Compound Interest: 3.11% Annually, 3x/Year

[WWGD]

I know I am right in practice, I do this for a living.

So much for my theory on peanut butter ( pb) ;)

 

[pbuk]

I just used common denominator =)

Yes of course

 

[etotheipi]

I think you mean discrete $n$. And it's not a 'very good approximation': at every time $t_i$ that the discrete model is defined it is equal to the value of the continuous model at $t_i$.

No, I mean finite $n$. If we take the function to be continuous on the interval containing all real $t$, then $A(t) = I(1+\frac{r}{n})^{nt}$ approaches $A_{2}(t) = Ie^{rt}$ in the limit of large $n$. However since the former is very close to the latter even for finite $n$, the latter is a very useful time-saving approximation. The former is still exactly an exponential function, $A(t) = Ie^{Rt}$, but with a slightly smaller rate constant $R =n\ln{\left(1 + \frac{r}{n} \right)}$. You can show that $\lim_{n \rightarrow \infty} n\ln{\left(1 + \frac{r}{n} \right)} = r$. My point is that an analyst would probably approximate discrete compounding with an exponential of unchanged rate constant

 

[pbuk]

No, I mean finite $n$. If we take the function to be continuous on the interval containing all real $t$, then $A(t) = I(1+\frac{r}{n})^{nt}$ approaches $A_{2}(t) = Ie^{rt}$ in the limit of large $n$. However since the former is very close to the latter even for finite $n$, the latter is a very useful time-saving approximation. The former is still exactly an exponential function, $A(t) = Ie^{Rt}$, but with a slightly smaller rate constant $R =n\ln{\left(1 + \frac{r}{n} \right)}$. You can show that $\lim_{n \rightarrow \infty} n\ln{\left(1 + \frac{r}{n} \right)} = r$. My point is that an analyst would probably approximate discrete compounding with an exponential of unchanged rate constant

I see. However we are not interested in large $n$, we are only interested in two values, $n = 3$ and $n = 1$. If we use $n = 3$ then the value of the investment after 1 year using the headline rate $r$ is $I (1 + \frac{r}{3} ) ^ 3$ and if we use $n = 0$ then the value of the investment after 1 year using the equivalent annual rate $r_a$ is $I ( 1+ r_a ) ^ 1$. From the definition of the equivalent annual rate $r_a = (1 + \frac{r}{n})^n - 1$ you can immediately see that these two calculations are exactly equal.

Because we only define the annual rate of return at integral numbers of years, the continuous model is not an approximation, it is exactly equal to the discrete model at every point that the discrete model is defined.

A model where a headline rate is compounded continuously to give an effective annual rate which you have correctly calculated as $e^r - 1$ is not useful either in investment theory or practice.

 

[etotheipi]

That is all fine, but when I wrote that I was referring to this

And as it happens the discrete model is not how it happens anyway - interest is normally accrued on a daily basis (using some conventional calculation) increasing the value of the deposit (almost) continuously. The accrued amount is not included in the amount that is used to calculate interest.

 

[pbuk]

That is all fine, but when I wrote that I was referring to this

Oh I see

Reading that back I could have phrased it better:

Interest is normally calculated on a daily basis (using some conventional calculation - see below) increasing the value of the investment (almost) continuously. However this does not mean that the interest compounds continuously, the calculated amount is added to a separate balance ("accrued interest") from the sum that is used to calculate interest (the "principal"). At each relevant date the balance of accrued interest is transferred to the principal ("rolled up") or paid to the investor.

The daily interest calculation is set by the contract and/or conventions of the particular financial instrument and will follow one of a number of surprisingly arcane procedures some of which are outlined here.

 

[Chestermiller]

Oh I see

Reading that back I could have phrased it better:

Interest is normally calculated on a daily basis (using some conventional calculation - see below) increasing the value of the investment (almost) continuously. However this does not mean that the interest compounds continuously, the calculated amount is added to a separate balance ("accrued interest") from the sum that is used to calculate interest (the "principal"). At each relevant date the balance of accrued interest is transferred to the principal ("rolled up") or paid to the investor.

The daily interest calculation is set by the contract and/or conventions of the particular financial instrument and will follow one of a number of surprisingly arcane procedures some of which are outlined here.

In line with this and what @etotheipi said in post #11, A(t) should really be a discontinuous function of t, with the discontinuities occurring on every compounding date (or whatever it's called), and A(t) should more properly be expressed with the nt in the exponent being replaced by $$\sum_{m=1}^{\infty}u\left(t-\frac{m}{n}\right)$$where u(x) is the unit step function, equal to 0 if its argument is less than zero and 1 if its argument is greater than zero. So, more properly, A(t) should read $$A(t) = I\left(1+\frac{r}{n}\right)^{\sum_{m=1}^{\infty}u\left(t-\frac{m}{n}\right)}$$