For this I-bond calculation there's a tiny adjustment to the rate (fixed rate x semiannual inflation rate) it barely changes the rate.Why is it there?

Question

From https://www.treasurydirect.gov/savings-bonds/i-bonds/i-bonds-interest-rates/#cerates

Composite rate formula: [Fixed rate + (2 x semiannual inflation rate) + (fixed rate x semiannual inflation rate)]

"fixed rate" is an annual rate.
"semiannual inflation rate" is a 6-month rate.

I feel like I never really understood why it's included in the calculation. In the example given it changes the rate on the bond from 4.28% to 4.29521%

Is it normal to multiply a 6-month rate by an annual rate?

For more information on this calculation

Calculating composite rate from fixed rate and inflation rate for I-bonds

Answer 1

The last term of the formula makes up for the loss of value of the fixed rate interest due to inflation.

The thread you linked to actually already contained that information, but probably not clear enough, so here is a hands-on example:

Let's assume the fixed rate is 4%. Note that that means 2% after half a year. Since you get semiannually compounding, the actual rate is a bit higher (as you get interest in the 2nd half of the year for the interest of the first half of the year), but this is not the reason for the last term in the calculation, this is covered by being semiannually.

Let's assume you have $100. Also let's assume an apple costs 1$.

Assume 0% inflation, which is basically what I bonds are trying to simulate for you. After half a year,

• you got 2% * $100 = $2 fixed rate interest
• $0 interest for the inflation
• you now have $102
• apples still cost $1
• you can buy 102 apples

Now assume 10% semiannual inflation. If we don't add the (fixed rate x semiannual inflation rate) term, after half a year,

• you got $2 fixed rate interest
• $10 interest for the inflation
• you now have $112
• apples now cost $1.10
• you can now buy $112 / $1.10 = 101.82 apples

So you cannot buy 102 apples anymore (as was the case with 0% inflation), because the $2 from the fixed rate lost value. To make up for it, you need another 10% * $2, which is 10% * ($100 * 2%), e.g. original amount * semianual infaltion rate * fixed interest rate.

So, including the (fixed rate x semiannual inflation rate) term, after half a year:

• you got $2 fixed rate interest
• $10 interest for the inflation
• 10% * $2 = $0.20 inflation interest for the fixed rate interest
• you now have $112.20
• apples now cost $1.10
• you can now buy $112.20 / $1.10 = 102 apples again

This was for half a year, but since interest rates refer to a full year, you just double it, e.g. 2 * 2% + 2 * 10% + 2 * 2% * 10%, and with 2 * 2% being the fixed interest, you get your

fixed rate + 2 * semiannual inflation rate + fixed rate * semiannual inflation rate

Answer 2

The composite rate is reported as an annual rate, but the interest is actually compounded semiannually. Thus at the end of the first half of the year, you get to collect interest on the amount at the start of the year including both the fixed interest and the inflation that occurred during the first half of the year. Then you get fixed interest plus inflation on that amount over the second half of the year.

If we let f be the fixed annual interest rate and s be the semiannual inflation rate (assume it's the same for both halves of the year), then every 6 months, the balance is multiplied by (1+f/2+s). Here, f/2 is the 6 month interest rate. If you multiply (1+f/2+s)(1+f/2+s), you get (1+f+2s+sf+s^2+(f/2)^2).

However, the s^2 and (f/2)^2 terms are quite small (e.g. 3% of 3% is 0.1%) After ignoring those terms, we see that the compounded interest multiplies the initial amount by (nearly exactly) (1+f+2s+sf). That matches the "composite interest rate" of

[Fixed rate + (2 x semiannual inflation rate) + (fixed rate x semiannual inflation rate)]

Don't worry about missing the s^2 and (f/2)^2 terms- you'll get that extra bit of interest from the actual computation which compounds semiannually.

If you think about it a bit, there doesn’t seem to be any reason to include the fs term and exclude the two squared terms in the composite interest formula. I can’t explain that choice.