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Question
A recurrence relation is an equation that recursively defines a sequence of values, whereby
each element of a sequence can be written as a function of preceding element(s);
the first element of the sequence will be uniquely defined by an initial value of the
recurrence relation.
Specifically, if a sequence un can be expressed as a function of only n and the immediate
preceding element un−1 , i.e., un = g(n, un−1 ), then we say that un is a recurrence relation of
order 1. The values of the entire sequence can be calculated recursively starting from the initial
value say u1 and then by u2 = g(2, u1 ) and more generally un = g(n, un−1 ) for n = 3, 4, 5, · · · .
(a) Write down a recurrence relation for än . Explain your thought process in words (e.g.,
using the timeline approach) OR prove the result mathematically by first principles. Also
write down the initial value for the sequence.
(4 marks)
(b) Given an effective discrete periodic rate of 3% per period, tabulate the values of än for
n = 1, 2, · · · , 30.
(3 marks)
(c) Repeat parts (a) and (b) for an instead of än .
(5 marks)
(d) Repeat parts (a) and (b) for (Iä)n instead of än .
(8 marks)
Suppose that for calculation purpose we only have the three tables of values obtained above,
calculate the following by making use of at least one of these three tables:
(e) A loan of $480,000 is to be fully repaid by 24 level annual repayments made in arrears
at an effective annual rate of 3% per annum. Calculate the amount of the level annual
repayment.
(4 marks)
(f) A loan of $350,000 is to be fully repaid by level semi-annual repayments made in arrears
for the next 10 years. The annual percentage rate charged for this loan is 6.09%. Calculate
the amount of the capital repayment between the 5th and the 9th year.
(4 marks)
(g) Calculate the present value at time 0 of a 16-year continuous annuity with a payment
rate of $2,000 per annum under a constant force of interest of 0.0295588.
(2 marks)
Calculate the following using at least two of the above constructed tables:
(h) Given an effective annual rate of 3%, calculate the present value at time 0 of a 15-year
arithmetically increasing annuity immediate whereby the first annual payment is $2,452
and subsequent annual increment is $450.
(4 marks)
(i) Consider a stream of 30-year continuous cash flows with a payment rate of $404t during
year t. For instance, the payment rate during year 5 is $2,020. Calculate its present value
at time 0 given an effective annual rate of 3%.
(6 marks)
2
Calculate the following using all of the above constructed tables:
(j) Consider a special annuity of a duration of 21 years with the following cash flows:
The payment rate during year 3k + 1 (i.e., between time 3k and time 3k + 1) is
$(80 + (k − 1) × ä3 |i=0.00990163 ) per annum.
The annual cash flow made in arrears in year 3k + 2 (i.e., at time 3k + 2) is $81.
Given an effective annual rate of 0.990163% and that the value of ä3 |i=0.00990163 is 2.97068247,
calculate the present value of this special annuity at time 0.
(10 marks)
3
ACST8081 Financial Mathematics
Lecture Notes 3
Learning Objectives
• calculate the present value and the future value of a sequence of equal or unequal discrete/continuous cash flows at any point in time using constant/varying specific rates
(nominal, effective, force) of interest/discount.
• construct the cash flows model of simple annuity certain contracts, including annuity
immediate, annuity due, continuous annuity, deferred annuity and perpetuities.
• define simple annuity certain functions an , sn , än , s̈n , an , sn , m| an , m| än , m| an
• derive the formulae above in terms of i, v, d, δ, n as well as derive the relationships between
these annuity functions.
Chong It Tan
1
1
Valuation of Cash flows
In this lecture we study the valuation of cash flows. In particular, we are interested in finding
the present values and future values of both discrete and continuous cash flows.
1.1 Discrete Cash flows
For a sequence of discrete cash flows ct1 , ct2 , . . . , ctn due at times t1 , t2 , . . . , tn , the present value
of these cash flows at time t is
X
P Vt =
ctj v(t, tj )
(1)
j:tj ≥t
Note that the rate of interest/discount is allowed to vary over time, as captured by the general
expression of discount factor v(t, tj ). If the rate of interest/discount is constant over the entire
period, that is, v(t, tj ) = v tj −t , the present value at time t can be simplified to
X
P Vt =
ctj v tj −t
(2)
j:tj ≥t
The future value of these cash flows at time T is
−1
X
X
F VT =
ctk A(tk , T ) =
ctk v(tk , T )
k:tk ≤T
(3)
k:tk ≤T
If we pick a time t that satisfies t ≤ tk ≤ T , from the principle of consistency we can write
v(t, T ) = v(t, tk )v(tk , T )
−1
−1
⇔ v(tk , T )
= v(t, tk ) v(t, T )
so equation (3) can be re-expressed into
−1
X
F VT =
ctk v(t, tk ) v(t, T )
k:tk ≥t
= A(t, T )
X
ctk v(t, tk )
k:tk ≥t
= A(t, T )P Vt
(4)
which can be simplified to
F VT = (1 + i)T −t P Vt
(5)
when the rate of interest is constant over the entire period, i.e., A(t, T ) = (1 + i)T −t .
1.2 Continuous Cash flows
For the case of continuous cash flows, suppose the rate of payment at time s is denoted by
ρ(s) for T1 ≤ s ≤ T2 , the present value of these cash flows at time t ≤ s ≤ T2 is
Z T2
P V (t, T2 ) =
ρ(s)v(t, s) ds
t
Z T2
Z s
=
ρ(s) exp −
δ(u) du ds
(6)
t
t
Note that the force of interest δ(u) is allowed to vary over time. If the force of interest is
constant over the entire period, that is, δ(u) = δ, the present value at time t can be simplified
to
Z
Z
T2
ρ(s)v s−t ds
ρ(s) exp(−δ(s − t)) ds =
P V (t, T2 ) =
t
Chong It Tan
T2
(7)
t
2
The future value of these cash flows at time T ≥ s ≥ T1 is
Z T
ρ(s)v(s, T )−1 ds
F V (T1 , T ) =
T1
T
Z
=
ρ(s) exp
Z
T1
T
δ(u) du ds
(8)
s
Since
Z
T
Z
s
δ(u) du =
T1
Z T
⇔
T
Z
δ(u) du +
T1
Z T
δ(u) du
s
Z
s
δ(u) du
T1
so equation (8) can be re-expressed into
Z T
Z
ρ(s) exp
F V (T1 , T ) =
T1
= exp
T
T1
T
= exp
T1
T
Z
Z
s
δ(u) du −
T1
Z
s
δ(u) du −
δ(u) du =
δ(u) du ds
T1
Z
δ(u) du
T
Z
s
ρ(s) exp −
T1
δ(u) du ds
T1
δ(u) du P V (T1 , T )
(9)
T1
which can be simplified to
F V (T1 , T ) = eδ(T −T1 ) P V (T1 , T )
(10)
under a constant force of interest δ(u) = δ.
2
Annuities I
In this section we consider a sequence of periodic cash flows (i.e., occur at fixed periodic time
points) called annuity. Practical examples include monthly mortgage installments, monthly
insurance premiums, regular pension benefits and regular savings.
We omit the possibilities of uncertainties associated with the cashflows and assume that the
payment amounts are guaranteed at known timings. This type of annuity is referred to as
annuity certain. An annuity in which each regular payment is 1 is known as a unit annuity.
2.1 Annuity Immediate
An annuity immediate (also known as ordinary annuity) is an annuity in which each payment
is made in arrears at the end of the period. Consider a sequence of n periodic payments of
1 payable at times t + 1, t + 2, . . . , t + n. The present value of these payments at a period
before the first payment is made1 (i.e., at time t) is denoted by an i (read a-angle-n-at-i),
or simply an when i is understood. Its formula under an effective interest rate of i 6= 0 can be
derived as follows
n
X
2
n
an = v + v + · · · + v =
vs
s=1
1 − vn
1 − vn
=v
=
1−v
i
(11)
1
Since the first payment of an annuity immediate occurs at the end of the first period, we are finding the
present value at the beginning of the first period.
Chong It Tan
3
Figure 2.1: PV of Annuity Immediate
vn
v n−1
..
.
v2
v
t
1
1
t+1
t+2
1
1
t+n−1 t+n
An annuity in which the payments continue infinitely is called a perpetuity. The PV of a unit
perpetuity immediate can be found via
1
1 − vn
=
n→∞
i
i
(12)
a∞ = lim
Alternatively, we may write
a∞ = v + v 2 + · · · =
∞
X
vs
s=1
1
1
=v
=
1−v
i
The future value of these payments at the time the last payment is made (i.e., at time
t + n) is denoted by sn i ,
n−1
sn = (1 + i)
n−2
+ (1 + i)
+ ··· + 1 =
n−1
X
(1 + i)s
s=0
n
n
(1 + i) − 1
(1 + i) − 1
=
(1 + i) − 1
i
n
1
−
v
= (1 + i)n
= (1 + i)n an
i
=
Chong It Tan
(13)
(14)
4
Figure 2.2: FV of Annuity Immediate
(1 + i)n−1
(1 + i)n−2
..
.
(1 + i)
t
1
1
t+1
t+2
1
1
t+n−1 t+n
Example 2.1. The PV of an annuity immediate with 50 payable annually for 10 years when
the interest rate is 7% per annum compounded annually is
1 − 1.07−10
50a10 i=0.07 = 50
= 351.18
0.07
Example 2.2. The FV of an annuity immediate with 300 payable quarterly for 5 years when
the interest rate is 8% per annum compounded annually is
1.085 − 1
= 7247.73
300s20 i=1.080.25 −1 = 300
1.080.25 − 1
Exercise 1
Given a nominal interest rate of 6% compounded semi-annually, calculate
(a) the FV of an annuity immediate with 100 payable monthly for 3 years.
(b) the PV of an annuity immediate with 1 payable daily forever.
2.2 Annuity Due
An annuity due is an annuity in which each payment is made in advance at the beginning of the
period. As before, we consider a sequence of n periodic payments of 1 except they are payable
at times t, t + 1, . . . , t + n − 1. The present value of these payments at the time the first
payment is made (i.e., at time t) is denoted by än i (read a-due-angle-n-at-i), or simply än
Chong It Tan
5
when i is understood. Its formula under an effective interest rate of i 6= 0 is given by
2
än = 1 + v + v + · · · + v
n−1
=
n−1
X
vs
s=0
1 − vn
1 − vn
=
=
1−v
d
n
i
i 1−v
= an = (1 + i)an
=
d
i
d
(15)
(16)
Figure 2.3: PV of Annuity Due
v n−1
..
.
2
v
v
1
1
1
1
t
t+1
t+2
t+n−1 t+n
Alternatively, a unit annuity due can be decomposed as the summation of
• 1 payable right now (at the beginning of the first period).
• an annuity immediate that starts right now (with the first payment at the end of the
period) with n − 1 payments.
so we have
än =
n−1
X
s
v =1+
s=0
n−1
X
v s = 1 + an−1
(17)
s=1
The PV of a unit perpetuity due can be found via
1 − vn
1
=
n→∞
d
d
ä∞ = lim
(18)
Alternatively, we may write
2
ä∞ = 1 + v + v + · · · =
∞
X
vs
s=0
1
1
=
=
1−v
d
Chong It Tan
6
The future value of these payments at a period after the last payment is made (i.e., at
time t + n) is denoted by s̈n i ,
n
s̈n = (1 + i) + (1 + i)
n−1
n
X
+ · · · + (1 + i) =
(1 + i)s
s=1
n
n
(1 + i) − 1
(1 + i) − 1
=
(1 + i) − 1
d
n
(1 + i) − 1
= (1 + i)
= (1 + i)sn
i
1 − vn
= (1 + i)n
= (1 + i)n än
d
(19)
= (1 + i)
(20)
(21)
Figure 2.4: FV of Annuity Due
(1 + i)n
(1 + i)n−1
(1 + i)n−2
..
.
(1 + i)
1
1
1
t
t+1
t+2
1
t+n−1 t+n
Example 2.3. The PV of an annuity due with 50 payable annually for 11 years when the
interest rate is 7% per annum compounded annually is2
1 − 1.07−11
=
401.18
=
50
+
351.18
=
50
1
+
a
50ä11 i=0.07 = 50
10
0.07
1.07
Example 2.4. The FV of an annuity due with 420 payable weekly for a year when the effective
quarterly interest rate is 3% is
!
1.034 − 1
420s̈52 i=1.03 524 −1 = 420
= 23209.92
1
1.03 13 −1
1
1.03 13
2
The value of 351.18 below comes from Example 2.1.
Chong It Tan
7
Exercise 2
Show that an = än = sn = s̈n = n
for i = 0.
2.3 Continuous Annuity
The payments in a continuous annuity are made continuously over infinitely small intervals
of dt. Consider a stream of continuous payments of 1 (also called a payment rate of 1) from
time t to time t + n, that is, the payment over the interval [t, t + dt] is dt. The present value
of these payments at the beginning of the period (i.e., at time t) is denoted by an i (read
a-bar-angle-n-at-i), or simply an when i is understood. Its formula under an effective interest
rate of i 6= 0 can be derived as follows
Z t+n
v(t, s) ds
an =
t
Z t+n
Z t+n
s−t
e−δ(s−t) ds
v ds =
=
t
t
−δ(s−t) t+n
=
−e
δ
t
1 − e−δn
1 − vn
=
=
δ
δ
i 1 − vn
i
=
= an
δ i
δ
d 1 − vn
d
=
= än
δ d
δ
The PV of a unit continuous perpetuity can be found via
(22)
(23)
(24)
1 − vn
1
=
(25)
n→∞
δ
δ
The future value of these continuous payments at the end of the period (i.e., at time t + n)
is denoted by sn i ,
Z t+n
sn =
(1 + i)t+n−s ds
t
Z n
u=s−t
eδ(n−u) du
=
a∞ = lim
0
δ(n−u) n
=
−e
δ
0
e −1
(1 + i)n − 1
=
=
δ
δ
n
1
−
v
= (1 + i)n
= (1 + i)n an
δ
δn
(26)
(27)
Example 2.5. Given an effective annual rate of i = 0.09, the PV of a unit continuous annuity
for a period of 15 years is
1 − 1.09−15
a15 i=0.09 =
= 8.42
ln 1.09
Chong It Tan
8
Exercise 3
Given an effective annual rate of i = 0.04, calculate the PVs of a one-year unit annuity
immediate, annuity due and continuous annuity.
2.4 Deferred Annuity
A deferred annuity is an annuity in which the payment starts in the future after a deferred
period. Consider a sequence of n periodic payments of 1 payable in arrears after a deferred
period of length m at times t + m + 1, t + m + 2, . . . , t + m + n. The present value of these
payments at time t is denoted by m| an i (read m-pipe-a-angle-n-at-i), or simply m| an when i is
understood. Its formula under an effective interest rate of i 6= 0 can be derived as follows
m| an
=v
m+1
+v
m+2
+ ··· + v
m+n
=
m+n
X
vt
t=m+1
= vm
n
X
v t = v m an
(28)
t=1
Also, we can express
m| an =
m+n
X
vt −
t=1
m
X
vt
t=1
= am+n − am
(29)
which implies that a deferred annuity is equivalent to the difference of two annuities immediate
with different number of payments. Moreover,
am+n = am + m| an = am + v m an
(30)
means that an annuity immediate with m + n payments can be decomposed into an annuity
immediate with m payments and an m-period deferred annuity immediate with n payments.
Figure 2.5: Deferred Annuity Immediate
t
1
···
1
1
···
1
t+1
1
···
1
1
···
1
t+m t+m+1
t+m+n
Example 2.6. The PV of a 3-year deferred annuity immediate of 500 payable annually for 7
years at an effective annual rate of i = 10% is
3| a7 i=0.1
= v 3 a7 = 1.1−3
1 − 1.1−7
= 3.66
0.1
Chong It Tan
9
Similarly, the following formulae hold for a corresponding m-period deferred unit annuity due
with n payments, in which its present value is denoted by m| än i ,
m| än
m
=v +v
m+1
+ ··· + v
m+n−1
=
m+n−1
X
vt
t=m
= vm
m| än
=
n−1
X
t=1
m+n−1
X
v t = v m än
t
v −
t=0
m−1
X
(31)
vt
t=0
= äm+n − äm
(32)
Finally, the following formulae hold for a corresponding m-period deferred unit continuous
annuity with n periods of continuous payments, in which its present value is denoted by m| an i ,
Z
m| an
t+m+n
=
e
−δ(s−t)
ds
u=s−t−m
Z
=
t+m
n
e−δ(u+m) du
0
Z
n
= e−δm
e−δu ds = v m an
0
Z t+m+n
Z t+m
−δ(s−t)
e
ds −
e−δ(s−t) ds
m| an =
t
(33)
t
= am+n − am
(34)
2.5 Summary: It’s All about Perpetuities
From equation (29), we see that a deferred annuity is equivalent to the difference of two
annuities immediate with different number of payments. We can also write
am = am+n − m| an
(35)
Substituting n → ∞ and re-writing m as n, we obtain
an = a∞ − n| a∞ =
1
1
1 − vn
− vn =
i
i
i
(36)
which means that an n-period annuity immediate is the difference between a perpetuity immediate today and an n-period deferred perpetuity immediate.
Exercise 4
Construct an m-period deferred n-period unit annuity immediate in terms of perpetuities
due by drawing a timeline. Hence, derive the present value of the cash flows involved at
the current time.
Since the present values of annuities an , än , an are interrelated as shown earlier in equations
(16), (23) and (24), and their corresponding future values can be found by simply applying
compounding factors, these imply that all the formulae presented so far can be expressed in
Chong It Tan
10
terms of the present value of a perpetuity immediate a∞ = 1i , a discount factor v n = (1 + i)−n
with relevant n, as well as the corresponding value(s) of i, d, δ.
More importantly, as we shall see in the next lecture, any complex annuity cash flows can be
decomposed into the summation or the difference of multiple sets of suitably defined simple
annuity functions, which in turn are just differences of deferred perpetuities.
Exercise 5
By using the inequality of d < δ < i, show that an ≤ an ≤ än and provide an intuitive
argument for the ranking.
Chong It Tan
11
...

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