Solving Continuous Interest Formulas: What Causes the Difference?

[Tiburon11`]

I was reviewing continuous interest formulas when something stumped me.

The accepted formula for continuous interest is A=Pe^rt, and the proof of it is simple enough to understand.

However, my math book solves the formula in a different way. It starts with the standard model for growth rate, P=P₀e^kt, and than solves for k.

Example: Determine how much money will exist in an account if Ed deposits 1000$ in an account with 5% interest for 5 years.


Case 1:
A=Pe^rt
A=1000e^(.05)(5)


Case 2:
P=P₀e^kt
P=1000e^kt

Evaluate the amount at one year to solve for k.

1000(1.05)=1050
1050=1000e^k(1)

Solve for k.

1050/1000=e^k(1)
k=ln(1.05)
P=1000e^ln(1.05)t


The growth constants are different in both cases. What is the cause of this?

 

[Redbelly98]

In Case 1, you are letting 5%-per-year be the growth constant (the "r" in the exponent).
This gives an annual yield (different than the growth rate) of
e^0.05 - 1
= 1.05127... -1
= 0.05127... or 5.127...%

In Case 2, you are saying the the annual yield (not the growth rate) is 5%.

The 5% represents a different parameter in the two cases: it's the growth constant in Case 1, and it's the annual yield in Case 2.