Section 4.4: Effective Yield
- Page ID
- 216981
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In the previous section, we saw when money is invested or borrowed, that interest can be calculated in different ways. Because different accounts use different compounding periods (such as annually, quarterly, monthly, or daily), comparing them directly can be confusing.
Effective yield, also called annual yield or effective rate or effective annual rate (EAR) provides a solution to this problem. It represents the actual annual percentage increase in an investment after accounting for compounding. In other words, the effective yield tells you how much your money truly grows in one year, regardless of how many times interest is added.
For example, an account that advertises a 5% annual interest rate compounded monthly does not actually grow by exactly 5%. Because interest is earned on previously earned interest throughout the year, the real growth is slightly higher. Effective yield captures this true rate of return. Understanding effective yield helps consumers compare financial products on a fair basis. Whether evaluating savings accounts, credit card rates, or loan offers, effective yield provides a single, meaningful number that reflects how interest really works over time.
In this section, we will explore how effective yield is calculated, why compounding matters, and how this concept can be used to make informed financial decisions.
The effective yield is the simple annual interest rate that would produce the same future value in one year as a given compound interest rate. It provides a way to compare different investments on an equal basis by converting any compounding schedule into an equivalent single‑year growth rate. The formula for the effective yield is:
\[E = \left(1 + \frac{r}{n}\right)^{n}-1\nonumber \]
where:
- \(E\) = Effective yield (the accumulated amount)
- \(r\) = Annual interest rate (expressed as a decimal)
- \(n\) = Number of compounds per year
An investment grows at 8% nominal interest, compounded twice per year. What is the effective annual yield?
✅ Solution:
Start with the effective yield.
\[E = \left(1 + \frac{r}{n}\right)^{n}-1\nonumber \]
We know that \(r=0.08\) and \(n=2\), so
\[\begin{align} E &= \left(1 + \frac{0.08}{2}\right)^{2}-1\nonumber \\ E &= (1 + 0.04)^{2}-1\nonumber \\ E &= (1.04)^{2}-1 \nonumber \\ E &= 1.0816-1 \nonumber \\ E &= 0.0816 \nonumber \\ \end{align}\]
\[\boxed{E = 8.16\%} \nonumber \]
So, the effective annual yield is 8.16%.
Now, here is what the value of 8.16% means:
- If you invest $100 at a 8% interest rate, compounded semiannually, for one year, then the future value would be $108.16 and the interest is $8.16, which is exactly 8.16% of $100.
- If you invest $1,000 at a 8% interest rate, compounded semiannually, for one year, then the future value would be $1081.60 and the interest is $81.60, which is exactly 8.16% of $1,000.
- If you invest $10,000 at a 8% interest rate, compounded semiannually, for one year, then the future value would be $10,816.00 and the interest is $816.00, which is exactly 8.16% of $1,000.
A savings account advertises a 6% nominal annual interest rate, compounded quarterly. Find the effective yield.
✅ Solution:
Start with the effective yield.
\[E = \left(1 + \frac{r}{n}\right)^{n}-1\nonumber \]
We know that \(r=0.06\) and \(n=4\), so
\[\begin{align} E &= \left(1 + \frac{0.06}{4}\right)^{4}-1\nonumber \\ E &= (1 + 0.015)^{4}-1\nonumber \\ E &= (1.015)^{4}-1 \nonumber \\ E &= 1.061363550625-1 \nonumber \\ E &= 0.061363550625 \nonumber \\ \end{align}\]
\[\boxed{E \approx 0.06136 = 6.14\%} \nonumber \]
So, the effective yield is 6.136% or 6.14%, depending on how much you decide to round.
Now, here is what the value of 6.14% (or 6.136%) means:
- If you invest $100 at a 6% interest rate, compounded quarterly, for one year, then the future value would be $106.14 and the interest is $6.14, which is exactly 6.14% of $100. (Effective Yield = 6.14%, if you round to the nearest hundredth of a percent).
- If you invest $1,000 at a 6% interest rate, compounded quarterly, for one year, then the future value would be $1061.36 and the interest is $61.36, which is exactly 6.136% of $1,000. (Effective Yield = 6.136%, if you round to the nearest thousandth of a percent).
Two investment funds advertise:
Fund X: 7.1% compounded semiannually
Fund Y: 7% compounded daily
Compute the effective annual rate for each. Which fund provides the better actual return?
✅ Solution:
Start with the effective yield.
\[E = \left(1 + \frac{r}{n}\right)^{n}-1\nonumber \]
For Fund X, we know that \(r=0.071\) and \(n=2\), so
\[\begin{align} E &= \left(1 + \frac{0.071}{2}\right)^{2}-1\nonumber \\ E &= (1 + 0.0355)^{2}-1\nonumber \\ E &= (1.0355)^{2}-1 \nonumber \\ E &= 1.07226025-1 \nonumber \\ E &= 0.07226025 \nonumber \\ \end{align}\]
\[\boxed{E \approx 0.0722 = 7.22\%} \nonumber \]
So, the effective yield for Fund X is 7.22%.
For Fund Y, we know that \(r=0.07\) and \(n=360\), so
\[\begin{align} E &= \left(1 + \frac{0.07}{360}\right)^{360}-1\nonumber \\ E &= (1 + 0.0001944444444)^{360}-1\nonumber \\ E &= (1.0001944444444)^{360}-1 \nonumber \\ E &= 1.0725008832111-1 \nonumber \\ E &= 0.0725008832111 \nonumber \\ \end{align}\]
\[\boxed{E \approx 0.0725 = 7.25\%} \nonumber \]
So, the effective yield for Fund Y is 7.25%.
Thus, Fund Y at an effective yield of 7.25% is a slighter a better return than Fund X at an effective yield of 7.22%.
The table belows shows how the same nominal rate produces different effective yields depending on how often interest is compounded.
| Nominal Rate | Annually | Semi-Annually | Quarterly | Monthly | Daily | Million Times Per Year |
|---|---|---|---|---|---|---|
| 2% | 2% | 2.01% | 2.0151% | 2.0184% | 2.0201% | 2.0201% |
| 3% | 3% | 3.0225% | 3.0339% | 3.0416% | 3.0453% | 3.0455% |
| 4% | 4% | 4.04% | 4.060% | 4.0742% | 4.0808% | 4.0811% |
| 5% | 5% | 5.0625% | 5.0945% | 5.1162% | 5.1267% | 5.1271% |
| 6% | 6% | 6.09% | 6.1364% | 6.1678% | 6.1831% | 6.1837% |
| 7% | 7% | 7.1225% | 7.1859% | 7.2290% | 7.2501% | 7.2508% |
| 8% | 8% | 8.16% | 8.2432% | 8.3000% | 8.3277% | 8.3287% |
| 9% | 9% | 9.2025% | 9.3083% | 9.3807% | 9.4162% | 9.4174% |
| 10% | 10% | 10.25% | 10.3813% | 10.4713% | 10.5156% | 10.5171% |
When you examine the table above of effective annual yields for different compounding frequencies, an important pattern becomes clear. For a fixed nominal interest rate, the effective yield increases as the number of compounding periods per year increases. However, this increase does not occur at a constant or proportional rate. For example, moving from semiannual to quarterly compounding does not double the gain, nor does monthly compounding produce a dramatically larger jump. Instead, each additional increase in compounding frequency results in a smaller improvement in the effective yield. The growth slows down because the effect of compounding more frequently has diminishing returns. In fact, even if interest were compounded a million or even a billion times per year, the effective yield would approach a limiting value and show almost no change. This illustrates an application called continuous compounding that represents the theoretical maximum yield for any given nominal rate. This application is usually introduced in a college algebra pr pre-calculus course when talking about exponential growth and decay.
- A bank pays 3.9% interest compounded quarterly. Compute the effective yield.
- A certificate of deposit (CD) offers 5.2% compounded monthly, while a bond fund advertises a simple annual return of 5.3% (no compounding). Compute the effective annual rate for each. Which fund provides the better actual return?
- Sad Sally is back looking at that Wells Fargo savings rate of 0.01%. Compounding daily, what is the APY? Round to the nearest hundred-thousandths of a percent.
- Answers
-
- E = 3.96%
- CD: E = 5.33%; Bond fund: E = 5.3%; Thus, the CD provides a better actual return by 0.03%
- E = 0.01%; (actual value: E = 0.01000049862764)