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Date & Time Handling and the Yield Curves

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Date & Time Handling and the Yield Curves Date & Time Handling and the Yield Curves Mathematics & Excel Modeling Joerg Hoerster Dr. Jan Rudl London, 2010 No part of this presentation may be circulated, quoted, or reproduced for distribution without prior written approval from Maravon. Maravon F I N E S T I N F I N A N C E © Copyright 2010 Maravon GmbH Day count convention determines how interest accrues over time MOTIVATION FOR DAY COUNT CONVENTIONS 2010 2022 Sep. 14 Sep. 27 Time Begin of interest an investment payment date How much interest decided by day should be transferred to account the payment date convention! What is the difference in years? Source: Maravon 1 There are three typical day count convention methods to calculate date differences DAY COUNT CONVENTION METHODS Typical methods 30/360 Actual Business day Source: Maravon 2 30/360 method assumes 30 days per month and 360 days per year GENERAL DESCRIPTION ∆(d m y ,d m y )=? 1 · 1 · 1 2 · 2 · 2 = y y +(m m )/12 + (d d )/360 2 − 1 2 − 1 2 − 1 Source: Maravon 3 There are three typical variants of the 30/360 method METHOD VARIANTS 30E/360 30I/360 30U/360 (European) (Italian) (US) • If d1=31, change d1 to 30 • The same as 30E/360 and • If d1=31, change d1 to 30 additionally, • If d2=31, change d2 to 30 • If d2=31 and d1≥30, Rules • if m1=2 and d1=28 or change d2 to 30 d1=29, change d1 to 30, and the same for m2 and d2 Source: Maravon 4 Actual method uses the actual number of days, but the number of days per year can differ GENERAL DESCRIPTION ∆(d m y ,d m y )=? 1 · 1 · 1 2 · 2 · 2 Days between(d m y ,d m y ) = 1 · 1 · 1 2 · 2 · 2 Days per year Source: Maravon 5 There are three basic variants for actual method, which differ in number of days per year ACT/360 ACT/365 ACT/ACT Days per year 360 365 365 or 366 Depending on whether y1 or y2 is a leap year or Feb 29 is covered by (d1⋅m1⋅y1, d2⋅m2⋅y2) Source: Maravon 6 Business day method uses the actual number of business days, where weekends and holidays are excluded GENERAL DESCRIPTION ∆(d m y ,d m y )=? 1 · 1 · 1 2 · 2 · 2 Business days between(d m y ,d m y ) = 1 · 1 · 1 2 · 2 · 2 Business days per year Basic rules: ➞ business days per year = 250 or 252 (mostly) ➞ weekends differ from country to country, e.g. United Arab Emirates, South Korea etc. ➞ holidays differ extremely from country to country Source: Maravon 7 Business day method requires day rolling conventions as the special handling of payment dates, which are non-business days DAY ROLLING CONVENTIONS FOLLOWING PREVIOUS payment date payment date → next business day → previous business day Day rolling conventions for payment dates as non-business days • If next business day is in the • If the previous business day is next month → PREVIOUS in the previous month → FOLLOWING • Otherwise → FOLLOWING • Otherwise → PREVIOUS MODIFIED FOLLOWING MODIFIED PREVIOUS Source: Maravon 8 Understanding Day Count xls Conventions 9 Interests can be calculated in various ways Variants of interest calculation Compound interest with Compound Continuous Simple interest several interest interest compounding periods per year Source: Maravon 10 In the following we use four mathematical symbols for interest calculation Symbol Description i Interest rate P Principal/invested amount of money at time 0 n Number of years Vn Value of P after n years Source: Maravon 11 Using simple interest, the annual interests stay the same EXAMPLE Value Vn is growing linearly with i=10% P=100 Vn = P (1 + n i) · · Calculation of simple interests n 1 2 3 … 10 annual 10 10 10 … 10 interest Vn 110 120 130 … 200 Source: Maravon 12 Using compounding interest, the interests are also compounded EXAMPLE Value Vn is growing expotentially n i=10% Vn = P (1 + i) P=100 · Calculation of compound interests n 1 2 3 … 10 annual 10 11=110 ⋅10% 12.1=121⋅10% … … interest 10 Vn 110 121 133.1 … 100 ⋅(1+10%) =259.37 Source: Maravon 13 There can also be several annual compounding periods EXAMPLE For fraction of i=10% i m n years, day P=100 Vn = P (1 + ) · count m=2 · m conventions are used Compounding frequency per year Calculation of compound interests with annual compounding periods n 0.5 1 1.5 … 10 annual 10% 10% 5 5.25=105 ⋅10% 5.25=110.25 ⋅10% … 10 interest 2 2 10% V 105 110.76 115.76 … 100 ⋅(1+ )10*2=265.33 n 2 Source: Maravon 14 If the number of compounding periods become more and more, we end up with the continuous compounding EXAMPLE m → ∞ Compounding Compounding m → ∞ m times a year times a year If the number of compounding periods is ∞, how valuable is an investment of 1 Euro m⋅n after 50 years " i % V = P ⋅ 1+ V = P ⋅en⋅i n $ m' n # & e: Euler's number: 2.71828… Source: Maravon 15 The effective interest rate allows to compare different compounding methods MOTIVATION OF EFFECTIVE INTEREST RATE Semi-annual n 1 2 3 … 10 How much compounding should the Vn 110.25 121.55 134.01 … 265.33 annual compounding interest rate be, in order to achieve the Continuous n 1 2 3 … 10 same interest compounding Vn 110.52 122.14 134.99 … 271.83 Effective interest rate Source: Maravon 16 Effective interest rate calculation is adapted for multi-annual and continuous compounding CALCULATION OF EFFECTIVE INTEREST RATE Multi-annual i: nominal interest rate Continuous Compounding ie: effective interest rate Compounding i m n n n i n Vn = P (1 + ) · = P (1 + ie) Vn = P e · = P (1 + ie) · m · n i · n· i m e · =(1+ie) 1+ie =(1+ ) ! i ! m e =(1+ie) i ! i i =(1+ )m 1 ie = e 1 ! e m − ! − Source: Maravon 17 Types of interest rates involved in the fixed-income jargon Taxonomy of Rates Coupon Rate Spot Yield to Forward Bond and current Zero-Coupon Maturity Rates Par Yield yield Rate Source: Maravon 18 Coupon rate and current yield Example The coupon rate is the stated interest rate on a security, referred to as an annual percentage of face value. It’s usually paid • How much is the current yield of a • Semi-annually (e.g. in the US), or bond with face value $1000, an annual coupon rate of 7% and a • Annually (e.g. in France and Germany) current price of $900? 7%×1000 Yc = = 7.78% Let c denote the coupon rate, N the nominal 900 value and P the current price of a bond. • Note the coupon rate of 7% does The current yield Yc is obtained from not change in any event. c × N Y = c P Source: Maravon 19 Yield to Maturity The yield to maturity (YTM) is the single rate that sets the present value of the cash flows Example equal to the bond price. Consider a $1000 face value 3-year bond with 10% annual coupon, wich sells for Semi-annual Annual $1010. The YTM Y of this bond is solved payment of payment of from coupon coupon 100 100 1000 + 100 2T CF T CF 1010 = + 2 + 3 t t 1 Y (1 Y ) (1 Y ) P = ∑ t Po = ∑ t + + + t=1 t=1 ⎛ Y2 ⎞ (1+ Y) ⎜1+ ⎟ Y = 9.601% ⎝ 2 ⎠ Where • P: Price of a bond Remark • T: Maturity Note the one-to-one correspondence • CFt: Cash flow at the date t between the price and the YTM of a bond. • Y2: YTM on a semi-annual basis Therefore, bonds are often quoted in YTM. • Y: YTM on an annual basis Source: Maravon 20 Spot Zero-Coupon (or Discount) Rate Example • Let B(0,t) denote the market price at time 0 of a bond paying off $1 at date t. Consider a 2-year zero-coupon bond that • Let R(0,t) denote the spot zero-coupon rate that trades at $92. How much is the 2-year zero is implicitly defined by coupon rate R(0,2)? 1 100 B(0,t) = t 92 = [1+ R(0,t)] [1+ R(0,2)]2 • In practice, if the spot zero-coupon yield curve 100 R(0,2) = −1= 4.26% t → R (0,t) and the future cash flows are known, 92 the spot prices for all fixed-income securities can be derived. Source: Maravon 21 Forward Rates Definition Characteristics Let R(0,t) denote the spot zero-coupon F(0,x,y-x) is a rate rate. An implied forward rate F(0,x,y-x) • that can be guaranteed on a (forward zero-coupon rate) between transaction occurring in the future years x and y is defined as (compare the example later on) 1 " [1+ R(0, y)]y % y− x • that can be viewed as a break-even F(0, x, y x) 1 point that equalizes the rate of return − = $ x ' − # [1+ R(0, x)] & on bonds across the entire maturity that is the forward rate as seen from spectrum date t=0, starting at date t=x, and with residual maturity y-x. Source: Maravon 22 Examples: Forward rates as a rate that can be guaranteed now on a transaction occurring in the future We simultaneously borrow and lend $1 repayable at the end of 2 years and 1 year, respectively. The cash flows generated by this transaction are as follows: Today In 1 Year In 2 Years Borrow 1 - [1+R(0,2)]2 Lend -1 [1+R(0,1)]1 Borrowing in 1 years repayable in 2 years at the Total 0 [1+R(0,1)] - [1+R(0,2)]2 amount of F(0,1,1) is the rate that can be guaranteed now for a loan starting in 1 year and repayable after 2 years with [1 + R(0, 2)]2 F (0, 1, 1) = 1 [1 + R(0, 1)] − Source: Maravon 23 Instantaneous forward rate: a particular forward rate Definition Characteristics Recall the forward rate F(t,s,T-s) as • f(t,s) is a continuously compounded seen from date t between years s and rate T.
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