ACI THE FINANCIAL MARKETS ASSOCIATION EXAMINATION FORMULAE 2009 VERSION page number INTEREST RATE..2 MONEY MARKET..... 3 FORWARD-FORWARDS & FORWARD RATE AGREEMENTS..4 FIXED INCOME.....5 FOREIGN EXCHANGE 7 OPTIONS 8 In all the formulae: interest s, yields, coupon s and s of discount are expressed as a decimal, eg 8.53% will be expressed as 0.0853 is the number of days in a year assumed under the appropriate convention term is the number of days from settlement to maturity of the instrument in question day count is the number of days from settlement to maturity of the instrument in question.
INTEREST RATE CONVERSIONS Converting between bond basis and money market basis (Act/360) bond basis = money market basis 365 360 money market basis = bond basis 360 365 Converting between annually and semi-annually compounding frequencies annually-compounded = 1 semi-annually compounded 2 2-1 semi-annually compounded = ( 1)2 annually compounded The formulae for converting between annually and semi-annually compounded apply only to s quoted on a bond basis, not a money market basis. 2
MONEY MARKET Certificates of deposit proceeds at maturity = face value ( 1 coupon x term ) secondary market proceeds = proceeds at maturity yield x day count Discount-paying instruments quoted as a true yield secondary market proceeds = face value yield x day count Discount-paying instruments quoted as a of discount discount amount = face value of discount x day count of discount x day count secondary market proceeds = face value 1- true yield = 1- of discount of discount x day count Forward price of sell/buy-back forward price = ( repurchase price - accrued interest on collateral at termination) 100 nominal price of collateral 3
FORWARD-FORWARDS & FORWARD RATE AGREEMENTS forward - forward interest interest = x day count x day count long period short period long period short period 1 day count forward-forward period FRA settlement amount = notional principal amount ( FRA - settlement ) 1 settlement x day count x d ay count 4
FIXED INCOME Clean and dirty price of bond with annual coupons on coupon date price = coupon 100 1 yield 1 ( ) ( ) remaining coupons remaining coupons yield yield 1 Dirty price of bond with annual coupons dirty price = first cashflow days to next coupon second cashflow days to next coupon L days to next coupon ( ) ( ) ( ) ( n-1 ) yield yield yield n th cashflow Duration at issue or on a coupon date Macaulay Duration = ( present value of first coupon amount x time to first coupon) (present value of second coupon amount x time to second coupon)... net present value of bond ( present value of ( last coupon amount nominal amount) x time to last coupon) Modified Duration = 1 Macaulay Duration yield compounding frequency 5
Calculating zero-coupon yield from an annual yield-to-maturity (bootstrapping) zero - coupon yield for n - = n year term final coupon amount nominal amount implied present value of final coupon and nominal amount 1100 The implied present value of the final coupon and nominal amount is calculated by subtracting from the net present value of the bond the sum of the present values of all coupons except the final one, where each present value is calculated using the appropriate zero-coupon yield. 6
FOREIGN EXCHANGE Forward FX forward = spot interest 1 interest 1 quoted currency base currency quoted currency x day count base currency x day count Covered interest arbitrage synthetic quoted currency interest = interest base currency x day count forward 1 1 base currency spot day count quoted currency synthetic base currency interest = interest quoted currency x day count 1 quoted currency spot forward 1 day count base currency 7
OPTIONS Standard deviation standard deviation = n t = 1 ( return at time t mean return) number of observations 1 2 Calculating the volatility over a period from annualised volatility volatility over period t = annualised volatility t Where t is in years or fractions thereof. In standard deviation calculations the ACI exams assume a year of 252 working days. 8