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

Page 2 January 2013 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)

Page 3 January 2013 Overview II Yield Changes and Performance of Bonds (Duration) A closer look at Duration (Performance Approximation) McCauley Duration (Average time, Price elasticity, Immunization)

Page 4 January 2013 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

Page 5 January 2013 Discounting and the time travelling machine Assume you invest today 100€ at 10% interest. Which amount can you expect after one year? Amount + Interest * 10% = 100 * (1 + 10%) = 110 And after 2 years ? Amount + Interest 100 * (1 + 10%) * (1 + 10%) * 10% = 100 * (1 + 10%) 2 = 121 And after n years ? 100 * (1 + 10%) n = Amount * (1 + interest) n

Page 6 January 2013 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

Page 7 January 2013 Value of a Bond A Bond represents the right to receive future Cash Flows. The Cash Flows consists out of Coupon and principal payment. Today 1 st Coupon date 2 nd Coupon date … Maturity Pay for bondCoupon Principal + Coupon Coupon

Page 8 January 2013 Example: Assume you buy a 5% Bond for 80 € with a maturity of 4.3 years

Page 9 January 2013 Idea of Valuation: Each individual Cash Flow can be valued and aggregated to the total value ! / (1+ 10%) 0.3 = 4,86 5 / (1+ 10%) 1.3 = 4,42 5 / (1+ 10%) 2.3 = 4,02 5 / (1+ 10%) 3.3 = 3, / (1+ 10%) 4.3 = 69,69 86,64

Page 10 January 2013 Pricing Formula The Value of the bond consists out of the sum of the individual values. Or in a more formal way. Notation PPrice (dirty) of Bond TTime to maturity tindex CF t Cash Flow at time t yinterest (yield) of bond

Page 11 January 2013 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 … 555 … Zero Coupon Bond Bond that pays no Coupon. Only at maturity the principal is repaid Maturity 00100

Page 12 January 2013 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 X 1 =3 Month-Rate Maturity X n / 4 X 1 / 4 is paid X 2 =3 Month-Rate X 2 / 4 is paid X 3 =3 Month-Rate... 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.

Page 13 January 2013 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 Next coupon payment CF Last coupon payment CF today Accrued Interest = (1-t) * CF t 1 - t

Page 14 January 2013 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 Price90 € Accrued Interest 6 €(1 – 0.25) * 8 Dirty Price96 €

Page 15 January 2013 Example Accrued Interest accrued interest

Page 16 January 2013 Example Accrued Interest continued Clean Price1.000 € * % = Accrued Interest(230 Days) € * 7.125% *(230+1)/365 = Dirty Price ( %) € Next coupon payment Last coupon payment today Days

Page 17 January 2013 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 : 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.

Page 18 January 2013 Example Day Count Conventions

Page 19 January 2013 Example: Pricing of a Bond , ,1257, ,699

Page 20 January 2013 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.

Page 21 January 2013 Premium Bond Bond which a coupon rate above the market interest rate. Consequently the Price of the bond is greater than 100.

Page 22 January 2013 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

Page 23 January 2013 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 … And calculate the first derivative with respect to the interest rate y. Changing to percentage change in Price gives:

Page 24 January 2013 Using the modified Duration we can approximate percentage price change. Dirty Price (P) Interest Rate (y)Current Interest

Page 25 January 2013 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€.

Page 26 January 2013 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 Time to Maturity Today Today - Δt ΔtΔt ΔyΔy

Page 27 January 2013 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%

Page 28 January 2013 McCauley Duration Besides the modified Duration, the McCauley Duration is often used as well. For its computation we start with the modified Duration: 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).

Page 29 January 2013 McCauley Duration – Calculation Example The following table is helpful to calculate the McCauley Duration:

Page 30 January 2013 McCauley Duration – Interpretation The McCauley Duration has various interpretations (average time, price elasticity etc.). Average Time to maturity (balances discounted Cash Flows) t Disc. CF 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.