n-year geometric increasing annuity
0
2
1
$K
3
n
$K(1+r) $K(1+r)2
PV
Imagine that payments increase every period by a
ratio (1+r). (beware - notation is inconsistent
between economics, business, math and actuarial
textbooks, especially once we get into inflation and
real rates of interest). Also, let’s have the first
payment of amount K (not yet increased though it’s
a year after the date at which we take the PV0). Use
PV0= K { 1/(1+i) + (1+r)/(1+i)2 + (1+r)2/(1+i)3
+…+ (1+r)n-1/(1+i)n}
= [K/(1+i)]
=K
1 – (1+𝑟)𝑛 /(1+𝑖)𝑛
1−(1+𝑟)/(1+𝑖)
1 – (1+𝑟)𝑛 /(1+𝑖)𝑛
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𝑖−𝑟
Geom increasing annuity in actuarial notation
0
2
1
$K
3
n
$K(1+r) $K(1+r)2
PV
Being careful about the first term, rewrite as
PV0 = [K/(1+r)] {(1+r)/(1+i) + (1+r)2/(1+i)2 +
+…(1+r)3/(1+i)3 +…+ (1+r)n/(1+i)n}
= [K/(1+r)]
an (at (1+i)/(1+r)-1)
Note that in this formulation we effectively say that
payments are based on a notional time 0 amount of
K/(1+r) , which to an actuary makes intuitive sense.
But finance people tend to think of the first actual
payment of K at time 1. Both are correct.
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Value of a stock: Gordon growth model (finance)
0
$K
∞
3
2
1
$K(1+r) $K(1+r)2
PV
PV0= K
1 – (1+𝑟)𝑛 /(1+𝑖)𝑛
𝑖−𝑟
= K/(i-r)
if the dividends go on forever. In reality we sell the
stock some time for a sale price which reflects the
present value then of the then-future dividends. This
ends up with the same answer as assuming we never
sell. The value of i might be taken as current market
Treasury bond yield rate, perhaps adjusted for risk,
and the analyst estimates next year’s dividend K and
the increase rate r. In real life this formula gets
some use as a rough-and-ready tool, maybe to
compare two stocks. Or to explain a big stock
market fall as a change in business confidence
resulting in a small change in the implicitly assumed
r.
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SECTION 7 - PAYMENTS FOLLOW A
GEOMETRIC PROGRESSION
SOA Exam FM/CAS Exam 2 Study Guide © S.
Broverman 2008 www.sambroverman.com 89
Example 27 (SOA): Common stock X pays a
dividend of 50 at the end of the first year, with each
subsequent annual dividend being 5% greater than
the preceding one. John purchases the stock at a
theoretical price to earn an expected annual effective
yield of 10%. Immediately after receiving the 10th
dividend, John sells the stock for a price of P . His
annual effective yield over the 10-year period was
8%. Calculate P.
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PROBLEM SET 7
Exam FM/CAS Exam 2 Study Guide © S.
Broverman 2008
12. Today is the first day of the month, and it is
Smith's 40th birthday, and he has just started a new
job today. He will receive a paycheck at the end of
each month (starting with this month). His salary
will increase by 3% every year (his monthly
paychecks during a year are level), with the first
increase occurring just after his 41st birthday. He
wishes to take c% of each paycheck and deposit that
amount into an account earning interest at an annual
effective rate of 5%. Just after the deposit on the day
before his 65th birthday, Smith uses the full balance
in the account to purchase a 15-year annuity. The
annuity will make monthly payments starting at the
end of the month of Smith's 65th birthday. The
monthly payments will be level during each year,
and will increase by 5% every year (with the first
increase occurring in the year Smith turns 66). The
starting monthly payment when Smith is 65 will be
50% of Smith's final monthly salary payment. Find c
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n-year arithmetic increasing annuity
0
1
• Take
PV
2
• $1
• $2
v + 2v2 + 3v3 + 4v4 + ……+ nvn
Ian =
(1+i) Ian =1 + 2v + 3v2 + 4v3 + ..…+ nvn-1
Subtract
i Ian = 1 + v + v2 + v3 + ..…+ vn-1 - nvn
Ian =
ä 𝑛 − 𝑛 𝑣 𝑛
𝑖
This arithmetic increasing annuity is less common in
real life than the geometric. But it can be useful if
e.g. regular coupon payments are accumulated in a
fund which pays interest (on the arithmetically
increasing balance) into another fund.
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Example 33 (SOA): (Broverman FM manual):
Coco invests 2000 at the beginning of the year in a
fund which credits interest at an annual effective rate
of 9%. Coco reinvests each interest payment in a
separate fund accumulating at an annual effective
rate of 8%. The interest payments from this fund
accumulate in a bank account that guarantees an
annual effective rate of 7%. Determine the sum of
the principal and interest at the end of 10 years.
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n-year arithmetic decreasing annuity
0
1
•
• PV
Dan =
2
• $n
• $n-1
nv + (n-1)v2 + (n-2)v3 ……+ vn
(n+1) an = (n+1)v + (n+1)v2 +(n+1)v3 ++ (n+1) vn
Subtract
Dan = (n+1) an - Ian
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