# 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.

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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