78
CHAPTER 4. ANNUITIES
Solutions
4.1
l56 + vl57 + + v 4 l60
= 4.451
l56
l[56] + vl57 + + v 4 l60
=
= 4.463
l[56]
a
56:5 =
a
[56]:5
4.2
ax = ax:n + n Ex ax+n =
n Ex
=
1
4
a
x:n = ax:n n Ex + 1 = 18.75
4.3
(i)
(ii)
ax = vpx (1 + ax+1 )
Note that with a one
13.5. EXERCISES
225
Exercises
13.1
(i) Define the following symbols in words, and give a formula in terms of an integral for each
of them:
(a) A
1
xy
(b) A 2
xy
(c) ayx
(ii) Consider the following sets of payments:
(1) 1 immediately on the death of (y) i
226
CHAPTER 13. REVERSIONARY ANNUITIES
13.5
A singlepremium policy provides the following benefits to a husband and wife each aged 40.
(1) An annuity of 5,000 per annum, payable continuously, commencing on the husbands
death within 25 years, or on his su
13.3. WIDOWS (OR SPOUSES) PENSION ON DEATH AFTER RETIREMENT
223
Men who are not married at age 65 need not buy spouses pension, but married men must buy
this. Calculate Mr Browns expected monthly pension if
(i) he is assumed to be single at age 65; and
(i
13.6. SOLUTIONS
229
Hence equation of value is
P = 24919 + 0.003719P
Therefore P = 25, 012.
13.6
Let annual premium be P .
Value of benefits = 10, 000A m
f
70:64
+ 5000(
am
f
7064
= 10, 000(1
am
f
70:64
' 10, 000[1 (a m
f
+a
f
70
f
Hence P =
23773.85
=
220
CHAPTER 13. REVERSIONARY ANNUITIES
Proof
Z
a
xy =
0
Z
=
v t t py (1 t px ) dt
f (t)g 0 (t) dt
0
where
f (t) = 1 t px = tRq x (which is such that f 0 (t) = t px x+t ) and
g(t) = t 
ay = t v r r py dr (which is such that g 0 (t) = v t t py )
Using int
Chapter 13
REVERSIONARY ANNUITIES
13.1
Reversionary Annuities Payable Continuously
Consider an annuity of 1 p.a. payable continuously to (y) after the death of (x). The present value
of this reversionary annuity is
(
T
if U > T
a
U a
Z=
0
if U T
where
T
U
228
CHAPTER 13. REVERSIONARY ANNUITIES
(c) value of benefit is value of
(1) an annuitycertain (of 3000 p.a., for 6 years) beginning at end of year of death of (x), plus
(2) 1500 p.a., payable at times t (t 7) if (y) alive and (x) dead 6 years previously
13.3. WIDOWS (OR SPOUSES) PENSION ON DEATH AFTER RETIREMENT
221
When m = 1, it may be omitted, giving
axy = ay axy
By Woolhouses formula
m1
m1
) (axy +
)
2m
2m
= ay axy
= axy .
(m)
axy ' (ay +
An alternative approach is to regard this reversionary annu
13.6. SOLUTIONS
227
Solutions
13.1
(i) (a) The m.p.v. of 1 payable immediately on the death of (y), if this occurs before that
of (x).
Z
A 1=
v t t pxy y+t dt
xy
0
(b) The m.p.v. pf 1 payable immediately
Z on the death of (x) if this occurs after that of
224
13.4
CHAPTER 13. REVERSIONARY ANNUITIES
Actuarial Reduction Factors
Suppose that, in a pension policy or scheme, the rules state that a reduction applies to the normal
widows pension if wife is more than 10 years younger than her husband. (This rule m
222
CHAPTER 13. REVERSIONARY ANNUITIES
where
y = average age of wife of member aged 65
= 65 d, where d = age difference between husband and wife
(approximately 3 years in practice)
Hence the value of widows pension for each member retiring at age 65 (mari
4.4. DEFERRED ANNUITIES
67
Proof.
(i)
By Maths. in Finance, a
mincfw_K+1,n =
1 v mincfw_K+1,n
.
d
Take expected values to obtain
a
x:n =
1 Ax:n
d
which gives the required result.
(ii)
Take expected values in the equation
a
mincfw_T,n =
4.4
1 v mincfw_T,n
96
CHAPTER 6. RESERVES
If expenses may be ignored, we have
tV
= M.P.V of future benefits
M.P.V of future premiums
(6.2.4)
The mortality, interest and expense assumptions used to evaluate t V are known as the reserving
basis. This may or may not agree wit
94
CHAPTER 5. PREMIUMS
(b) (1)
(2)
E[g(T )] = P a
x:n S Ax:n
1 h(T )
Write g(T ) = P
Sh(T )
(
vT
where h(T ) =
vn
=
P
Var[g(T )] =
=
if T < n
if T n
P
+ S h(T )
P
+S
P
+S
2
Var[h(T )]
2 h
2 i
Ax:n Ax:n
where indicates the rate of interest 2i + i2 p.a.
S
70
CHAPTER 4. ANNUITIES
Temporary mthly annuities
Define
(m)
a
x:n = M.P.V. of 1 p.a. payable mthly in advance to (x)
for at most n years
=a
(m)
n 
a(m)
x
x
D
x+n
(m)
=a
(m)
a
x
Dx x+n
m1
Dx+n
m1
+ a
x
a
x+n
2m
Dx
2m
m1
Dx+n
=a
x:n
1
2m
Dx
cfw_z
4.7. VARYING ANNUITIES
75
Define
(I
a)x:n = M.P.V. of payments of t + 1 at time t
(t = 0, 1, 2, . . . , n 1), provided that (x) is
then alive
"n1
#
n1
X
X
t
=
(t + 1)v t px =
(t + 1)Dx+t /Dx
t=0
t=0
Dx + 2Dx+1 + + (n 1)Dx+n2 + nDx+n1
=
Dx
(Dx + 2Dx+1 + .
5.8. FAMILY INCOME BENEFITS
89
Try n = 7
f (n) = 6.12136 +
1, 289.7567
7.772 11.2903
2, 144.171
<0
Try n = 6
f (n) = 5.34626 +
1, 401.6093
8.112 9.6774
2, 144.171
>0
Hence 6 < n < 7
5.8
Family income benefits
A family income benefit of term n years is a
4.7. VARYING ANNUITIES
final
payment
(on death of (x) ) 6
1
1
0
71
ax = ax +
2
3
Z
X
1
t

t=0
v t px
cfw_z
0
4
.
.
.
.
 time (years)
rv r r px+t x+t+r dr
cfw_z

pure endowment
factor to age x + t
value at time t
of death benefit
in year t + 1
By U.D.
4.7. VARYING ANNUITIES
Ia
Ia
v t t px dt
t
x
x
0
v t t px dt
0
ax
1
2
1 jax
Nx
2 jax
Nx+1
Dx
.
.
Sx
Dx
Sx
Nx+t
t=0
x
bt
t
t
, , ,.
K
vt b t
g K
t=0
K
Eg K
b
>
dx+1
g k k jqx
k=0
lx
lx
fg
dx
g
fg
dx
dx+1
g
g
lx
Eg K
b
lx
g
g
vb
.
dx+1
dx+2
lx+1
vt
bt
t=0
t
Career Paths in Finance*
For someone new to the world of finance, understanding the available career pathsand
corresponding career opportunitiescan be confusing. This is especially true for those in the middle of
the finance recruiting season, since decis
Financial management application & strategy
American
Grain
Company
Equipment Replacement
Produced by
Weiju Lin
Yihan Chen
Background:
American Grain Company (AGC) is a multinational firm with total revenues in excess of $5 billion per
year. The firm is
Home Search Assignment Due: Wed. Dec. 7th by midnight
Find a home ad that is priced at least 4% under 2 times the most recent SEV value
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MKT 432 and MKT 536 term project
Term Paper and PPT
Group project
Paper length: approximately 812 doublespaced pages if done if a group.
Group size: 23 people (see professor for deviations in class).
DUE: per syllabus or TBA
This is marketing, so more
MKT338.W1
Professor Banerjee
October 3, 2016
Student name: WEIJU, LIN
PRODUCT CATEGORY: SMART PHONES
PART B: What are the different motivations consumers have to buy that category of
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The use of smartphones has seen a steady and rapid increase o
STIMULATION STOCK
GAME
Kenny Lin portfolio
Weiju Lin (FIN269)
RANK 57From September 9th 2016 to December 7th 2016
iThe most profitable stock in my portfolio is JNJ. I bought it twice. First time is on
September 29th and I sold it on October 10th. The buy
121
Investing in Stocks
Chapter Objectives
1. Identify the most important features of
common and preferred stock
2. Explain how you can evaluate stock
investments
3. Analyze the numerical measures that cause
a stock to increase or decrease in value
4. De
131
Investment in Mutual Funds
Chapter Objectives
1. Describe the characteristics
of mutual fund investments
2. Classify mutual funds by
investment objectives
3. Evaluate mutual funds
4. Describe how and why mutual
funds are bought and sold
132
What is
The National Taiwan Universitys accounting is ranked 21st position of top
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Case two: Replacement Project Analysis
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The Innovative Sporting Goods Company was founded in 1975 in
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and Beyond: Testing
the Commute of the
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Student name: Weiju, Lin
Public transportation
A new subway line is nearly ready
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How do people get to and from
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Use the picture named ch3 HW pic to answer the following questions. Assume the output is for a
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If the profit contribution (objective function coefficient) for variable E increases from 63 to 69, will the
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