For I-bonds:
Composite rate = [Fixed rate + (2 x Semiannual inflation rate) + (Fixed rate x Semiannual inflation rate)]
I wonder how to understand the above way of calculating the composite rate?
Thanks and regards!
The goal of the I-bond is to provide a return corrected for inflation.
The fixed rate is the annual rate of return paid out on the bond, with inflation taken out. It's the "real return" on the bond.
The rate of inflation is recalculated every six months (semiannually).
The annual rate of inflation at a particular time is twice this.
The last term takes into account the inflation of the interest earned.
So the I-Bond rate is the sum of
I am not 100% sure... it could be something like this:
Let f = annual fixed rate, g = semi-annual inflation rate.
Then a $1 investment will earn (1+f/2) * (1+g) in 6 months. So
Composite interest (for 1 year)
= 2 * [(1 + f/2) * (1+g) - 1]
= 2 * [1 + f/2 + g + (fg)/2 - 1]
= 2 * [f/2 + g + (fg)/2]
= f + 2g + fg
Comparing http://treasurydirect.gov/news/pressroom/currentibondratespr.htm and https://www.treasurydirect.gov/indiv/research/indepth/ibonds/IBondRateChart.pdf we learn that the Semiannual Inflation Rate is a six-month rate, not an annualized rate.
Fixed Rate and Composite Rate are annual rates, so they're mixing 6 month rates and annualized rates in one formula. That's confusing.
Using data from the chart I reversed engineered how the calculator at https://www.treasurydirect.gov/BC/SBCPrice works.
It does indeed use the Composite Rate as shown in the calculation, Composite rate = [Fixed rate + (2 x Semiannual inflation rate) + (Fixed rate x Semiannual inflation rate)]
2 x Semiannual inflation rate is just annualizing the Semiannual inflation (6 month) rate.
I don't understand why they don't annualize that value in the next part of the formula. In any case, Fixed rate x Semiannual inflation rate is going to be very small, a few hundredths of a percent. For the note I calculated, after 17 years the difference in value was less than $1 ($134.04 versus $133.08).
Since the interest is compounded every 6 months, the Composite Rate is almost immediately divided by 2. I wonder if there's a different way to write the formula the would make it clearer what the intention is.
Some caveats:
All calculations are done on a hypothetical $25 bond. A $5,000 bond is calculated like two hundred $25 bonds.
They appear to use a 30/360 Day count convention.
Before compounding, they round the interest to the closest penny.
If you cash the bond in before 5 years, you lose the most recent 3 months of interest.
Final caveat:
I have no idea if their online calculator accurately reflects how they calculate the value of a bond when you actually go to cash it in.
@mbhunter provided the correct answer. Yet, it seems there is still confusion about how this formula is used, or where it comes from. It is not just how it is defined. Although fixed income can be confusing, the conventions and computations of products are usually never based on random choice. The formula is designed to always provide the fixed rate of the contract as the real return.
Short answer
1 ) The fixed rate never changes throughout the life of the bond. For new bonds, the fixed rate is determined twice a year, just like inflation (May 1:and November 1)
2 ) The semi-annual inflation rate is multiplied by two because the entire formula is compounded semiannually, meaning that every 6 months the bond's interest rate is applied to a new principal value and inflation is reset. Since the fixed rate is quoted annualized, you simply multiply by two. Quoting as annualized is in line with pretty much all other interest rates, implied volatility quotes, computed historical volatility values at date vendors etc.
3 ) Fixed rate x Semiannual inflation is a product of the design of the formula which guarantees that your real return will be equal to the fixed rate.
Long answer
Origin of the formula:
@Kamaraju Kusumanchi derives the origin of the formula. Some more details:
r= (1+i)/(1+π) – 1
where
Therefore, i = (1+r)(1+π)-1. The rest is shown nicely at @Kamaraju Kusumanchi's answer.
In particular, the annual inflation rate is not equal to
(1+semiannual inflation rate)^2-1
because the entire formula is compounded semiannually, meaning that every 6 months the bond's interest rate is applied to a new principal value and inflation is reset.
Some more details about I bonds
How the inflation rates are computed are illustrated in this answer.
How the interest itself is determined is shown here.
How treasury direct quotes the current value of an I bond on the website is computed here.
Definitions on TreasuryDirect
Numerical example:
I always think numerical examples help a lot in getting a good understanding of theoretical results.
Starting values (in line with the actual values of the I bond in this recent question):
Therefore, the composite rate using the following formula
[Fixed rate + (2 x Semiannual inflation rate) + (Fixed rate x Semiannual inflation rate)]
results in 6.89%, computed as (0.004 + (2 × 0.0324) + (0.004 × 0.0324), rounded to two decimal places. That is the correct value as shown in the link above.
The inflation rate used in this computation is valid for 6 months, after which it resets to the new inflation rate, which is computed as explained here.
After 6 months, investing $1 in I bonds yields in (1+0.0689/2) = $1.03445.
At the same time, what used to cost $1 is now (on average, as measured by the inflation series) (1+0.0324) = $1.0324.
In total, you have $1.03445, but items now cost $1.0324, which means you can effectively buy 1.03445/1.0324 = 1.0019856644711. Your real return is, again rounded to two decimal places, 0.2%., computed as round((1.0019856644711-1)*100,2).
Annualizing this semi-annual real return gives 0.4%, which is, by design, the fixed rate of the bond.
Unsurprisingly, this works all the time, even for seemingly unrealistic number with say 10% fixed rate and 25% semi-annual inflation.
