Introduction to Financial Markets Unit 2 PDF

Introduction to Financial Markets Unit 2 Fixed Income Markets Prof. Dr. M. De Ceuster Prof. Dr. M. De Ceuster Introduction to Financial Markets 1 / 39 Financing Sources Equity instruments D...

Introduction to Financial Markets Unit 2 Fixed Income Markets Prof. Dr. M. De Ceuster Prof. Dr. M. De Ceuster Introduction to Financial Markets 1 / 39 Financing Sources Equity instruments Debt instruments Loans Debt securities Loans and debt securities are known as “fixed income instruments”. Unit 2 and 3 will focus on these kind of instruments. Prof. Dr. M. De Ceuster Introduction to Financial Markets 2 / 39 Financial History Debt instruments are the oldest instruments in the world: IOU goes back to the time of the hunters-gatherers. The codex of Hammurabi (1780 BC) provided amongst others in loan concepts (including interest charges) and insurance/risk sharing contracts. Prof. Dr. M. De Ceuster Introduction to Financial Markets 3 / 39 Financial History 16th century loan book Prof. Dr. M. De Ceuster Introduction to Financial Markets 4 / 39  Interest Rates Section 1 Interest Rates Prof. Dr. M. De Ceuster Introduction to Financial Markets 5 / 39  Interest Rates Interest In a world with positive interest rates, you will have to pay back more than the amount you borrowed. The rate of interest is the price of money. the price to ‘rent’ money the reward for the lender to postpone consumption Example: You can borrow 100K at 2.34% per annum for the next year. Alternatively you can state that your borrow at 234 basis points per year. This means you receive 100K now and that you will have to repay 2340 next year and 102340 within a year’s time. Prof. Dr. M. De Ceuster Introduction to Financial Markets 6 / 39  Interest Rates Interest The interest rate charged depends on the risks taken by the lender. This risks depend (at least) on the credit worthiness of the borrower. maturity of the debt. Hence, we cannot speak about the interest rate. There will be a different interest rate per maturity and per borrower. Prof. Dr. M. De Ceuster Introduction to Financial Markets 7 / 39  Interest Rates Term Structure of Interest Rates What kind of graph do we get if we plot the “interest rate” that would be charged vis-a-vis a certain counterparty for maturities ranging from 1 day to 30 years? Notice this is a graph on a particular day, keeping all the loan charac- teristics constant except for the maturity of the loan. What would you get for a different, less credit worthy borrower? This graph is named the term structure of interest rates. Often people talk about a yield curve but this term as such is less precise as we will explain. Prof. Dr. M. De Ceuster Introduction to Financial Markets 8 / 39  Interest Rates Term Structure of Interest Rates (aka yield curve) Prof. Dr. M. De Ceuster Introduction to Financial Markets 9 / 39  Interest Rates Term Structure of Interest Rates Prof. Dr. M. De Ceuster Introduction to Financial Markets 10 / 39  Interest Rates Term Structure of Interest Rates Shapes Prof. Dr. M. De Ceuster Introduction to Financial Markets 11 / 39  Interest Rates Decomposition of an Interest Rate Demand and supply for loanable funds determine a short term risk free interest rate. This interest rate rewards the delay in consumption. The lender faces risks for which a compensation (premium) is required. 1 A longer maturity of the debt instrument delays consumption more than a shorter maturity and hence the lender will ask for a maturity premium. This gives rise to a term structure of the real riskless interest rates. 2 There is no guarantee that the purchasing power of the funds repaid in the future will be the same as the purchasing power of the funds lent out. Therefore the lender adds a(n expected inflation) premium to each risk free rate to obtain the term structure of nominal risk free interest rate. Of course the expected inflation premium can be different for different maturities. 3 The lender will charge the borrowers with a credit spread for expected credit losses. Often this credit spread also embeds a liquidity premium. Of course the credit spread can be different for different maturities and for different borrower qualities. This results in a term structure of nominal rates for risky assets. Prof. Dr. M. De Ceuster Introduction to Financial Markets 12 / 39  Interest Rates Decomposition Other factors that we conveniently ignored. Special contractual provisions (e.g. seniority, embedded options,...) Collateral arrangements Differential tax treatment ··· Prof. Dr. M. De Ceuster Introduction to Financial Markets 13 / 39  Interest Rates Decomposition Irvin Fisher stated the often used relationship between nominal and real interest rates. (1 + rnominal ) = (1 + rreal )(1 + fi e ) or rnominal ¥ rreal + fi e Remarks: fie denotes the expected inflation, r stands for the interest rate. Most of the time, this Fisher equation is used to determine the real interest rate, given the observed nominal interest rates and an estimate of the expected inflation. P rof. D r. M. D e Ceuster Introduction to Financial M arkets 14 / 39  Interest Rates Time Variation Source: https://ftalphaville.ft.com/2017/10/20/2195061/harking-back-to-prestiti-stock-eight-centuries-of-the-risk-free-rate/ P rof. D r. M. D e Ceuster Introduction to Financial M arkets 15 / 39  Time Value of Money Section 2 Time Value of Money Prof. Dr. M. De Ceuster Introduction to Financial Markets 16 / 39  Time Value of Money Preliminary Remark In order to understand fixed income instruments, we need to understand the time value of money. We introduce both simple interest rates compounded interest rates since both of them are used in practice. Prof. Dr. M. De Ceuster Introduction to Financial Markets 17 / 39  Time Value of Money Timeline A timeline is a linear representation of the timing of potential cash flows. Inflows are positive cash flows. Outflows are negative cash flows. Example: Assume that you are lending 10K today and that the loan will be repaid in two annual 6K payments. Prof. Dr. M. De Ceuster Introduction to Financial Markets 18 / 39  Time Value of Money Single Cash Flow Compounding (simple interest rate) Suppose that we invest 100 today at an interest rate of 10% p.a. (i.e. per annum). What will be the future value of our investment within 1 year? 100 ◊ (1 + 10%) = 110 What will be the future value of our investment within 6 months time? 6 100 ◊ (1 + 10%◊ ) = 105 12 100 ◊ (1 + 10%◊ 0.5) = 105 Prof. Dr. M. De Ceuster Introduction to Financial Markets 19 / 39  Time Value of Money Single Cash Flow Compounding (simple interest rate) In general we get V T = V 0 ◊ (1 + r ◊ T ) Remarks: V 0 represents the present value, V T refers to the future value at time T. Both time and the interest rate are measured in years. Prof. Dr. M. De Ceuster Introduction to Financial Markets 20 / 39  Time Value of Money Single Cash Flow Discounting (simple interest rate) Let’s reverse the question. Suppose you receive 105 within 6 months and on the market the interest rate is 10% p.a. What is the present value of this future cash flow? We simply rewrite our equation V T = V 0 ◊ (1 + r ◊ T ) as VT V0 = (1 + r ◊ T ) Hence, 105 = 100 (1 + 10%◊ 0.5) Prof. Dr. M. De Ceuster Introduction to Financial Markets 21 / 39  Time Value of Money Single Cash Flow Annual Compounding Suppose you deposit 1,000 for one year at a rate of 10% p.a. How much will it amount to in one year? V 1 = V 0 ◊ (1 + r ◊ 1) = 1 000 ◊ (1 + 10%) = 1 100. What happens if you leave the money in the account for another year? V 2 = V 1 ◊ (1 + r) = 1 100 ◊ (1.1) = 1 210 You earned an EXTRA 10 in year 2 with compound over simple interest. Prof. Dr. M. De Ceuster Introduction to Financial Markets 22 / 39  Time Value of Money Single Cash Flow Annual Compounding In general we get V T = V 0 ◊ (1 + r)T Graphically, we see the exponential nature of compounded interest rates arise. Prof. Dr. M. De Ceuster Introduction to Financial Markets 23 / 39  Time Value of Money Single Cash Flow Discounting (with Annual Compounding) To determine the present value of a future cash flow, we compute VT V0 = (1 + r)T Consequently, the present value of 1 210 which we receive within 2 years equals 1 000 if the interest rate on a two year investment equals 10% p.a. with annual compounding (i.e. 10% p.a.a.c.). Prof. Dr. M. De Ceuster Introduction to Financial Markets 24 / 39  Time Value of Money Single Cash Flow Compounding (with a different compounding frequency) Annual compounding implies that we add the interest amount to the capital so that interest on interest can be earned. We do not have to wait for a full year in order to add the interest rate to the capital. We can change the compounding frequency to e.g. semi-annual compounding so that we get 3 4T ◊ 2 rsac VT = V0 ◊ 1+ 2 Any compounding frequency m can be chosen: 3 4T ◊ m rm VT = V0 ◊ 1+ m Prof. Dr. M. De Ceuster Introduction to Financial Markets 25 / 39  Time Value of Money Single Cash Flow Compounding (with a different compounding frequency) Example: 100 at r = 10% over 1 year (T=1) 110.00 if m=1 (annual compounding) 110.25 if m=2 (bi-annual compounding) 110.38 if m=4 (quarterly compounding) 110.47 if m=12 (monthly compounding) 110.52 if m=365 (daily compounding) Prof. Dr. M. De Ceuster Introduction to Financial Markets 26 / 39  Time Value of Money Single Cash Flow Discounting (with a different compounding frequency) VT V0 = 3 4T ◊ m. 1+ r m Prof. Dr. M. De Ceuster Introduction to Financial Markets 27 / 39  Multiple Cash Flows Section 3 Multiple Cash Flows Prof. Dr. M. De Ceuster Introduction to Financial Markets 28 / 39  Multiple Cash Flows Value Additivity Cash flows can only be summed if their values are expressed at the same moment! Example: Paul Draper has won a crossword competition and will receive the following set of cash flows over the next two years: Mr. Draper can currently earn 6 percent in his money market account. What is the present value of the cash flows? Source: Hillier, Ross, Westerfield, Jaffe & Jordan Prof. Dr. M. De Ceuster Introduction to Financial Markets 29 / 39  Multiple Cash Flows Value Additivity Computation: Prof. Dr. M. De Ceuster Introduction to Financial Markets 30 / 39  Multiple Cash Flows Bond Pricing 1 A financial instrument that gives rise to a stream of cash flows (CF) can be interpreted as a portfolio of single cash flow instruments. 2 We know that present values are additive. 3 We also know that each cash flow has to be discounted at it’s own interest rate. (Technically, we will use spot rates (see later). That’s why we use the symbol s.) CF1 CF2 V0 = + (1 + s1 )1 (1 + s2 )2 Prof. Dr. M. De Ceuster Introduction to Financial Markets 31 / 39  Multiple Cash Flows Bond Pricing We can apply these principles to a coupon bond. What is the price of a 4 year 2% coupon bond with a face value of 1000? The coupons are paid annually. Assume a FLAT term structure with a discount rate of 6% p.a. P = 20 + 20 + 20 + 1020 = 861.40 (1 + 6%) (1 + 6%)2 (1 + 6%)3 (1 + 6%)4 In Excel: =pv(rate,nper,pmt,fv) i.e. =pv(0.06,4,20,1000) Prof. Dr. M. De Ceuster Introduction to Financial Markets 32 / 39  Multiple Cash Flows Bond Pricing: The New Normal Prof. Dr. M. De Ceuster Introduction to Financial Markets 33 / 39  Multiple Cash Flows Bond Pricing with a Negative Discount Rate In a world with negative interest rates. 20 20 20 P = + + (1 - 0.5%) (1 - 0.5%)2 (1 - 0.5%)3 1020 + = 1101.26 (1 - 0.5%)4 In Excel: =pv(rate,nper,pmt,fv) i.e. =pv(-0.005,4,20,1000) Prof. Dr. M. De Ceuster Introduction to Financial Markets 34 / 39  Multiple Cash Flows Bond Pricing: Laws 1 Prices and the interest rate are inversely related. If the interest rate increases/decreases, bond prices decrease/increase. 2 If the coupon rate equals the (single) discount rate, the bond prices at par. c=y à par cy à above par Prof. Dr. M. De Ceuster Introduction to Financial Markets 35 / 39  Yields Section 4 Yields Prof. Dr. M. De Ceuster Introduction to Financial Markets 36 / 39  Yields Yields of Single Cash Flow Instruments VT V0 = (1 + sT )T leads to Û VT sT = T —1 (V 0) The return on a single cash flow that will be received at the end of period T is called a T-year spot rate. That explains why we use the symbol s. Prof. Dr. M. De Ceuster Introduction to Financial Markets 37 / 39  Yields Yields of Multiple Cash Flows Instruments CF 1 CF 2 We know that V 0 = (1+s 1 ) 1 + (1+s 2 ) 2. But what is the single “yield” that equates the present value of the future cash flows to the current market value of the instrument? CF1 CF2 V0 = + (1 + y)1 (1 + y)2 So we assume that V 0 is the current market price. We also assume that the cash flows CF1 and CF2 and their timing are given. Hence the ‘yield’ is the only unknown. The single return on a stream of cash flows is called the yield-to-maturity aka the internal rate of return, the actuarial rate of return or the overall yield. Prof. Dr. M. De Ceuster Introduction to Financial Markets 38 / 39  Yields Determining the Yield to Maturity Suppose an investment of 339.03 will yield the following cash flows: CF1 = 50 ; CF2 = 150 ; CF3 = 70, CF4 = 100. We need to solve 50 150 70 100 339.03 = + + + (1 + irr) (1 + irr)2 (1 + irr)3 (1 + irr)4 In Excel: =irr(values) i.e. =irr(-339.03,50,150,70,100) = 3.45% Alternatively you can use goal seek. Prof. Dr. M. De Ceuster Introduction to Financial Markets 39 / 39