Why is nominal interest defined the way it is?

Question

So if nominal interest is 12% compounded monthly, it is actually 1% compounded each month. It is not 12% effective year, though it is close (It is 12.7%) So why don't we/they say 1% compounded monthly?

Or just use effective yearly rate? Nothing is actually 12%, so why is this number used?

I asked my professor this, and her answer was so that we could compare these nominal interest rates, but that doesn't really "sit" with me, because you cannot directly compare these rates. Example: How do you compare 12% compounded monthly, or 13% compounded three times a year. It is not immediately obvious which is greater. So why is nominal interest defined the way it is?

Answer 1

The nominal interest rate is defined the way it is because, along with the compounding interval, it is a succinct way of describing how interest is computed.

If, for example, the nominal interest rate is $6$ percent, and it is compounded monthly, then we can simply divide the nominal rate by the number of months to obtain $0.5$ percent, and now we know that each month, the principal goes up by a factor of $1.005$.

The actual effective interest rate is about $6.1678$ percent, since $1.005^{12} \approx 1.0061678$, but it would be a rather ungainly way of expressing the same thing. What's more, it's likely to be only approximately correct, unless you want to carry this out to $28$ places or whatever.

To be sure, of course, we could have started with the effective interest rate, and then worked out what the nominal interest rate must be. But this requires us to compute a twelfth root, and people back in the day of hand calculators (and before that, hand computation) were understandably loath to do that. And just imagine what would happen if you were to go to daily compounding. (In many ways, compounding continuously is easier, though it requires taking a logarithm.) It was simply easier to deal with the nominal interest rate.

Also, from a marketing perspective, it was easier to tell people that their effective rate was higher than their nominal rate (sounds like they're getting a compounding bonus) than that the rate they actually got each month was less than the effective rate divided by $12$ (sounds like compounding costs them money).

Answer 2

It's simply a convention. It doesn't allow for precise comparison of nominal rates for different compounding frequencies, but the nominal rate is roughly the same order of magnitude as the actual annual rate. This provides some justification, but beyond that it's arbitrary.

Answer 3

You are right. The statement from your Prof is simply wrong. You need a common convention for comparing rates:

1. Daycount Conventions. How do you count the days over which period you accrue interest?: 30/360, ACT/360, ACT/ACT-ISDA, ACT/ACT-ICMA251, BUS/252 (these "latin-american" conventions require a business day calendar), (there are bazillion others).

2. Compounding convention: Simple interest, Compounded monthly, Compounded Daily, Compounded Continuously.

I'd say (but this is a matter of personal taste) it is instructive to use ACT/365 continuously compounded. Then a discount factor is computed as $\exp(-\text{DCF}(d_1,d_2)\cdot r)$ where $r$ is quoted with the above convention. I usually denote this as EXP ACT/365.

Answer 4

Nothing is actually 12%, so why is this number used?

You are right. The used monthly interest rate is not equivalent to the yearly interest rate. The equivalent monthly interest rate $i_m$ can be evaluated by solving the following equation

$\left(1+i_m\right)^{12}=1+i$

But we can see that the monthly interest $\frac{i}{12}$ is a good approximation. For this purpose the binomial theorem can be applied.

$$\left(1+\frac{i}{12} \right)^{12}=\sum_{t=0}^{12} {12 \choose t} \cdot \left(\frac{i}{12} \right)^t\cdot 1^{12-t}$$

The first five summands are

$$=\color{blue}{1+i}+\frac{11}{24}i^2+\frac{55}{432}i^3+\frac{55}{2304}i^4\ldots$$

Since $i<1$ the terms $i^3,i^4,i^5,\ldots$ get smaller and smaller the greater the exponent is. Additionally $i$ is commonly much smaller than $1$.