cHAPTER 4
Annuities with different payment and
conversion periods
(4.2) Level annuities with payments less frequent than each interest period
(1) (a) The effective monthly interest rate is 4.85% / 12 and there are three months in a quarter. so the quarter
Assignment 1
ACTSC 231
Winter 2015
Due on Thursday January 28, 2015 by the end of the class.
1. The problem is to prove that limn!1 nrn = 0 for
1 < r < 1. Do this in three steps:
(a) For r = 0, the result is obvious.
(b) For 0 < r < 1, use l
Hospital rule
Tutorial Problems
Tutorial 2
Q1. On July 1, 2014, a person invested $ 700 in a fund for which the force of interest at
time t is given by t = 0.05 + 0.02t, where t is the number of years since january 1,
2014. Determine the accumulated value of the invest
ACTSC 231 (Mathematics of Finance) Spring 2015
Midterm Review Formula Sheet
The function a(t) is dened as the accumulated value (AV) of the fund at time t of an
initial investment of $1, and is called Accumulation Function.
The amount function, AK (t),
Tutorial Problems
Tutorial 4
Q1. A annuity pays $975 at the end of each quarter for 13 years. The payments are made
directly to a saving account with a nominal interest rate of 2.88% payable monthly,
and they are left in the account. Find the balance in t
Lectures week 8
4.6 Continuous Annuities
1
Notation a n| lim m m a mn|i im
m
1
v e and ln1 i
1v n
, where
This is the PV of an n-year annuity which pays
continuously at a rate of $1 per year.
Example: Find the PV of an annuity which pays
continuously at
ACTSC 231 (Mathematics of Finance) Spring 2015
Midterm 2 Review Formula Sheet
n1
a(rn 1)
a(1 rn )
=
.
ar =
1r
r1
i
Formula for sum:
i=0
e.g. If |r| < 1, then limn rn = 0 and
ari =
i=0
rst term
a
=
1r
1 common ratio
Formulas for level annuities:
1 vn
a
Tutorial Problems
Tutorial 1
Q1. Rank the following interest rates from lowest to highest: i, i(2) , i(12) . Suppose i =
11.62%, calculate the other values.
Q2. Given the amount function A(t) = 1, 100(1 + 0.16t), nd the eective rate of discount
in the 8-t
Tutorial Problems
Tutorial 3
Q1. Esteban borrows $ 19,834, and the loan is governed by compound interest at an annual
eective interest rate of 4%. Esteban agrees to repay the loan by making a payment of
$8,000 at the end of T years and a payment of $ 15,0
ACTSC 231 (Mathematics of Finance)w Spring 2015
Midterm Review Problem Set (2)
1. An annuityimmediate has 20 annual payments. The odd numbered payments are
$100 and the even payments are $200. Find the present value of this annuity, under
the following ra
ACTSC 231 (Mathematics of Finance) Spring 2015
Midterm Review Problem Set
Please do. ms FE)de ovum iWVMiiarl HLPB. \Taf Askew
Wamg, VOW/ix
ag 45w Moe/5+: W ; W W nummk
M - w! \ N' \GIXIAX .
oxnbNQ/(Sl? (chumulatigl? mtion 2 Q1110 nt Function) Suppose that
7/16/2015 Maple TA
Back Close
B/MapleIA;
TM rcm'm &
Description:Finding a band's price
Grade: 0.0
An eight-year $2,000, 6% bond with semiannual coupons and redemption value $1,800 is
bought at a price to give the investor an annual effective yield rate o
1. Rank the following interest rates from lowest to highest.
1: 11.62%
1(2) = 11.36%
102) = 11.07%
2. Given the amount function A(t) = 1,100 (l + 0.16 t), find the effective rate of discount
in the 8 " year.
3. At a certain rate of simple interest $2,000
ACTSC 231 (Mathematics of Finance) Spring 2015
Midterm Review Problem Set
1. (Accumulation Function & Amount Function) Suppose that an account is governed by a quadratic accumulation function a(t) = t2 + 0.1t + and the eective
interest rate i2 for the sec
ACTSC 231 (Mathematics of Finance) Spring 2015
Midterm Review Problem Set (2)
1. An annuity-immediate has 20 annual payments. The odd numbered payments are
$100 and the even payments are $200. Find the present value of this annuity, under
the following ra
Tutorial Solutions
Tutorial 2
Q1. The equation of value:
19, 834 8, 000v T 15, 000v 2T = 0
Let x = v T
= 15x2 + 8x 19.834 = 0
8 64 4 15 19.834
= x =
2 30
Since x = v T > 0, discarding the negative root we have v T = 0.9137. Hence,
T =
log(0.9137)
= 2.3.
l
Lectures week 7
3.9 Payments in Arithmetic Progression
Payments starting at P and each subsequent
payment is Q more.
PV Pa n| Qv 2 2Qv 3 n 1Qv n
(1)
1 iPV 1 iPa n| Qv 2Qv 2 n 1Qv n1 (
1 2
iPV iPa n| Qv v 2 v n nQv n
iPa n| Qa n| nQv n
a n| nv n
PV Pa n|
Lectures week 9
5.2 Amortization of a Debt
Consider a loan of amount L which is repaid by n
payments of R at the end of each period.
L Ra n|i
Let B t be the outstanding loan balance at time t, so
B 0 L Ra n|i
The interest due at time t is I t iB t1
If the
Lectures week 6
3.3 Annuities Due
Payments of R at the beginning of each period for n
periods.
PV
R n| R Rv Rv 2 Rv n1
1 vn
R
1v
1 vn
R
d
AV
Rs n| R1 i n R1 i n1 R1 i
R1 i 1 i n1 1 i n2 1 i 1
R1 is n|
1 i n 1
R1 i
i
1 i n 1
R
d
a 5|
6| 1 a 5|
n1| 1 a
ActSc 231
Oct 26 and Oct 28 2015
3.8 Payments in Geometric Progression
1. Determine the discounted value of a series of 20 annual payments of $500 if i(1) = 5% and we
want to allow for an ination factor of r(1) = 2%. (assume the payments are at the end of
ActSc 231
Oct 26 and Oct 28 2015
3.8 Payments in Geometric Progression
1. Determine the discounted value of a series of 20 annual payments of $500 if i(1) = 5% and we
want to allow for an ination factor of r(1) = 2%. (assume the payments are at the end of
ActSc 231
Oct 14 and Oct 16 2015
Examples
1. Mr. Haywood deposits $100 in a fund on Jan 1. On Mar 1, the account balance is $102 and then
$50 is withdrawn. The account balance becomes $52.5 on May 1 and Mr. Haywood deposits $50. At
the end of year Mr. Hay
ActSc 231
Oct 19 and Oct 21 2015
3.3 Annuities due
1. A couple wants to accumulate $10,000 by Dec 31, 2015. They make 10 annual deposits starting Jan 1, 2006. If the annual eective rate of interest is 2%, what annual deposits are needed?
Sol: The rst paym
ActSc 231
Oct 19 and Oct 21 2015
3.3 Annuities due
1. A couple wants to accumulate $10,000 by Dec 31, 2015. They make 10 annual deposits starting Jan 1, 2006. If the annual eective rate of interest is 2%, what annual deposits are needed?
2. Starting on hi
ActSc 231
Nov 23 and Nov 25 2015
8.3 Term Structure of Interest
1. Consider the following $1000 zero coupon bonds:
Term (in year) Price
1
934.58
2
865.35
3
797.58
Find their yields (i.e. spot rates) and use them to calculate the price of a $500 bond that