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
e the face value of the one-year 6% bond. With notation as '
r usual, th
hporn rate a — 6:6, and a coupon rate r = 2 = 3%. At [héseplznct’lfllalsem = 2 coupons
t e ace amount (smce the bond (S a par value bond). along with the if the homer
pon amount
z “'
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$50 is withdrawn.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
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
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
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
Lectures week 10
Example: Complete the following amortization
schedule. Note that there is no payment at time 2 and
that the interest at time 3, I 3 , represents the interest
earned between time 1 and 3.
Time Payment Interest Principal Balance
0
/
1
R1
20
Lectures week 11
6.5 Bond Amortization
A bond can be thought of as a loan. The amount
borrowed is the price.
The outstanding loan balance is called the book value
and is the PV of the remaining payments using the
interest rate that the bond was purchased
Equations of value and yield rates
(22) Equations of value for investments involving a single deposit made under compound interest
(1) The accumulation function is “(1‘) = (I — -04)_', and a time 3 equation of value is $K(1 — .04)_3 = 3982‘
Therefore, K =
CHAPTER 1
The growth of money
(1.3) Accumulation and amount functions
H) In nrdcrtmlclcrminc K. use the prnpcrly A140) = K, we have Aldo) = I,000 = 10,80 K = 10’ Therefore,
100-0
"(20) Mg! .2 [_25_
I
ll
(3) ﬁrstly, uhsurvc "HI! «(0) f l) forces [1 = l. Sc
(71)common and preferred stock
(1) since Bridget purchased $200 shares and the stock pays $.28 pershare each quarter, she will receive 200x528 =
$56.00 at the end of each quarter. The price is based on an effective quarterly interest rate of J = (1 .062)%
ActSc 231
Oct 10, 2015
Here are some end-of-chapter questions (mathematical interest theory, 2nd edition) you can
practice on. No solution will be provided. Please come to instructors and TAs office hours if you
want to have a discussion. I suggest you fo
Marking Scheme:
Marks will be given for your solution to each problem. If your solution is difficult to follow (unorganized,
messy, etc.) then you will not receive full credit.
Question Score Question
Score
1. [5 pts]
6. [5 pts]
2. [5 pts]
7. [4 pts]
3. [
ActSc 231 Fall 2015 midterm 1 solution
1. (a) Show that the annual eective rate of interest for a simple discount account
is increasing over time, i.e. im > in if m > n. [4]
Sol:
The accumulation function is
a(t) =
1
0t< .
d
1
1 dt
The eective rate of int
Lectures week 4
Example: The PV of $1000 due in 2n years plus the
PV of $2000 due in 4n years is $1388.68. If
i 12 9. 6% find n.
i .096 0. 008 eff. rate of int. per month
12
1388. 68 1000 1. 008 24n 2000 1. 008 48n
0 2000X 2 1000X 1388. 68
where X 1. 008
Lectures week 5
Approximate Dollar-Weighted Yield over 1 year
Let A opening balance at t 0
Let B closing balance at t 1
Let C t k net contribution at time t k , where
0 t1 t2 tn 1
I
i
n
k1 C tk 1t k
A
Approximate Dollar-Weighted Yield over 2 years
Let
ACTSC 231 Mathematics
of Finance:
Course overview
Chapter 1 The growth of money
Accumulated value
Present Value
Simple Interest
Compound Interest
Rates of interest
Speeds can be expressed using different units (100
km/hr, 60 mph, 27.78 m/s, 1 min./m
Lectures week 3
Graphing at 1 i t
i m
m
at
1
m
at
at
1
t m
1 i t
1 i
1 i t
1 i
1
m
1
d m
m
1
at m at
1
at m
1 1 i
1
m
Mark the following on the graph:
1 , d 2 , i 4 , i 2 , i, 2i 1
2
d
2
4
2
i 1
1
i
1
2
i 2
2
i 4
4
2i
1 i 1
1
4
1 i 1
1
2
d
d 2
2
Lectures Week 2
1
a 1 t at is called the discount function or present
value function. It moves money from time t to time 0.
What if we want to move money from time t 1 to time
t2?
First move it from t 1 to 0 and then from 0 to t 2
Xa 1 t 1 at 2
Example:
Lectures week 12
Chapter 9 Asset-Liability Management
In an ideal world we would pick our assets, such that
the inflow of income from these assets exactly
matches our outflow of expenses and other liabilities.
This is called exact matching and is not alwa
ActSc 231
Nov 16 and Nov 18 2015
6.2 Basic Price Formula
1. A $4000 bond is redeemable at 103 on Nov 10, 2025. It pays semi-annual coupons at i(2) = 5%.
Determine the price of the bond on May 10, 2010, to yield i(2) = 4%.
Sol: F = 4000, C = 4000 1.03 = 41
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Examples for Chapter 6
ACTSC 231 Mathematics of Finance
Department of Statistics and Actuarial Science
University of Waterloo
Instructor: Fan Yang
Fan Yang ( UWaterloo )
Chapter 5
1/7
1. A 10,000 par value 10-year bond with 8% annual coupons is bought
at
Examples for Chapter 7
ACTSC 231 Mathematics of Finance
Department of Statistics and Actuarial Science
University of Waterloo
Instructor: Fan Yang
Fan Yang ( UWaterloo )
Chapter 7
1/6
1. (SOA, May, 2003, #26) 1000 is deposited into Fund X, which earns
an
Examples for Chapter 5
ACTSC 231 Mathematics of Finance
Department of Statistics and Actuarial Science
University of Waterloo
Instructor: Fan Yang
Fan Yang ( UWaterloo )
Chapter 5
1/6
1. (SOA, Nov. 00, #12) Seth borrows X for four years at an annual
effec