Thread: quick interest question ...

To provide an annual scholarship for 25 years, a donation of $50,000 is invested in an account for a scholarship that will start a year after the investment is made. If the money is invested at 5.5% per annum, compounded annually; determine the amount of each scholarship?

I would greatly appreciate it if I could be given some help.
Thanks so much in advance.

Originally Posted by asiankatt

To provide an annual scholarship for 25 years, a donation of $50,000 is invested in an account for a scholarship that will start a year after the investment is made. If the money is invested at 5.5% per annum, compounded annually; determine the amount of each scholarship?

I would greatly appreciate it if I could be given some help.
Thanks so much in advance.

I don't know if there is a ready formula for this. Let us derive one.

Let X = amount of scholarship to be deducted every year
A = amount of investment after any year
P = principal = $50,000 here.
r = rate of interest = 0.055 here.
n = number of years

In compounded annually,
A = P(1+r)^n

After year 1,
A = P(1+r)
Atfer withdrawal of X,
A = P(1+r) -X

After year 2,
A = [P(1+r) -X](1+r) = P(1+r)^2 -X(1+r)
After withdrawal of X,
A = P(1+r)^2 -X(1+r) -X

After year 3,
A = [P(1+r)^2 -X(1+r) -X](1+r)
A = P(1+r)^3 -X[(1+r)^2 +(1+r)]
After withdrawal of X,
A = P(1+r)^3 -X[(1+r)^2 +(1+r)] -X
.
.
After year 25,
A = P(1+r)^25 -X[(1+r)^24 +(1+r)^23 +(1+r)^22 +....+(1+r)]
After withdrawal of X,
A = P(1+r)^25 -X[(1+r)^24 +(1+r)^23 +(1+r)^22 +....+(1+r)] -X ------(i)
And that is now equal to zero.

The [(1+r)^24 +(1+r)^23 +(1+r)^22 +....+(1+r)] can be rewritten as
[(1+r) +(1+r)^2 +(1+r)^3 +...+(1+r)^24].
It is a geometric series where
common ratio = (1+r)
a1 = (1+r) also
n = 24
So, since (1+r) = (1+0.055) = 1.055, then,
Sn = (a1)[(1 -r^n)/(1-r)]
S(24) = (1.055)[(1 -(1.055)^24) / (1 -(1.055)]
S(24) = (1.055)[(-2.61459)/(-0.055)]
S(24) = 50.15259 ----------------------***

Substituting that and the givens into (i),
A = P(1+r)^25 -X[(1+r)^24 +(1+r)^23 +(1+r)^22 +....+(1+r)] -X ------(i)
which is zero, so,
0 = (50,000)(1.055)^25 -X(50.15259) -X
0 = 190,670 -X(51.15259)
X = (190,670)/(51.15259)
X = $3727.47 -----------------answer.

Hello, ticbol!


There is a "Sinking Fund" formula . . . which you derived from scratch.

. . Lovely work!


One of its forms looks like this: .

where: .

Originally Posted by Soroban

Hello, ticbol!


There is a "Sinking Fund" formula . . . which you derived from scratch.

. . Lovely work!


One of its forms looks like this: .

where: .

Good morning, Soroban.

So that is a Sinking Fund. Umm.
I just had time last night to play with it. I thought the question was interesting.

Hello again, ticbol!

I can appreciate the reasoning and the algebra
. . that went into your derivation.

Every year or so, I can't my list of favorite formulas,
. . and I've been forced to derive the Amortization Formula.
So I've had considerable practice.

If that was your first attempt at such a derivation, well done!
. . And an excellent explanation, too.