur if e.g. there is a temporary price difference between similar financial instru- ments. It means that an arbitrageur can make profits without having to risk his or her own money. An illustrative example of this would be if the govern- ment is purchasing gold at the price x whilst a mining company is selling at the price y where y < x. An arbitrageur can then simultaneously purchase at a price y and sell at the price x for an immediate profit of x−y. Arbitrage can happen in various ways, especially when constructing new data points, which is why a good method for constructing a yield curve should prevent this. Since the yield curve can be constructed with different algorithms, with many of them being intended for engineering or physics [4], it is interesting to look at a popular method developed for the financial market. This article will study the monotone convex method for constructing the yield curve. 5 2 Background 2.1 Bonds A bond is a type of obligation usually issued by the government or large corporations. The basic principle of a bond is that an investor lends money to e.g. one of the above mentioned. The bond has a face value F which is the amount of money the bond pays at the date of maturity. A bond usually has coupon payments C which is annual payments to the investor consisting of a percentage of the face value. The coupon payments can also be divided up in coupon periods m during the year which then would pay C out m at each m in time. If we for example look at a 10% coupon bond with a face value of 500$. The coupon bond will yield the value of 50$ per year [8]. A bond without coupons is called a zero-coupon bond. The difference com- pared to a regular coupon bond is that the zero-coupon bond has no periodic coupon payments which means that the investor solely buys the face value of the bond. Bonds are categorized into different ratings which shows the risk of the investment. In USA there are primarily two companies that post these ratings, Moody’s and Standard & Poor’s. The ratings is classified as: Grade Standard & Poor’s Moody’s High grade AAA Aaa AA Aa Medium grade A A BBB Baa Speculative grade BB Ba B B Default danger CCC Caa CC Ca C C D Table 1: Bond rating classification [8, p. 52] As seen in the table above the bonds rates are quite similar in their notations 6 from triple-A (AAA) to C or D. The grades provide a reflection of the in- vestment security and cost of bonds. Higher rated bonds will be more secure to invest in but will also have lower yield. David Luenberger (1998) denotes the high and medium grades as investment grade and the speculative grades and lower as junk bonds. From now on this article will use the notations of Standard’s and Poor’s. [8] 2.2 Yield to maturity When investing in bonds, yield to maturity (YTM) is used to calculate the rate of return of the bonds current price P [8, p. 52]. As previously mentioned a bond can either consist of coupons being paid annually over m periods or with the face value as the only cash flow (zero-coupon). Regardless of what bond it is, the interest on all of the cash flows should be equal to the bonds current price. We can look at this as a compounded interest. If we purchase a 10–year coupon bond and hold it until maturity. The simple interest of this would be to add all the stream of payments and divide it with the current price. This will however not account for the embedded interest rates from the cash flows, or perhaps more commonly known as interest on interest. Thus when calculating the YTM, it is assumed that each cash flow is reinvested at the current interest rate. The following formula can be used to calculate the YTM from a bond with face value F , coupon payments C, m number of coupon payments and n remaining payments: F C 1 ! P − = YTM n + 1 YTM n (1) [1 + m ] YTM [1 + m ] 2.3 Yield curve A yield curve is a plot depicting yield as a function of time to maturity. When looking at the yield curve for the highest rated zero-coupon bonds it is denoted as the term structure of interest rates. The most important usage for the yield curve is to evaluate the bond market. If we revisit table 1 and consider an AAA-rated bond. There exists a yield curve associated 7 with this bonds quality which in turn can be used to evaluate and compare it to other AAA-rated bonds [8]. For instance if you find a 30 year bond on the secondary market that has F = 1000$ and C = 2%. Since the bond was issued the interest rates has increased and the market now offers similar 30-year bonds but with C = 3%. It would thus be valued at a lower price compared to the similar bonds at present date. It is not only the present market investors want to evaluate. The yield curve also gives a good indication of what will happen in the future. As later will be explained the yield curve can have different shapes which indicates what the future market will look like [3]. An example of this is predicting future recessions. There is a clear correlation between the shape of the curve and recessions, which can be shown empirically. For instance if a government wants to slow down the economy because they anticipate a recession they will flatten the yield curve [1]. A good example of this would be to look at the yield curves issued by the US Treasury before, during and after the financial crisis 2008. In the next three subsections, the three most common shapes of the yield curve will be outlined. There are three types of shapes generally discussed when analyzing a yield curve, normal, inverted or flat. 8 2.3.1 Normal The normal shape is a gradually rising curve and the most common of the three. It can be summarized with the short term rates being lower and long term rates higher. It is typically a sign of a growing economy.
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