2)What is the relationship between the present-value factor and the annuity pres
ID: 2821920 • Letter: 2
Question
2)What is the relationship between the present-value factor and the annuity present-value factor?
3)What will $5,000 invested for 10 years at 8 percent compounded annually grow to? How many years will it take $400 to grow to $1,671 if it is invested at 10 percent compounded annually? At what rate would $1,000 have to be invested to grow to $4,046 in 10 years?
7)What is the future value of an ordinary annuity of $1,000 per year for 7 years compounded at 10%? What would be the future value if it were an annuity due?
Explanation / Answer
2) Relationship between the present-value factor and the annuity present-value factor
We have
present-value factor (PVF) = (1+r)^-n and
annuity present-value factor = (1-(1+r)^-n)/r
annuity present-value factor = (1-PVF)/r
3)What will $5,000 invested for 10 years at 8 percent compounded annually grow to?
Future Value = Present Value * (1+r)^n
= 5000 * (1.08)^10
= 5000*2.15892499727
= 10794.62
-How many years will it take $400 to grow to $1,671 if it is invested at 10 percent compounded annually?
Let n be the no. of years
Future Value = Present Value * (1+r)^n
1671 = 400 * (1.1)^n
(1.1)^n = 1671/400 = 4.1775
n = log4.1775 / log1.1
= 0.62091645856/0.04139268515
n = 15 years
-At what rate would $1,000 have to be invested to grow to $4,046 in 10 years?
Future Value = Present Value * (1+r)^n
4046 = 1000 * (1+r)^10
(1+r)^10 = 4046/1000 = 4.046
1+r = 4.046^(1/10) = 1.15001257129
r = 1.15001257129-1
=.15001257129
r = 15%
7)What is the future value of an ordinary annuity of $1,000 per year for 7 years compounded at 10%?
Future value of Annuity = A [((1+r)n-1) / r]
Where
A - Annuity payment = 1000
r - rate per period = 10%
n - no. of periods = 7
Future value of Annuity = 1000 [((1.1)^7 -1) / .1]
= 1000*9.487171
= 9487.17
Future value of Annuity Due = A [((1+r)n-1) / r] * (1+r)
= 1000 [((1.1)^7 -1) / .1] * 1.1
= 1000 * 10.4358881
= 10,435.89
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