Converting an Effective Annual Rate to Quarterly Interest
Summary
The document explains why a stated effective annual rate cannot simply be divided by four to get the equivalent quarterly rate. The quarterly rate must compound over four quarters to reproduce the annual growth factor, so it is found by taking the fourth root of that factor and subtracting one. Dividing the annual rate evenly instead produces a slightly higher effective annual return when compounded quarterly.
It applies the conversion to quarterly deposits made over ten years, then carries their accumulated value forward to year fifteen and equates it to the value of four beginning-of-year withdrawals. The calculation uses the quarterly equivalent rate for deposit accumulation and the effective annual rate for the annual withdrawals. The answer also flags a likely notation error in the stated answer key. This is an actuarial finance example, not a trading strategy, and its usefulness to quantitative traders is mainly the general lesson about matching compounding periods and rate conventions.
Key ideas
- An effective annual rate represents the growth over a full year, not a nominal rate divided among periods.
- The equivalent quarterly rate is the rate whose fourth power of one-period growth matches the annual growth factor.
- Quarterly deposits accumulate using the equivalent quarterly rate, while annual withdrawals use the effective annual rate.
- Rate conversions must respect compounding conventions to avoid overstating returns.
Tags
Full text
# Can please help me understand the proper measurement of interest for this problem
# Can please help me understand the proper measurement of interest for this problem
I know this should appear simple but I cannot really wrap my head on how to know which measurement of interest should be used.
So, I am reviewing for a FM exam and tried this problem.
John makes deposits of 450 at the end of each quarter for 10 years. At the end of 15 years, he will use the fund to make annual payments of Y at the beginning of each year for 4 years, after which the fund is exhausted. Find Y if i=7% per annum.
I checked the answer key and it indicates that the interest used to compute for the future value of 450 is equal to “r" where (1+r/m)^40=1.07
My question is why can’t I simply use 0.07/4 as the rate of interest to compute for the future of 450 at the end of each quarter for 10 years?
## Answer by Greg (score 1)
https://quant.stackexchange.com/a/82268
The confusion you're experiencing relates to the difference between nominal and effective interest rates, which is a fundamental concept in financial mathematics.
When given an annual interest rate of 7%, this is an effective annual interest rate, meaning money grows by a factor of 1.07 after one full year. When working with quarterly periods, we need the equivalent quarterly rate that, when compounded over four quarters, gives us exactly that same 7% annual growth.
Why 0.07/4 Is Not Correct
If you simply divide 0.07 by 4 to get 0.0175 (1.75%) as your quarterly rate, compounding this over a year would give: $$(1 + 0.0175)^4 = 1.07186...$$
This equates to approximately 7.186% annually - higher than our stated 7%. This small difference compounds significantly over many periods!
The Correct Conversion
The proper formula to find the equivalent quarterly rate r is: $$(1 + r)^4 = 1.07$$
Solving for r: $$r = (1.07)^{1/4} - 1 ≈ 0.017033... ≈ 1.7033\%$$
This quarterly rate, when compounded over four quarters, gives exactly 7% annual growth.
The formula from your answer key $$(1+r/m)^{40}=1.07$$ appears to be mixing notation. The correct interpretation is that r is the quarterly interest rate, and $$m = 4$$ (quarters per year), so: $$(1 + r)^4 = 1.07$$
Solving Your Given Problem:
- Calculate the future value of 450 paid quarterly for 10 years using the quarterly rate $$r = 0.017033$$
- Let that accumulate for another 5 years (from years 10 to 15)
- Find the annuity due value Y that this accumulated amount can support for 4 years
Step 1: Accumulation of deposits The accumulated value at year 10 is: $$450 \cdot s_{\overline{40}|r} = 450 \cdot \frac{(1+r)^{40}-1}{r}$$
Step 2: Growth for 5 more years The value at year 15 is: $$450 \cdot s_{\overline{40}|r} \cdot (1+r)^{20}$$
Step 3: Finding the annual payment Y $$450 \cdot s_{\overline{40}|r} \cdot (1+r)^{20} = Y \cdot \ddot{a}_{\overline{4}|i}$$
where $$\ddot{a}_{\overline{4}|i}$$ is the present value of an annuity due for 4 years at the annual effective rate $$i = 0.07$$.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.