CL s Handy Formula Sheet. (Useful formulas from Marcel Finan s FM/2 Book) Compiled by Charles Lee 8/19/2010

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1 CL s Handy Frmula Sheet (Useful frmulas frm Marcel Finan s FM/2 Bk) Cmpiled by Charles Lee 8/19/2010

2 Interest a(t) Perid when greater Interest Discunt Simple Cmpund Simple Cmpund The Methd f Equated Time The Rule f 72 The time it takes an investment f 1 t duble is given by Interest Frmulas Frce f Interest Date Cnventins Recall knuckle memry device. (February has 28/29 days) Exact actual/actual Uses exact days 365 days in a nnleap year 366 days in a leap year (divisible by 4) Ordinary 30/360 All mnths have 30 days Every year has 360 days Banker s Rule actual/360 Uses exact days Every year has 360 days Basic Frmulas

3 Basic Equatins Immediate Due Annuities Perpetuity Annuities Payable Mre Frequently than Interest is Cnvertible Let = the number f payments per interest cnversin perid Let = ttal number f cnversin perids Hence the ttal number f annuity payments is Perpetuity Cefficient f is the ttal amunt paid during n interest cnversin perid Immediate Due Annuities Payable Less Frequently than Interest is Cnvertible Let = number f interest cnversin perids in ne payment perid Let = ttal number f cnversin perids Hence the ttal number f annuity payments is Immediate Due Perpetuity

4 Cntinuus Annuities a = 1 r = 1+k k i If k = i a = 1 r = 1-k k i Gemetric Varying Annuities If k=i General P, P+Q,, P+(n-1)Q Arithmetic Immediate Due Perpetuity Increasing P = Q = 1 Decreasing P = n Q = -1 Cntinuusly Varying Annuities Cnsider an annuity fr n interest cnversin perids in which payments are being made cntinuusly at the rate and the interest rate is variable with frce f interest. Under cmpund interest, i.e., the abve becmes Perpetuity

5 Rate f Return f an Investment Rate f Return f an Investment Yield rate, r IRR, is the interest rate at which Hence yield rates are slutins t NPV(i)=0 Interest Reinvested at a Different Rate Invest 1 fr n perids at rate i, with interest reinvested at rate j Invest 1 at the end f each perid fr n perids at rate i, with interest reinvested at rate j Discunted Cash Flw Technique Invest 1 at the beginning f each perid fr n perids at rate i, with interest reinvested at rate j Uniqueness f IRR Therem 1 Therem 2 Let B t be the utstanding balance at time t, i.e. Then

6 Dllar-Weighted Interest Rate A = the amunt in the fund at the beginning f the perid, i.e. t=0 B = the amunt in the fund at the end f the perid, i.e. t=1 I = the amunt f interest earned during the perid c t = the net amunt f principal cntributed at time t C = c t = ttal net amunt f principal cntributed during the perid i = the dllar-weighted rate f interest Nte: B = A+C+I Exact Equatin Simple Interest Apprximatin Summatin Apprximatin The summatin term is tedius. Define Expsure assciated with i"= A+ c t (1-t) If we assume unifrm cash flw, then Time-Weighted Interest Rate Des nt depend n the size r timing f cash flws. Suppse n-1 transactins are made during a year at times t 1,t 2,,t n-1. Let j k = the yield rate ver the kth subinterval C t = the net cntributin at exact time t B t = the value f the fund befre the cntributin at time t Then The verall yield rate i fr the entire year is given by

7 Bnds Ntatin P = the price paid fr a bnd F = the par value r face value C = the redemptin value r = the cupn rate Fr = the amunt f a cupn payment g = the mdified cupn rate, defined by Fr/C i = the yield rate n = the number f cupns payment perids K = the present value, cmpute at the yield rate, f the redemptin value at maturity, i.e. K=Cv n G = the base amunt f a bnd, defined as G=Fr/i. Thus, G is the amunt which, if invested at the yield rate i, wuld prduce peridic interest payments equal t the cupns n the bnd Quted yields assciated with a bnd 1) Nminal Yield a. Rati f annualized cupn rate t par value 2) Current Yield a. Rati f annualized cupn rate t riginal price f the bnd 3) Yield t maturity a. Actual annualized yield rate, r IRR Pricing Frmulas Basic Frmula Premium/Discunt Frmula Base Amunt Frmula Makeham Frmula Yield rate and Cupn rate f Different Frequencies Let n be the ttal number f yield rate cnversin perids. Case 1: Each cupn perid cntains k yield rate perids Case 2: Each yield perid cntains m cupn perids Amrtizatin f Premium r Discunt Let B t be the bk value after the tth cupn has just been paid, then Let I t dente the interest earned after the tth cupn has been made Let P t dente the crrespnding principal adjustment prtin Date June 1, 1996 Dec 1, 1996 June 1, 1997 Cupn Interest earned Amunt fr Amrtizatin f Premium Apprximatin Methds f Bnds Yield Rates Where Bk Value Exact Apprximatin Bnd Salesman s Methd Pwer series expansin Equivalently

8 Valuatin f Bnds between Cupn Payment Dates The purchase price fr the bnd is called the flat price and is dented by The price fr the bnd is the bk value, r market price, and is dented by The part f the cupn the current hlder wuld expect t receive as interest fr the perid is called the accrued interest r accrued cupn and is dented by Frm the abve definitins, it is clear that Theretical Methd The flat price shuld be the bk value B t after the preceding cupn accumulated by (1+i) k Practical Methd Uses the linear apprximatin Semi-theretical Methd Standard methd f calculatin by the securities industry. The flat price is determined as in the theretical methd, and the accrued cupn is determined as in the practical methd. $ Bk value Flat price Premium r Discunt between Cupn Payment Dates Callable Bnds The investr shuld assume that the issuer will redeem the bnd t the disadvantage f the investr. If the redemptin value is the same at any call date, including the maturity date, then the fllwing general principle will hld: 1) The call date will be at the earliest date pssible if the bnd was sld at a premium, which ccurs when the yield rate is smaller than the cupn rate (issuer wuld like t stp repaying the premium via the cupn payments as sn as pssible) 2) The call date will be at the latest date pssible if the bnd was sld at a discunt, which ccurs when the yield rate is larger than the cupn rate (issuer is in n rush t pay ut the redemptin value) Serial Bnds Serial bnds are bnds issued at the same time but with different maturity dates. Cnsider an issue f serial bnds with m different redemptin dates. By Makeham s frmula, where

9 Lan Repayment Methds Amrtizatin Methd Prspective Methd The utstanding lan balance at any time is equal t the present value at that time f the remaining payments Retrspective Methd The utstanding lan balance at any time is equal t the riginal amunt f the lan accumulated t that time less the accumulated value at that time f all payments previusly made Cnsider a lan f at interest rate i per perid being repaid with payments f 1 at the end f each perid fr n perids. Perid Payment amunt Interest paid Principal repaid Outstanding lan balance Sinking Fund Methd Whereas with the amrtizatin methd the payment at the end f each perid is, in the sinking fund methd, the brrwer bth depsits int the sinking fund and pays interest i per perid t the lender. Example Create a sinking fund schedule fr a lan f $1000 repaid ver fur years with i = 8%. If R is the sinking fund depsit, then Perid Interest paid Sinking fund depsit Interest earned n sinking fund Amunt in sinking fund Net amunt f lan Ttal

10 Measures f Interest Rate Sensitivity Stck Preferred Stck Prvides a fixed rate f return Price is the present value f future dividends f a perpetuity Cmmn Stck Des nt earn a fixed dividend rate Dividend Discunt Mdel Value f a share is the present value f all future dividends Shrt Sales In rder t find the yield rate n a shrt sale, we intrduce the fllwing ntatin: M = Margin depsit at t=0 S 0 = Prceeds frm shrt sale S t = Cst t repurchase stck at time t d t = Dividend at time t i = Peridic interest rate f margin accunt j = Peridic yield rate f shrt sale Duratin Methd f Equated Time (average term-t-maturity) made at times 1,2,,n Macaulay Duratin where R 1,R 2,,R n are a series f payments is a decreasing functin f i Vlatility (mdified duratin), where if P(i) is the current price f a bnd, then Cnvexity Mdified Duratin and Cnvexity f a Prtfli Cnsider a prtfli cnsisting f n bnds. Let bnd K have a current price, mdified duratin, and cnvexity. Then the current value f the prtfli is The mdified duratin f the prtfli is Inflatin Given i' = real rate, i = nminal rate, r = inflatin rate, Similarly, the cnvexity f the prtfli is Fischer Equatin A cmmn apprximatin fr the real interest rate: Thus, the mdified duratin and cnvexity f a prtfli is the weighted average f the bnds mdified duratins and cnvexities respectively, using the market values f the bnds as weights.

11 Redingtn Immunizatin Effective fr small changes in interest rate i Cnsider cash inflws A 1,A 2,,A n and cash utflws L 1,L 2,,L n. Then the net cash flw at time t is Interest Yield Curves The k-year frward n years frm nw satisfied where i t is the t-year spt rate Immunizatin cnditins We need a lcal minimum at i The present value f cash inflws (assets) shuld be equal t the present value f cash utflws (liabilities) The mdified duratin f the assets is equal t the mdified duratin f the liabilities The cnvexity f PV(Assets) shuld be greater than the cnvexity f PV(Liabilities), i.e. asset grwth > liability grwth Full Immunizatin Effective fr all changes in interest rate i A prtfli is fully immunized if Full immunizatin cnditins fr a single liability cash flw 1) 2) 3) Cnditins (1) and (2) lead t the system where δ=ln(1+i) and k=time f liability Dedicatin Als called abslute matching In this apprach, a cmpany structures an asset prtfli s that the cash inflw generated frm assets will exactly match the cash utflw frm liabilities.

12 Optin Styles Eurpean ptin Hlder can exercise the ptin nly n the expiratin date American ptin Hlder can exercise the ptin anytime during the life f the ptin Bermuda ptin Hlder can exercise the ptin during certain prespecified dates befre r at the expiratin date Buy Write Call Put Flr wn + buy put Cap shrt + buy call Cvered Call stck + write call = write put Cvered Put shrt +write put = write call Cash-and-Carry buy asset + shrt frward cntract Synthetic Frward a cmbinatin f a lng call and a shrt put with the same expiratin date and strike price F,T = n arbitrage frward price Call(K,T) = premium f call Put-Call Parity Lng Frward Lng Call Payff Prfit Price at Maturity Lng Put Shrt Frward Shrt Call Shrt Put Derivative Maximum Lss Maximum Gain Psitin wrt Strategy Payff Psitin Underlying Asset Lng -Frward Price Unlimited Lng(buy) Guaranteed price P T -K Frward Shrt Unlimited Frward Price Shrt(sell) Guaranteed price K-P T Frward Lng Call -FV(Premium) Unlimited Lng(buy) Insures against high price max{0,p T -K} Shrt Call Unlimited FV(Premium) Shrt(sell) Sells insurance against high price -max{0,p T -K} Lng Put -FV(Premium) Strike Price FV(Premium) Shrt(sell) Insures against lw price max{0,k-p T } Shrt Put FV(Premium) Strike Price FV(Premium) Lng(buy) Sells insurance against lw price -max{0,k-p T }

13 (Buy index) + (Buy put ptin with strike K) = (Buy call ptin with strike K) + (Buy zer-cupn bnd with par value K) (Shrt index) + (Buy call ptin with strike K) = (Buy put ptin with strike K) + (Take lan with maturity f K) Spread Strategy Creating a psitin cnsisting f nly calls r nly puts, in which sme ptins are purchased and sme are sld Bull Spread Investr speculates stck price will increase Bull Call Buy call with strike price K 1, sell call with strike price K 2 >K 1 and same expiratin date Bull Put Buy put with strike price K 1, sell put with strike price K 2 >K 1 and same expiratin date Tw prfits are equivalent (Buy K 1 call) + (Sell K 2 call) = (Buy K 1 put) + (Sell K 2 put) Prfit functin Bear Spread Investr speculates stck price will decrease Exact ppsite f a bull spread Bear Call Sell K 1 call, buy K 2 call, where 0<K 1 <K 2 Bear Put Sell K 1 put, buy K 2 put, where 1<K 1 <K 2 K 2 -K 1 Payff K 1 K 2 P T K 2 -K 1 -FV[ -FV[ Prfit P T Lng Bx Spread Bull Call Spread Bear Put Spread Synthetic Lng Frward Buy call at K 1 Sell put at K 1 Synthetic Shrt Frward Sell call at K 2 Buy put at K 2 Regardless f spt price at expiratin, the bx spread guarantees a cash flw f K 2 -K 1 in the future. Net premium f acquiring this psitin is PV(K 2 -K 1 ) If K 1 <K 2, then lending mney Invest PV(K 2 -K 1 ), get K 2 -K 1 If K 1 >K 2, then brrw mney Get PV(K 1 -K 2 ), pay K 1 -K 2 Butterfly Spread An insured written straddle Let K 1 <K 2 <K 3 Written straddle Sell K 2 call, sell K 2 put Lng strangle Buy K 1 call, buy K 3 put Prfit Let F Asymmetric Butterfly Spread

14 Cllar Used t speculate n the decrease f the price f an asset Buy K 1 -strike at-the-mney put Sell K 2 -strike ut-f-the-mney call K 2 >K 1 K 2 -K 1 = cllar width Prfit Functin Cllared Stck Cllars can be used t insure assets we wn Buy index Buy at-the-mney K 1 put Buy ut-f-the-mney K 2 call K 1 <K 2 Prfit Functin Zer-cst Cllar A cllar with zer cst at time 0, i.e. zer net premium Straddle A bet n market vlatility Buy K-strike call Buy K-strike put Prfit Functin Strangle A straddle with lwer premium cst Buy K 1 -strike call Buy K 2 strike put K 1 <K 2 Prfit Functin

15 Equity-linked CD (ELCD) Can financially engineer an equivalent by Buy zer-cupn bnd at discunt Use the difference t pay fr an at-the-mney call ptin Prepaid Frward Cntracts n Stck Let F P 0,T dente the prepaid frward price fr an asset bught at time 0 and delivered at time T If n dividends, then F P 0,T = S 0, therwise arbitrage pprtunities exist If discrete dividends, then If cntinuus dividends, then Let δ=yield rate, then the and 1 share at time 0 grws t e δt shares at time T Financial Engineering f Synthetics (Frward) = (Stck) (Zer-cupn bnd) Buy e -δt shares f stck Brrw S 0 e -δt t pay fr stck Payff = P T F 0,T (Stck) = (Frward) + (Zer-cupn bnd) Buy frward with price F 0,T = S 0 e (r-δ)t Lend S 0 e -δt Payff = P T (Zer-cupn bnd) = (Stck) (Frward) Buy e -δt shares Shrt ne frward cntract with price F 0,T Payff = F 0,T If the rate f return n the synthetic bnd is i, then S 0 e (i-δ)t = F 0,T r Implied rep rate Frward Cntracts Discrete dividends Cntinuus dividends Frward premium = F 0,T / S 0 The annualized frward premium α satisfies If n dividends, then α=r If cntinuus dividends, then α=r-δ

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