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A deferred perpetuity-due begins payments at time n with annual payments of $1000 per year. If the present value of this perpetuity-due is equal to $6561 and ...
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The form for the sum of a geometric sequence is:
Sum(n) ≡ a + ar + ar 2 + ar 3 + · · · + ar n−^1
Here a = (the first term) n = (the number of terms) r = (the multiplicative factor between adjacent terms) Note that
rSum(n) = ar + ar 2 + · · · + ar n^ and therefore
a + rSum(n) = a + ar + ar 2 + · · · + ar n−^1 + ar n = Sum(n) + ar n.
Solving this equation for Sum(n) produces
(r − 1 )Sum(n) = a(r n^ − 1 ).
Therefore
Sum(n) =
a( 1 −r n) ( 1 −r ) =^
a(r n− 1 ) (r − 1 ) if^ r^6 =^1 na if r = 1
We will refer to this formula with the abbreviation SGS.
Example
100 ν + 100 ν^2 + 100 ν^3 + · · · + 100 ν^30 = 100 ν( 1 − ν^30 ) ( 1 − ν)
So , for example, if ν = (^11). 1 , then the above sum is
100 ( 1. 1 )−^1 ( 1 − ( 1. 1 )−^30 ) ( 1 − ( 1. 1 )−^1 )
In the annuity-Immediate setting
Generic Setting The amount of 1 is paid at the end of each of n payment periods.
The present value of this sequence of payments is
an| ≡ an|i ≡ ν + ν^2 + ν^3 + · · · + νn
( 1 − νn) i
because ν 1 − ν
( 1 + i)−^1 i( 1 + i)−^1
i where i is the effective interest rate per payment period.
Invest 1 for n periods, paying i at the end of each period. If the principal is returned at the end, how does the present value of these payments relate to the initial investment?
1 = iν + iν^2 + · · · + iνn^ + 1 νn = ian| + νn.
A relationship that will be used in a later chapter is
1 sn|
i ( 1 + i)n^ − 1
i + i( 1 + i)n^ − i ( 1 + i)n^ − 1
= i 1 − (^) ( 1 +^1 i)n
= i 1 − νn^
an|
Example Auto loan requires payments of $300 per month for 3 years at a nominal annual rate of 9% compounded monthly. What is the present value of this loan and the accumulated value at its conclusion?
The cash price of an automobile is $10,000. The buyer is willing to finance the purchase at 18% convertible monthly and to make payments of $250 at the end of each month for four years. Find the down payment that will be necessary.
If an| = x and a 2 n| = y, express i as a function of x and y.
The present value of this sequence of payments is
¨an| ≡ ¨an|i ≡ 1 + ν + ν^2 + · · · + νn−^1
( 1 − νn) d
Note also that
¨an| = 1 + an− 1 |
Now view these payments from the end of the last payment period.
The accumulated value at t = n is
¨sn| ≡ ( 1 + i) + ( 1 + i)^2 + · · · + ( 1 + i)n
( 1 + i)
( 1 + i)n^ − 1
( 1 + i) − 1
] (^) (by SGS) = ( 1 + i)n^ − 1 d
Find the present value of payments of $200 every six months starting immediately and continuing through four years from the present, and $100 every six months thereafter through ten years from the present, if i(^2 )^ = .06.
A worker age 40 wishes to accumulate a fund for retirement by depositing $3000 at the beginning of each year for 25 years. Starting at age 65 the worker plans to make 15 annual equal withdrawals at the beginning of each year. Assuming all payments are certain to be made, find the amount of each withdrawal starting at age 65 to the nearest dollar, if the effective interest rate is 8% during the first 25 years but only 7% thereafter.
At the end of period 9 what is the value of these future payments? Here the answer is
ν^3 a 8 | = ν^4 ¨a 8 | = ν^11 s 8 | = ν^12 ¨sn|
A deferred annuity is one that begins payments at some time in the future. Using the setting above, we could describe this stream of payments from the time t = 0 as
12 |a 8 | =^ (8 payment annuity immediate deferred 12 periods.)
It could also be viewed as an annuity-due deferred 13 periods
13 |a¨ 8 | =^ ν^13 ¨a 8 | =^ a¨ 21 | −^ a¨ 13 |
What is the accumulated value of this stream of payments at the end of period 24?
( 1 + i)^4 s 8 | = s 12 | − s 4 | (viewed as an annuity-immediate)
( 1 + i)^3 ¨s 8 | = ¨s 11 | − ¨s 3 | (viewed as an annuity-due)
What is the value of this stream of payments at time t = 17?
ν^3 s 8 | = ν^4 ¨s 8 | = ( 1 + i)^5 a 8 | = ( 1 + i)^4 a¨ 8 |
The value at t = 17 can also be expressed as, for example,
s 5 | + a 3 | = ¨s 4 | + ¨a 4 |
