AS 2553a — Mathematics of finance Formula sheet November 29, 2010 This document contains some of the most frequently used formulae that are discussed in the course. As a general rule, students are responsible for all definitions and results appearing in propositions in the lecture notes.
1
Inte Intere rest st rate rate meas measur urem emen entt
To convert from one type of compound interest/discount rate to another, one may use the relationships i(m) 1+i = 1+ m
−1
=(1 − d)
m
d(m) = 1− m
=eδ .
−m
As a result, we have i 1+i 1 v= 1+i ln(1 + i). δ = ln(1
d=
For a general force of interest δ t , S (t) d = S (t). δ t = (0) S (0) dt When inflation is taken into account, the inflation-adjusted rate of interest is i − r and the real rate of interest is (i (i − r )/(1 + r ), where r is the inflation rate.
1
2
Valuation of annuities Present value
Level annuity-immediate
an i
Level annuity-due
¨n i a
Level continuous annuity m-thly payable
Future value
1 − vn = i 1 − vn i = = an i d d
1 − vn i ¯n i = = an i a δ δ n 1−v i (m) an i = (m) = (m) an i i i
sn i s¨n i
(1 + i)n − 1 = i (1 + i)n − 1 i = = sn i d d
(1 + i)n − 1 i = sn i s¯n i = δ δ n (1 + i) − 1 i (m) = sn i = sn i i(m) i(m)
annuity-immediate m-thly payable
(m) ¨n i a
1 − vn i = (m) = (m) an i d d
(m) s¨n i
(1 + i)n − 1 i = = sn i d(m) d(m)
annuity-due k-period deferred annuity Perpetuity(-immediate) Perpetuity-due
k| an i
= vk an i
1 i 1 d
¨n i − nvn a i ¨n i − nvn a i = = (Ia)n i d d n ¯n i − nv a = δ
s¨n i − n i s¨n i − n i = = (Is)n i d d s¯n i − n = δ
Increasing annuity-immediate
(Ia)n i =
(Is)n i =
Increasing annuity-due
(I a ¨)n i
(I ¨ s)n i
Continuously increasing
¯a (I ¯)n i
¯s)n i (I ¯
continuous annuity Decreasing annuity-immediate (Da)n i Decreasing annuity-due
(D¨a)n i
Continuously decreasing
¯ a)n i (D¯
continuous annuity
n − an i = i n − an i i = = (Da)n i d d ¯n i n−a = δ
(Ds)n i (D¨ s)n i ¯ s)n i (D¯
n(1 + i)n − sn i = i n(1 + i)n − sn i i = = (Ds)n i d d n n(1 + i) − s¯n i = δ
If the force of interest varies in time, then the present and future values of a continuous annuity paying h(t) at time t are respectively n
¯n δ = a u
h(t)e−
0
2
R t 0
δu du
dt,
and
n
s¯n δ = u
R n
h(t)e
t
δu du
dt,
0
and they are related by
R n
s¯n δ = e
0
δu du
u
a¯n δ . u
We also have the following relationships by placing the relevant payments on the time line. (Students are not expected to memorize the next set of formulae but to be able to derive them when needed.) v k an =an+k − ak =k| an i
sn (1 + i)k =sn+k − sk
¨n i =(1 + i)an i a ¨n i =1 + an−1 i a
s¨n i =(1 + i)sn i s¨n i =sn+1 i − 1 (Da)n i + (Ia)n i =(n + 1)an i ¯ a)n i + (I ¯a (D¯ ¯)n i =n¯an i
3
Loan repayment
Under the amortization method of a loan repayment, we have for t = 1, 2, . . . , n where n is the term of a loan of amount L I t =iOBt−1 , P Rt =K t − I t , and OBt =OBt−1 − P Rt =(1 + i)OBt−1 − K t . Once the loan is completely payed off, OBn = 0. Thus, the total principal repaid is n
n
P Rt =
t=1
and the total interest paid is
(K t − I t ) = L,
t=1
n
n
I t =
t=1
K t − L.
t=1
The retrospective form of the outstanding balance is t t
OBt = (1 + i) L −
(1 + i)t− j K j ,
j=1
and its prospective form is
n
OBt =
j=t+1
3
v j−t K j .
When payments are level, I t =K 1 − v n−t+1 , P Rt =Kv n−t+1 ,
and OBt = Ka n−t i . Under the sinking-fund method, I t =iL − jS F t−1 = iL − jL P Rt = and
st−1 j , sn j
L (1 + j)t−1 , sn j
st j OBt = L − SF t = L 1 − sn j
.
If a loan is repaid under the sinking-fund method and there are n remaining payments, the present value of the loan may be calculated through Makeham’s fomula i A = M + (L − M ) j where M = Lv jn .
4
Bond valuation
The present value of a bond that has face value F, redemption amount C, effective yield to maturity j per coupon period, coupon rate r, and term to maturity n is P = F ran j + Cv jn . When F = C, we also have P = F + F (r − j)an j . Letting M = F v jn , Makeham’s formula may be obtained r P = M + (F − M ). j The price-plus-accrued at time t = k + s is P t = (1 + j)s P k = v 1−s (F r + P k+1 ), where
# of days since last coupon paid . # of days between coupon payments The respective price quoted in the newspapers is defined by s=
Pricet = P t − sFr. 4
When amortizing a bond, we have OBt =(Book value)t =Cv jn−t + F ran−t j , I t = jOBt−1 , and P Rt =
t = 1, 2, . . . , n − 1,
t = 1, 2, . . . , n
F r − I t , t = 1, 2, . . . , n − 1 . F r + C − I t , t = n
Using Makeham’s formula, the price of a serial bond is calculated by r P = M + (F − M ), j where M =
5
m k=1
M k and F =
m k=1
F k .
Measuring the rate of return of an investment
The following are three ways of determining the rate of return on an investment yielding cashflows (positive or negative) C 0 , C 1 , . . . , Cn occurring at times t0 , t1 , . . . , tn : 1. the internal rate of return i = v −1 − 1 > −1 is such that v is a positive real root to the equation n
v t C k = 0; k
k=0
2. if A is the initial amount of the portfolio, B is its final amount, and 0 < t0 < t1 < · · · < tn < 1, then the net interest earned in the fund is I = B − [A + nk=1 C t ] and the dollar-weighted rate of return is I ; i= n A + k=1 C t (1 − tk )
k
k
3. if V t is the value of the portfolio at time tk just after interest has been credited but before contributions or withdrawals have been made, then the time-weighted rate of return is k
n
i=
6
V t + C t k
V t k=1
k−1
k−1
1/(tn −t0 )
− 1.
Term structure of interest rates
Let the term structure of zero-coupon bonds be {s[0,t] }t≥0 , then the one-year forward rate of interest for n − 1 years from now is i[n−1,n]
(1 + s[0,n] )n = − 1. (1 + s[0,n−1] )n−1
5
7
Cashflow duration and immunization
If a series of n payments K 1 , K 2 , . . . , Kn occurring at times 1, 2, . . . , n , respectively, is evaluated at price P at time 0, then the Macaulay duration (also called just duration) is calculated through the formula dP n t dj t=1 tK t v j D = − P = , P 1+ j
and the modified duration is found by MD = −
dP dj
P
=
n t=1
tK t v jt+1 . P
As a result, the Macaulay and the modified durations of a coupon bond are respectively F r(Ia)n j + nCv jn D= F r an j + Cv jn and
F rv j (Ia)n j + nCv jn+1 MD = . F r an j + Cv jn
(Remembering the last two formulae is optional for this course.) If the current term structure {s[0,t] }t≥0 is used to evaluate the modified duration of cashflows K 1 , K 2 , . . . , Kn occurring at times 1, 2, . . . , n , respectively, then we have n
P =
K t (1 + s[0,t] )−t .
t=1
and MD = −
dP dj
P
=
n −(t+1) t=1 tK t (1 + s[0,t] ) . n −t (1 + ) K s t [0,t] t=1
Redington immunization is in place, if 1. P V A (i0 ) = P V L (i0 ); d 2. P V A (i) di
i=i0
d2 3. 2 P V A (i) di
d = P V L (i) di
i=i0
i=i0
d2 > 2 P V L (i) di
;
i=i0
.]
6
