8. Valuation of Known Cash Flows: Bonds

Transcription

1 8. Valuation of Known Cash Flows: Bonds 8. Using PV Factors to Value Known Cash Flows We use interest rate to value known cash flow. Which discount rate to use? In practice, you do not usually know which discount rate to use in the present value formula. The interest rates are not the same for all maturities. A word "yield" has many similar but different meanings. In the following, a word, "yield" means interest rate per year denoted as annual percentage rate or in decimal numbers. It is not effective annual rate. Yield curve is a curve depicting the relation between interest rate and maturity. An example of yield curve is Figure 8. on page 7. More precisely speaking, what the curve tells is interest rates on fixed-income instruments of a given risk and the maturity. Fixed-income instruments are securities which promise cash flows of fixed amounts. Flat yield curve implies that interest rates are the same for all maturities. Usually yield curve is not flat; different interest rates for different maturities. Usually it is upward sloping. Risk of Price Change Yield curve may change its shape. This implies that interest rates associated with the fixed-income instruments may change before the maturity. Such a change is not predictable. Interest rate and price of bond move in the opposite directions. So the prices of fixed-income securities are uncertain up to the time of maturity. Example: Risk of Price Change Suppose you buy a fixed income security which promises to pay $00 each year for the next three 00 year. This is an ordinary annuity. PV of this annuity is given by t=, where y is yield. First (+y) t we assume yield is %. Then we assume yield increases to 7%, right after you bought. As yield increases, the PV of the annuity decreases. Clear[p00, t]; 00 p00 = (+0.0) t; t= Clear[p007, t]; 00 p007 = (+0.07) t; t= Print["Price change is -$", Abs[p007-p00]] Price change is -$.899 Why do we write -$,Abs[p007-p00], instead of $,p007-p00 in the above? 07-0Ch08a.nb

2 Value of PV as a function of interest rate 00 Clear[f, y, t]; f[y_] := (+y) t t= Plot[f[y], {y, 0.0, 0.09}, PlotLabel "value of annuity", AxesLabel {"int. rate % ", "$"}, ImageSize 00] $ value of annuity int. rate % 8. Pure Discount Bonds Pure discount bonds are bonds which promise single payments of cash at maturity. They are also called zero coupon bonds. Face value: It is a promised cash payment on maturity. It is also called par value. Yield on pure discount bond is rate of return to investors who buy it and hold it until it matures. Yield has many meanings. Here it is the same as interest rate. Interest rates for pure discount bonds are called spot rates or zero rates. Suppose that we have pure discount bond with one year to maturity. Let F be face value and P be price. Assume annual compounding. Then, yield satisfies the following equation. P = F +x Solution x is interest rate for pure discount bond. So it is zero rate by definition. case one year of remaining life, F=00, P=9 Yield satisfies the following equation. Let x be APR based on annual compounding. Then variable x satisfies the following; today s price = x = face value - today' s price today' s price face value. +x = 00-9 =0.0. Interest rate is.% as APR. 9 What if we have a maturity different from one year? case two years of remaining life, F=00, P=9 We apply annual compounding. Clear[F, P, x, ans]; F = 00; P = 9; F ans = NSolve P {x} (+x), x = x /. ans[[]] {{x -.0}, {x 0.0}} Ch08a.nb

3 Print["Zero rate as APR is ", 00x, " %"] Zero rate as APR is. % case three months of remaining life, F=00, P=98 We have months to maturity. So interest rate over month is given by of APR. Clear[P, x, ans]; F = 00; P = 98; F ans = NSolve P, {x} x = x /. ans[[]]; Print["x = ", x] {{x 0.087}} x = x 8. The Basic Building Blocks: Pure Discount Bonds Pure discount bonds are the basic building blocks for valuing all contracts promising streams of known cash flows. We can always decompose any contract into its component cash flows and value each one separately. Each cash flow is considered as a pure discount bond. Always we can consider series of cash flows as a set of individual pure discount bonds. They may have the same face values but different maturities. We evaluate PV of each one of them and sum them up. The sum is equal to value of the set of cash flows. Valuing of Three-Year Annuity as a Set of Pure Discount Bonds We have a security which promises to make three payments of $00; $00 each year for the next three years. In other words, this security is an ordinary three year annuity. What is the value of this ordinary annuity? We apply annual compounding. We consider each of three $00 payments as separate pure discount bonds. We evaluate each one separately in the following. Prices of pure discount bonds Suppose that, in the bond market, we observe a set of prices of pure discount bonds. also called zero rate or spot rate. Yield here is Maturity price per $ of face value yield (per year) year 0.9.% years % years % Prices of pure discount bonds Print "Price of one year bond per $ of face value = ", Price of one year bond per $ of face value = Annuity as a Set of Pure Discount Bonds 07-0Ch08a.nb

4 We consider that each cash flow of the three year annuity is pure discount bond. So we have three pure discount bonds with different maturity. We evaluate each separately and sum them up. Total is equal to $.0. Clear[v] 00 v = (+0.0) (+0.077) ; Print["Value of annuity as a set of pure discount bonds = ", v" dollars."] Value of annuity as a set of pure discount bonds =.007 dollars. Sum of present values of cash flows is equal to $.007 We consider that each cash flow of the three year annuity is pure discount bond. We evaluate each separately and sum them up. It is $.0. IRR and zero rates If you buy this annuity at $.007, then you will have three cash flows. We like to find IRR of this investment project. "Zero rate or spot rate" is not the same as IRR. IRR is a single interest rate. It is not directly observable. We calculate IRR as follows. IRR is interest rate which makes NPV =0. It is.88%. Clear[ans, y] ans = NSolve t= y = y /. ans[[]]; 00 -v 0, {y} t (+y) Print["IRR as APR is ", 00y, " %"] {{y }, {y }, {y }} IRR as APR is.880 % IRR is interest rate which makes NPV =0. It is.88%. In the above annuity, we can consider we have three pure discount bonds. Zero rate or spot rate is IRR of a single pure discount bond. IRR of the annuity is not necessarily the same as IRR of each bond. Homework No., due June st Q. Problem 8. Hint: This question checks your understanding of definitions of terms. Hint: Treasury bill is pure discount bond issued by government. Pure discount instrument means that it is a security such that an issuer pays promised amount at maturity and that there is no interest payment before the maturity. Instrument means bond here. The promised amount is called face value. When you buy, you pay lower than this face value. Difference between the price you pay and the face value you receive on maturity is equivalent to interest payment. Price of pure discount bond is equal to present value of the face value. Actually, interest rate called zero rate is calculated from the face value and price of pure discount bond so that present value becomes equal to its price. In other words, we solve the following and find value of x. PV = face value +x and price = PV Variable x in the above is interest rate per period. It is not necessarily APR. It depends on the context. Q. Problem 8. Suppose you buy bond at P0 and sell it later at P. Then holding period rate of return is given by Rate of return is per period while you own. It has to be changed to APR or Eff(effective annual rate) for comparison. P - P0. P0 07-0Ch08a.nb

5 comparison. Q : Zero rates are given in the table below. () Evaluate year ordinary annuity of $00. () Find IRR as APR of this investment. Apply annual compounding. "maturity(year)" "zero rate" Q: Prices of pure discount bonds are given below. Their face values are all $00. () Find zero rates quoted as APR. Apply appropriate compounding rule. It means that apply monthly, month, biannual compounding for, and month maturity. However, as for month maturity, apply biannual compounding. () Draw yield curve of zero rates. Hint: To draw yield curve, refer to Plot data at the end of this page. "maturity(month)" "price" Plot data Let s show data graphically. For this purpose, we use ListPlot[ ] and ListLinePlot[ ] When you want to use arrow in the code, be sure to input hyphen - and inequality >. An arrow sign from the palette does not work. The Simplest Case Clear[A, graph]; A = ; graph = ListPlot[ A, PlotLabel "Use of ListPlot[ ]", ImageSize 00] Use of ListPlot[ ] You can modify figure by the following options. PlotLabel title of the graph : You can show title of the graph. ImageSize-> 00 : You can adjust size of a graph. 07-0Ch08a.nb

6 Clear[graph] graph = ListLinePlot[ A, PlotLabel "Use of ListLinePlot[ ]", ImageSize 00] Use of ListLinePlot[ ] You can save space if you arrange figures horizontally using Row[ ] as follows. In the following, without space here, two figures are shown without space. Row[{graph, " space here ", graph}] Use of ListPlot[ ] Use of ListLinePlot[ ] space here Scatter Diagram In the above example, horizontal axis is measured by {,,...}. They are ID numbers of the samples. However, in many cases, we want to show graph of {pair of x and y} ; {x,y}, {x, y},... Suppose we have following data; ID score score And suppose we want to show data as pairs of score and score. Our data must prepare data in the following form of list ; { {9,80}, {8,},... }. To do this, we take advantage of Table[ ]. Clear[A, AA]; A = "ID" "score " "score " ; AA = Table[{A[[, +i]], A[[, +i]]}, {i,, }] {{9, 80}, {8, }, {0, 7}, {70, 89}} Grid[AA, Frame All] You can show names of variables of the axes by AxesLabel { name of variable, variable } 07-0Ch08a.nb