AS 2553a Mathematics of finance

Transcription

1 AS 2553a Mathematcs of fnance Formula sheet November 29, 2010 Ths ocument contans some of the most frequently use formulae that are scusse n the course As a general rule, stuents are responsble for all efntons an results appearng n propostons n the lecture notes 1 Interest rate measurement To convert from one type of compoun nterest/scount rate to another, one may use the relatonshps As a result, we have For a general force of nterest t, ] m 1 + = [1 + (m) m =(1 ) 1 = [1 (m) m =e = 1 + v = = ln(1 + ) t = S (t) S(0) = t S(t) When nflaton s taken nto account, the nflaton-auste rate of nterest s r an the real rate of nterest s ( r)/(1 + r), where r s the nflaton rate ] m 1

2 2 Valuaton of annutes Present value Future value Level annuty-mmeate Level annuty-ue Level contnuous annuty m-thly payable annuty-mmeate m-thly payable annuty-ue k-pero eferre annuty Perpetuty(-mmeate) Perpetuty-ue Increasng annuty-mmeate Increasng annuty-ue Contnuously ncreasng contnuous annuty a n = 1 vn ä n = 1 vn ā n = 1 vn a (m) n ä (m) n s n = (1 + )n 1 = a n s n = (1 + )n 1 = a n s n = (1 + )n 1 = 1 vn (m) = (m) a n = 1 vn (m) = (m) a n k a n = v k a n 1 1 (Ia) n = än nv n (Iä) n = än nv n (Īā) n = ān nv n = s n = s n s (m) n = (1 + )n 1 (m) = (m) s n s (m) n = (1 + )n 1 (m) = (m) s n (Is) n = s n n = (Ia) n (I s) n = s n n (Ī s) n = s n n = (Is) n Decreasng annuty-mmeate (Da) n = n a n (Ds) n = n(1 + )n s n Decreasng annuty-ue (Dä) n = n a n = (Da) n (D s) n = n(1 + )n s n = (Ds) n Contnuously ecreasng contnuous annuty ( Dā) n = n ā n ( D s) n = n(1 + )n s n If the force of nterest vares n tme, then the present an future values of a contnuous annuty payng h(t) at tme t are respectvely ā n u = n 0 h(t)e R t 0 uu t, 2

3 an an they are relate by s n u = n 0 h(t)e R n t uu t, s n u = e R n 0 uu ā n u We also have the followng relatonshps by placng the relevant payments on the tme lne (Stuents are not expecte to memorze the next set of formulae but to be able to erve them when neee) v k a n =a n+k a k = k a n s n (1 + ) k =s n+k s k ä n =(1 + )a n s n =(1 + )s n ä n =1 + a n 1 s n =s n Loan repayment (Da) n + (Ia) n =(n + 1)a n ( Dā) n + (Īā) n =nā n Uner the amortzaton metho of a loan repayment, we have for t = 1, 2,, n where n s the term of a loan of amount L an I t =OB t 1, P R t =K t I t, OB t =OB t 1 P R t =(1 + )OB t 1 K t Once the loan s completely paye off, OB n = 0 Thus, the total prncpal repa s an the total nterest pa s P R t = (K t I t ) = L, I t = K t L The retrospectve form of the outstanng balance s OB t = (1 + ) t L t (1 + ) t K, =1 an ts prospectve form s OB t = v t K =t+1 3

4 When payments are level, I t =K ( 1 v n t+1), P R t =Kv n t+1, an OB t = Ka n t an Uner the snkng-fun metho, I t =L SF t 1 = L L s t 1 s n, P R t = L s n (1 + ) t 1, ( OB t = L SF t = L 1 s ) t s n If a loan s repa uner the snkng-fun metho an there are n remanng payments, the present value of the loan may be calculate through Makeham s fomula A = M + (L M) where M = Lv n 4 Bon valuaton The present value of a bon that has face value F, reempton amount C, effectve yel to maturty per coupon pero, coupon rate r, an term to maturty n s P = F ra n + Cv n When F = C, we also have P = F + F (r )a n Lettng M = F v n, Makeham s formula may be obtane P = M + r (F M) where The prce-plus-accrue at tme t = k + s s s = P t = (1 + ) s P k = v 1 s (F r + P k+1 ), # of ays snce last coupon pa # of ays between coupon payments The respectve prce quote n the newspapers s efne by Prce t = P t sf r 4

5 When amortzng a bon, we have an OB t =(Book value) t =Cv n t + F ra n t, t = 1, 2,, n 1, I t =OB t 1, t = 1, 2,, n P R t = { F r I t, t = 1, 2,, n 1 F r + C I t, t = n Usng Makeham s formula, the prce of a seral bon s calculate by where M = m k=1 M k an F = m k=1 F k P = M + r (F M), 5 Measurng the rate of return of an nvestment The followng are three ways of etermnng the rate of return on an nvestment yelng cashflows (postve or negatve) C 0, C 1,, C n occurrng at tmes t 0, t 1,, t n : 1 the nternal rate of return = v 1 1 > 1 s such that v s a postve real root to the equaton v t k C k = 0; k=0 2 f A s the ntal amount of the portfolo, B s ts fnal amount, an 0 < t 0 < t 1 < < t n < 1, then the net nterest earne n the fun s I = B [A + k=1 C t k ] an the ollar-weghte rate of return s I = A + k=1 C t k (1 t k ) ; 3 f V tk s the value of the portfolo at tme t k ust after nterest has been crete but before contrbutons or wthrawals have been mae, then the tme-weghte rate of return s = ( n k=1 V tk V tk 1 + C tk 1 6 Term structure of nterest rates ) 1/(tn t 0 ) Let the term structure of zero-coupon bons be {s [0,t] } t 0, then the one-year forwar rate of nterest for n 1 years from now s [n 1,n] = (1 + s [0,n]) n 1 (1 + s [0,n 1] ) n 1 1 5

6 7 Cashflow uraton an mmunzaton If a seres of n payments K 1, K 2,, K n occurrng at tmes 1, 2,, n, respectvely, s evaluate at prce P at tme 0, then the Macaulay uraton (also calle ust uraton) s calculate through the formula D = an the mofe uraton s foun by an P P 1+ = P MD = P = tk tv t, P tk tv t+1 P As a result, the Macaulay an the mofe uratons of a coupon bon are respectvely D = F r(ia) n + ncv n F r a n + Cv n MD = F rv (Ia) n + ncv n+1 F r a n + Cv n (Rememberng the last two formulae s optonal for ths course) If the current term structure {s [0,t] } t 0 s use to evaluate the mofe uraton of cashflows K 1, K 2,, K n occurrng at tmes 1, 2,, n, respectvely, then we have P = K t (1 + s [0,t] ) t an P MD = P = Rengton mmunzaton s n place, f 1 P V A ( 0 ) = P V L ( 0 ); 2 P V A() = =0 P V L() ; =0 3 2 P V A() > 2 2 =0 P V L() ] 2 =0 tk t(1 + s [0,t] ) (t+1) K t(1 + s [0,t] ) t 6