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
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top