MHB What is the formula for calculating future value with increasing interest rates?

AI Thread Summary
The discussion focuses on calculating the future value of an investment with increasing interest rates. An example illustrates how $1,000 grows at a starting rate of 3%, increasing by 10% each year. The formula for future value in this scenario is expressed as FV(n)=P_0 ∏(1+r_0 ρ^(k-1)), where P_0 is the principal, r_0 is the initial interest rate, and ρ is the growth factor. The conversation also touches on the complexity of finding a closed-form solution, suggesting the use of a differential equation for continuously compounded interest. Ultimately, the participants conclude that while an average rate approach seems intuitive, it does not yield accurate results.
Wilmer
Messages
303
Reaction score
0
Code:
YR    RATE      INTEREST      BALANCE
0                             1000.00
1    .03         30.00        1030.00
2    .033        33.99        1063.99
3    .0363       38.62        1102.61
4    .03993      44.03        1146.64
Above is an example of future value of an amount at an incresing rate:
$1000.00 at rate 3% 1st year, then the rate increasing by .10 each year.
As example, year2 rate = .03 * 1.10 = .033

What is the formula to calculate the future value in such circumstances?
 
Mathematics news on Phys.org
Wilmer said:
Code:
YR    RATE      INTEREST      BALANCE
0                             1000.00
1    .03         30.00        1030.00
2    .033        33.99        1063.99
3    .0363       38.62        1102.61
4    .03993      44.03        1146.64
Above is an example of future value of an amount at an incresing rate:
$1000.00 at rate 3% 1st year, then the rate increasing by .10 each year.
As example, year2 rate = .03 * 1.10 = .033

What is the formula to calculate the future value in such circumstances?

I can't see an obvious closed form (rather than a product with one term for each year), but this can be tackled by setting up the differential equation for continuously compounded interest with a linearly increasing interest rate.

The solution is then of the form:

\[ FV(t)=P_0 e^{\frac{r_0*\rho^t}{\log(\rho)}} \]

Where \(P_0,\ r_0\) and \(\rho\) are related to but not quite the principle, the initial interest rate and the annual interest growth factor.

In this case \(P_0\approx 741.228\), \( r_0\approx 0.0281893\) and \( \rho\approx 1.09871\)

CB
 
Thanks CB.
I thought there was a way, since the rates themselves can be "summed" by formula,
(like in example: .03 + .033 + .0363 + .03993 = .13923; .13923 / 4 = ~.0348)
then use an average...but that doesn't quite work...
 
Wilmer said:
Thanks CB.
I thought there was a way, since the rates themselves can be "summed" by formula,
(like in example: .03 + .033 + .0363 + .03993 = .13923; .13923 / 4 = ~.0348)
then use an average...but that doesn't quite work...

We want:

\[ FV(n)=P_0 \prod_{k=1}^n (1+r_0 \rho^{k-1}) , \ \ n\ge 1\]

where \(P_0\) is the principle, \(r_0\) the initial interest rate and \(\rho\) the annual groth factor for the rate.

Now there may be a way to express the product in a "nice" form but I can't see it.

CB
 
Agree. In "looper words":
a=1000:r=.03:i=.10:n=4

FOR y = 1 TO n

k = a * r [this period's interest]

a = a + k [this period's resulting principle]

PRINT y,k,a

r = r * i [update rate]

NEXT y
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Is it possible to arrange six pencils such that each one touches the other five? If so, how? This is an adaption of a Martin Gardner puzzle only I changed it from cigarettes to pencils and left out the clues because PF folks don’t need clues. From the book “My Best Mathematical and Logic Puzzles”. Dover, 1994.
Back
Top