Accumulation And Amount Functions Flashcards
What is the amount function?
A function ( A(t) ) that gives the amount of money at any time “t” in an account.
What is the accumulation function?
a(t) = A(t)/A(0)
Where A(0) is the principal and A(t) is the amount function.
What 3 properties do all accumulation functions ( a(t) ) have?
1) a(0) = 1
2) a(t) is increasing
3) if interest accrues for any non-integer values of t, then a(t) is a continuous function.
Show that a(t) = t^2 + 2t + 1 where t>=0 is a real number and satisfies the 3 properties of an accumulation function.
a) a(0) = 0^2 + 2(0) + 1 = 1
b) a’(t) = 2t + 2
a’(0) = 2(0) + 2, a’(t) > 0 for t>=0 so it is increasing
c) the function is quadratic so it is continuous.
What formula is used to find the amount of interest earned on an investment.
I(n) = A(n) - A(n-1)
What formula do you use to find the accumulated value of some value k that is deposited at some time other than 0?
If k is deposited at time s, then the accumulated value of $k at time t>s is k(a(t)/a(s)), and a(t)/a(s) is called the accumulation factor or growth factor.
What formula(s) is used to calculate the effective rate if interest?
i(n) = ( A(n) - A(n-1) / A(n-1) )
Or
i(n) = ( a(n) - a(n-1) / a(n-1) )
When discussing effective rates of interest, when during the time period is interest paid?
The effective rate of interest i is the amount of money that one unit invested at the beginning of a period will earn during the period, where interest is paid at the end of the period.
