When combining multiple loans with separate interest rates into one interest rate, is it possible to reason about if the resulting interest rate will always be less than that of the combined investments?
To elaborate with equations:
Say we have an investment, with value $V_1$ and interest rate $R_1$, and another with $V_2$ $R_2$.
The value of the investment using the weighted average interest rate would be:
$$ (V_1 + V_2)((V_1 R_1 + V_2 R_2) / (V_1 + V_2)) ^ T$$
Where T is the time periods passed.
The actual value of these investments kept separate would be:
$$ V_1 R_1 ^ T + V_2 R_2^T$$
2 Questions about this:
- Does there exist some perfect interest rate such that
$ V_1 R_1 ^ T + V_2 R_2^T = (V_1 + V_2)R_?^T$
that perfectly combines the two interest rates into one rate? My gut intuition is that this is impossible, but I have struggled to prove why.
- Given #1 is impossible, can I reason that the weighted average interest rate is always going to be less than or equal to the true interest rate of the investments?
I've convinced myself that taken out to $T = infinity$ the true interest rates $ V_1 R_1 ^ T + V_2 R_2^T$ will outpace the weighted average, given $V_1 \ne V_2$, since the larger of $V_1$ or $V_2$ will outpace the weighted average rate, which must be somewhere between $V_1$ and $V_2$.
The question is though, is this true from T=1 onward, or is there some set of values for which the weighted sum is larger than the true investment value in the short term?
I've run a few examples with plugging in numbers, but I haven't been able to find any such values. I'm also not really sure how to approach proving or disproving this.
Any thoughts on this would be greatly appreciated! For some context, I'm a developer looking at learning about the Aave smart contract system, and they use the weighted sum interest approximation inside of their contracts. I'm curious what effect this has on the overall system, and if there are any problems that could be caused by the weighted average interest rate being lower than the true interest rate loans were supposed to accrue.
Edit:
Here is an example:
I have one loan for \$1000 and is at a 10% interest rate. I have another loan for \$500 at a 20% interest rate. Both loans will be combined into one loan for $1500 with the following interest rate:
$$(1000 * 1.1 + 500 * 1.2) / (1000 + 500)$$
or $$(1100 + 600) / 1500 = 1.13333333333$$
So I will pay 13.3% interest on my loan. But this is not exactly what I would have paid if I kept the 2 loans separate.
In 2 years, the value of the weighted sum of rates would be:
$$1500 * 1.13333^2 = $1926.66$$
However if kept seperate, these loans would yield:
$$1000 * 1.1 ^ 2 + 500 * 1.2^2 = $1930$$
I've run lots of numerical examples, but in all of them I've seen that the weighted average interest rate ends up lower than the actual rate on the investments. My question is, can this be proved true or false?
Running algebra I quickly run into complexity of binomial expansions and cant seem to make much progress.
Also, I've come to know this method is also referred to as the weighted average cost of capital.
Best Answer
As I I've written in the comments the average interest/return rate is $$\overline R=\sqrt[T]{\frac{V_1 (1+R_1) ^ T + V_2 (1+R_2)^T}{V_1+V_2} }-1$$,
where $R_i$ are the corresponding interest rates of the investments.
Numerical example
With $R_1=0.1, R_2=0.2, V_1=1000, V_2=500$ and $T=3$ the average return rate is
$$\overline R= \sqrt[\Large 3 ]{\frac{1000\cdot 1.1 ^ 3 + 500 \cdot 1.2^3}{1500} }-1=0.13530989...$$
See here for the computed result.
The two investments have an value at $T=3$ of $1000\cdot 1.1^3+500\cdot 1.2^3=2195$
Let's see if we can calculate the value of the two investments at $T=3$ with this average interest rate: $1500\cdot 1.13530989^3=2195$. See here again for the computed result.
General remarks
The average interest rate is changing with $T$. The minimum of $\overline R$ is when $T=1$: $\frac{V_1 (1+R_1) + V_2 (1+R_2)}{V_1+V_2}-1=1+\frac{V_1R_1+V_2R_2}{V_1+V_2}-1=\frac{V_1R_1+V_2R_2}{V_1+V_2}=0.1333...$
In this case $R_2>R_1$. The more periods have past, the faster $V_2 $ grows in comparison to $V_1$. If we let T go to infinity we obtain $\lim\limits_{x \to \infty} \overline R=R_2=0.2$, the upper bound.
continued
That the average return increases when T increases can be seen if we look at the power mean inequality which is
$$\sqrt[q]{\sum_{i=1}^nw_ix_i^q} \geq \sqrt[p]{\sum_{i=1}^nw_ix_i^p} \quad \forall \ \ p <q$$
In the numerical example we have $n=2, w_1=\frac{1000}{1500}=\frac23, w_2=\frac{500}{1500}=\frac13, x_1=1.1, x_2=1.2, p=1, q=3$.
$$\sqrt[\Large 3 ]{\frac{1000\cdot 1.1 ^ 3 + 500 \cdot 1.2^3}{1500} } \geq \sqrt[\Large 1 ]{\frac{1000\cdot 1.1 ^ 1 + 500 \cdot 1.2^1}{1500} } $$
$$\sqrt[\Large 3 ]{\frac{1000\cdot 1.1 ^ 3 + 500 \cdot 1.2^3}{1500} } \geq \frac{1100 + 600}{1500}$$
$1.13530989>1.133333$
With the power mean equality we can conclude as well, that the average interest rate is not constant, if $T$ changes.