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
Interest rate measurement
To convert from one type of compound interest/discount rate to another, one may use the relationships
m
i(m)
1+i= 1+
m
−m
d(m)
−1
=(1 − d) = 1 −
m
δ
=e .
As a result, we have
i
1+i
1
v=
1+i
δ = ln(1 + i).
d=
For a general force of interest δt ,
δt =
S 0 (t)
d
= S(t).
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 − r)/(1 + r), where r is the inflation rate.
1
2
Valuation of annuities
Present value
1 − vn
i
1 − vn
i
=
= an i
d
d
Future value
(1 + i)n − 1
i
(1 + i)n − 1
i
=
= sn i
d
d
Level annuity-immediate
an i =
sn i =
Level annuity-due
än i
s̈n i
Level continuous annuity
ān i =
m-thly payable
an i
(m)
i
1 − vn
= an i
δ
δ
1 − vn
i
= (m) = (m) an i
i
i
i
(1 + i)n − 1
= sn i
δ
δ
(1 + i)n − 1
i
=
=
sn i
i(m)
i(m)
s̄n i =
(m)
sn i
annuity-immediate
m-thly payable
1 − vn
i
= (m) an i
(m)
d
d
(m)
än i =
(1 + i)n − 1
i
= (m) sn i
(m)
d
d
(m)
s̈n i =
annuity-due
k-period deferred annuity
Perpetuity(-immediate)
Perpetuity-due
k| an i
= v k an i
1
i
1
d
än i − nv n
i
än i − nv n
i
=
= (Ia)n i
d
d
ān i − nv n
=
δ
s̈n i − n
i
s̈n i − n
i
=
= (Is)n i
d
d
s̄n i − n
=
δ
Increasing annuity-immediate
(Ia)n i =
(Is)n i =
Increasing annuity-due
(Iä)n i
(I s̈)n i
Continuously increasing
¯ ni
(Iā)
(I¯s̄)n i
continuous annuity
n − an i
i
n − an i
i
=
= (Da)n i
d
d
n − ān i
=
δ
n(1 + i)n − sn i
i
n(1 + i)n − sn i
i
=
= (Ds)n i
d
d
n
n(1 + i) − s̄n i
=
δ
Decreasing annuity-immediate
(Da)n i =
(Ds)n i =
Decreasing annuity-due
(Dä)n i
(Ds̈)n i
Continuously decreasing
(D̄ā)n i
(D̄s̄)n i
continuous annuity
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
Z n
Rt
ān δu =
h(t)e− 0 δu du dt,
0
2
and
n
Z
Rn
s̄n δu =
h(t)e
t
δu du
dt,
0
and they are related by
Rn
s̄n δu = e
0
δu du
ān δu .
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
än i =(1 + i)an i
än i =1 + an−1 i
s̈n i =(1 + i)sn i
s̈n i =sn+1 i − 1
(Da)n i + (Ia)n i =(n + 1)an i
¯ n i =nān i
(D̄ā)n i + (Iā)
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
It =iOBt−1 ,
P Rt =Kt − It ,
and
OBt =OBt−1 − P Rt
=(1 + i)OBt−1 − Kt .
Once the loan is completely payed off, OBn = 0. Thus, the total principal repaid is
n
X
P Rt =
n
X
t=1
(Kt − It ) = L,
t=1
and the total interest paid is
n
X
It =
t=1
n
X
Kt − L.
t=1
The retrospective form of the outstanding balance is
OBt = (1 + i)t L −
t
X
(1 + i)t−j Kj ,
j=1
and its prospective form is
OBt =
n
X
j=t+1
3
v j−t Kj .
When payments are level,
It =K 1 − v n−t+1 ,
P Rt =Kv n−t+1 ,
and
OBt = Kan−t i .
Under the sinking-fund method,
It =iL − jSFt−1 = iL − jL
P Rt =
and
st−1 j
,
sn j
L
(1 + j)t−1 ,
sn j
st j
.
OBt = L − SFt = 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 = Lvjn .
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 + Cvjn .
When F = C, we also have
P = F + F (r − j)an j .
Letting M = F vjn , Makeham’s formula may be obtained
r
P = M + (F − M ).
j
The price-plus-accrued at time t = k + s is
Pt = (1 + j)s Pk = v 1−s (F r + Pk+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 = Pt − sF r.
4
When amortizing a bond, we have
OBt =(Book value)t
=Cvjn−t + F ran−t j ,
It =jOBt−1 ,
and
t = 1, 2, . . . , n − 1,
t = 1, 2, . . . , n
(
F r − It ,
t = 1, 2, . . . , n − 1
P Rt =
.
F r + C − It , t = n
Using Makeham’s formula, the price of a serial bond is calculated by
r
P = M + (F − M ),
j
where M =
5
Pm
k=1
Mk and F =
Pm
k=1
Fk .
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) C0 , C1 , . . . , 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
X
v tk Ck = 0;
k=0
2. if A is the initial amount of the portfolio, B is its final amount,
and 0 < t0 < t1 < · · · < tn < 1,
Pn
then the net interest earned in the fund is I = B − [A + k=1 Ctk ] and the dollar-weighted
rate of return is
I
Pn
i=
;
A + k=1 Ctk (1 − tk )
3. if Vtk 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
i=
6
n
Y
Vtk
Vtk−1 + Ctk−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 K1 , K2 , . . . , 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
Pn
dP
tKt vjt
dj
,
D = − P = t=1
P
1+j
and the modified duration is found by
MD = −
dP
dj
P
Pn
=
t=1
tKt vjt+1
.
P
As a result, the Macaulay and the modified durations of a coupon bond are respectively
F r(Ia)n j + nCvjn
D=
F r an j + Cvjn
and
MD =
F rvj (Ia)n j + nCvjn+1
.
F r an j + Cvjn
(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
K1 , K2 , . . . , Kn occurring at times 1, 2, . . . , n, respectively, then we have
P =
n
X
Kt (1 + s[0,t] )−t .
t=1
and
dP
dj
Pn
−(t+1)
t=1 tKt (1 + s[0,t] )
.
MD = −
= P
n
−t
P
t=1 Kt (1 + s[0,t] )
Redington immunization is in place, if
1. P VA (i0 ) = P VL (i0 );
d
d
2.
P VA (i)
= P VL (i) ;
di
di
i=i0
i=i0
d2
d2
3. 2 P VA (i)
> 2 P VL (i) .]
di
di
i=i0
i=i0
6