Converting Annual Effective Rates for Monthly Cash Flow Discounting
Summary
The question compares two ways to discount 48 monthly payments when the available investment return is stated as 2% per year. One approach divides the annual rate by twelve and then applies another fractional-period exponent; the proposed solution instead converts the annual effective rate into an equivalent monthly growth factor by taking its twelfth root.
The response endorses the latter conversion: a monthly rate consistent with a 2% annual effective return is the twelfth root of 1.02, less one. That rate can then be used to discount each monthly payment. The exchange highlights a period-consistency issue, but gives no full present-value comparison with the cash offer and does not discuss alternative conventions such as nominal annual rates compounded monthly; the correct conversion depends on how the quoted annual rate is defined.
Key ideas
- Monthly discounting must use a monthly rate consistent with the stated annual rate convention.
- For an annual effective rate, the equivalent monthly growth factor is the twelfth root of the annual growth factor.
- Dividing an annual effective rate by twelve and then applying a fractional exponent does not preserve the annual return.
- Discount each payment using periods that match the cash flow frequency.
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Full text
# Correct form of discount rate
# Correct form of discount rate
I'm solving the following problem:
> Two dealers compete to sell you a new Hummer with a list price of \$45,000. Dealer C offers to sell it for \$40,000 cash. Dealer F offers “0-percent financing:” 48 monthly payments of \$937.50. (48x937.50=45,000) (a) You can finance purchase by withdrawals from a money market fund yielding 2% per year. Which deal is better?
So I need to calculate the present value of the financing option with a yearly rate of $r=0.02$, and monthly cashflows of $C=\\\$937.50$. My logic for this is that we need to first convert the annual interest rate to the monthly rate so that $r\to r/12$. Moreover, we need to ensure that our discounting is consistent, in that $(1+r)^T$ represents $T$ years of the time horizon. Therefore, the exact expression for the present value is
$$ PV = C \sum_{n=1}^{48} \left(\frac{1}{\left(1+\frac{r}{12}\right)^{1/12}}\right)^n. $$
However, the official solution to the problem states that
$$ PV = C \sum_{n=1}^{48} \left(\frac{1}{\left(1+r \right)^{1/12}}\right)^n. $$
So the only difference is that the official solution compounds the yearly interest for each payment, and mine compounds on a monthly basis. So my question is which one is the correct solution.
## Answer by Bob (score 1, accepted)
https://quant.stackexchange.com/a/75159
The official solution is correct. Consider the case where $r = 0.02$. The monthly rate in this case is $1.02^{1/12}$ or about $1.0016516$.
By the way, $(1 + 0.02/12)^{1/12}$ is about $1.0001388$. That is not going to annualize to 2%.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.