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Bond Arithmetic

Bond Arithmetic. Adapted version with the permission of Dr. Gunther Hahn, CFA. Frankfurt, January 2013. Overview. Discounting and the time travelling machine ( compounding vs. discounting ) Value of a Bond ( pricing formula ) Special Bonds ( Zero Coupon, Consol, Floater )

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Bond Arithmetic

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  1. Bond Arithmetic Adapted version with the permission of Dr. Gunther Hahn, CFA Frankfurt, January 2013

  2. Overview • Discounting and the time travelling machine (compounding vs. discounting) • Value of a Bond (pricing formula) • Special Bonds (Zero Coupon, Consol, Floater) • Price Quotation in the market (Clean vs. Dirty Price, Day Count Conventions) • Price Behaviour of bonds (Discount vs. Premium Bond, Price vs. time, Price vs. yield)

  3. Overview II • Yield Changes and Performance of Bonds (Duration) • A closer look at Duration (Performance Approximation) • McCauley Duration (Average time, Price elasticity, Immunization)

  4. Literature Bond Basic: Fabozzi, F. (1993): „Fixed Income Mathematics“, McGraw-Hill Bonds and Yield Curves : Luenberger, D. (1998): „Investment Science“, Oxford, pp. 40 – 101 Bonds and xls examples: Benninga, S. (2008): „Financial Modelling“, 3rd edition, MIT press, pp. 669-717

  5. Discounting and the time travelling machine • Assume you invest today 100€ at 10% interest. Which amount can you expect after one year? Amount + Interest 100 + 100 * 10% = 100 * (1 + 10%) = 110 • And after 2 years ? Amount + Interest 100 * (1 + 10%) + 100 * (1 + 10%) * 10% = 100 * (1 + 10%)2 = 121 • And after n years ? 100 * (1 + 10%)n = Amount * (1 + interest)n

  6. Now assume you receive 110€ in 1 year from today. How much is this worth today, if the interest level is at 10% ? Amount + Interest = 110 ? + ? * 10% = ? * (1 + 10%) = 110 ?= 110 / (1 + 10% ) = 100 • Assume you receive X € in n years. How much is this worth at y % interest? Todays Value = X / (1 + y)n

  7. Value of a Bond A Bond represents the right to receive future Cash Flows. The Cash Flows consists out of Coupon and principal payment. Principal + Coupon Pay for bond Coupon Coupon Today … 1 st Coupon date 2 nd Coupon date Maturity

  8. -80 5 5 5 5 105 0 0.3 4.3 1.3 2.3 3.3 Example:Assume you buy a 5% Bond for 80 € with a maturity of 4.3 years.

  9. -80 5 5 5 5 105 5 / (1+ 10%)0.3= 4,86 0 0.3 4.3 1.3 2.3 3.3 5 / (1+ 10%)1.3= 4,42 5 / (1+ 10%)2.3 = 4,02 5 / (1+ 10%)3.3 = 3,65 105 / (1+ 10%)4.3 = 69,69 86,64 Idea of Valuation:Each individual Cash Flow can be valued and aggregated to the total value !

  10. Or in a more formal way. Notation P Price (dirty) of Bond T Time to maturity t index CFt Cash Flow at time t y interest (yield) of bond Pricing Formula The Value of the bond consists out of the sum of the individual values.

  11. 5 5 5 5 … 0 0.3 … 1.3 2.3 3.3 Special Bonds • British Consol • Bond that never matures. The Bond pays its coupon forever and needs to be bought back by the issuer in order to mature. • Zero Coupon Bond • Bond that pays no Coupon. Only at maturity the principal is repaid. 0 0 0 100 0 0.3 Maturity 1.3 2.3

  12. On each of the reset days the value of the floater is 100. The idea behind this logic is that the cash flow from a floater can be duplicated easily. On each of the reset days a fixed term deposit for 3 Month earning the 3 Month-Rate is opened. At the end of the period the 3 Month-Rate is earned and the 100 are recieved back. On each of the reset days the value of the floater is 100. The idea behind this logic is that the cash flow from a floater can be duplicated easily. On each of the reset days a fixed term deposit for 3 Month earning the 3 Month-Rate is opened. At the end of the period the 3 Month-Rate is earned and the 100 are recieved back. Special Bonds II • Floater • Bond that pays a floating rate (on a quartely basis) depending on the level of the interest rate. At the beginning of the period the rate is observed and at the end the rate is paid and the new rate is observed. 0 0.25 0.5 Maturity ... X1=3 Month-Rate X1 / 4 is paid X2=3 Month-Rate X2 / 4 is paid X3=3 Month-Rate ... 100 + Xn / 4

  13. CF Accrued Interest = (1-t) * CF CF today Next coupon payment Last coupon payment Price Quotation in the market • So far the valuation was equal to the amount which needs to be paid. This amount is called the dirty price. • The price which is quoted on Bloomberg or in the newspaper is the clean price of the bond, which accounts for the accrued interest. Dirty Price = Clean Price + Accrued Interest t 1 - t

  14. Example:Assume you buy a 8% Coupon Bond with 4.25 years to maturity. The clean Price is 90€. How much do you pay to receive the Bond? Clean Price 90 € Accrued Interest 6 € (1 – 0.25) * 8 Dirty Price 96 €

  15. Example Accrued Interest accrued interest

  16. 230 Days 230 Days 230 Days 20.04.2011 6.12.2011 20.04.2012 Example Accrued Interest continued today Next coupon payment Last coupon payment Clean Price 1.000 € * 103.19% = 1031.90 Accrued Interest (230 Days) 1.000 € * 7.125% *(230+1)/365 = 45.09 Dirty Price (107.699%) 1076.99 €

  17. Day Count Conventions The difference between two dates can be calculated according to different market standards. • Actual / Actual : real Number of days are counted. • Actual / 365 : real Number of days are counted; the number of days in a year is counted as 365 (even if it is a leap year). • Actual / 360 : real Number of days are counted; the number of days in a year is counted as 360. • 30 / 360 : every month is counted as 30 days and every year as 360 days; - If the period starts on the 31st then the start is moved on the 30th - If the period ends on the 31st then the end is moved on the 1st - If the period ends on the 31st and starts on the 31st then the end is moved on the 30th. • 30 E / 360 : every month is counted as 30 days and every year as 360 days; - If the period starts on the 31st then the start is moved on the 30th - If the period ends on the 31st then the end is moved on the 30th.

  18. Example Day Count Conventions

  19. Example: Pricing of a Bond 7.12.2011 20.04.2014 20.04.2015 7.12.2016 20.04.2012 20.04.2013 -107,699 7,125 7,125 7,125 7,125 107,125

  20. Discount vs. Premium Bonds • Discount Bond • Bond which a coupon rate below the market interest rate. Consequently the Price of the bond is cheaper than 100.

  21. Premium Bond • Bond which a coupon rate above the market interest rate. Consequently the Price of the bond is greater than 100.

  22. Yield Changes and Performance of Bonds The following picture shows how the dirty price changes if we vary the market interest rate. Dirty Price Interest Rate

  23. And calculate the first derivative with respect to the interest rate y. • Changing to percentage change in Price gives: In order to compute the price change approximately, we calculate the first derivative of the dirty price function. Using the derivative we can approximate the change in price. • We start with the pricing function …

  24. Using the modified Duration we can approximate percentage price change. Dirty Price (P) Current Interest Interest Rate (y)

  25. Example:Assume you have a bond with a modified duration of 6. The dirty price is 120€. Suddenly the yield decreases from 4% to 3.5%. Will you gain or loose? How much is the percentage change in price and absolute change? • Since the yield decreases the price of the bond will increase. This way investors are compensated for a lower yield level. • Percentage change in dirty price = - modified Duration * change in yield • Percentage change in dirty price = - 6 * -0,5% = 3% • Absolute change in dirty price = 3% * 120€ = 3,6€ • The Price will increase approximately from 120€ to 123,6€.

  26. A closer look at Duration Using the modified Duration and yield curve we can approximate the Performance of a bond over a period of time. Interest Rate / yield curve Δy Δt Today - Δt Time to Maturity Today

  27. Example:Assume you hold a bond for half a year. When you buy the bond, the Duration was 6 and the yield 3%. At the end of the period the yield increased to 3.5%. Which approximate Performance did you earn? • The formula gives: • Performance = 3% * 0,5 - 6 * 0,5% = 1,5% - 3% = -1,5%

  28. Now we multiply both sides with (1+y) to obtain the McCauley Duration: The McCauley Duration represents the percentage price change over the percentage yield change. So the McCauley Duration is an elasticity (% Change / % Change). McCauley Duration Besides the modified Duration, the McCauley Duration is often used as well. For its computation we start with the modified Duration:

  29. McCauley Duration – Calculation Example The following table is helpful to calculate the McCauley Duration:

  30. Price elasticity (can be used to calculate percental price changes) If yields rise from 5% to 6% the denominator is not 1%, but 1%/1.05 = 0.95%. For this reason of complexity modified duration is more often used. McCauley Duration – Interpretation The McCauley Duration has various interpretations (average time, price elasticity etc.). • Average Time to maturity (balances discounted Cash Flows) Disc. CF t

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