OT: Fixed Income & Bond Yields (Highest)

Investing Strategies Mechanical Investing
1

Bond Yields after new inflation numbers released

AS OF 9:58 AM ET 11/10/2022. 3mo 6mo 9mo 1yr 2yr 3yr 5yr 10yr 20yr 30yr+
CDs (New Issues) 3.90% 4.50% 4.60% 4.75% 4.90% 5.10% 4.85% 5.30% – –
BONDS
U.S. Treasury 4.27% 4.49% 4.61% 4.59% 4.39% 4.23% 4.02% 3.88% 4.35% 4.13%
U.S. Treasury Zeros 3.67% 4.19% 4.35% 4.45% 4.24% 4.33% 4.04% 4.06% 4.50% 4.04%
Agency/GSE 4.21% 4.62% 4.65% 4.77% 4.72% 5.19% 5.47% 5.75% 5.55% 5.17%
Corporate (Aaa/AAA) 3.97% 4.12% 4.14% 4.26% 4.64% 4.24% 4.60% – 4.87% 5.51%
Corporate (Aa/AA) 4.25% 4.57% 4.13% 4.59% 5.02% 4.72% 5.09% 5.34% 5.51% 6.17%
Corporate (A/A) 4.70% 5.34% 6.18% 5.14% 5.66% 5.78% 6.35% 6.81% 6.69% 6.14%
Corporate (Baa/BBB) 4.77% 5.70% 6.24% 6.33% 6.57% 6.69% 7.42% 7.30% 7.54% 7.62%
Municipal (Aaa/AAA) 3.03% 3.08% 3.36% 3.11% 3.91% 3.84% 4.23% 4.83% 5.19% –
Municipal (Aa/AA) 3.48% 3.47% 3.69% 4.00% 4.22% 4.02% 4.57% 5.10% 5.21% 4.88%
Municipal (A/A) 3.07% 3.55% 3.47% 3.36% 3.75% 3.98% 4.21% 5.10% 5.20% 5.13%
Taxable Municipal* – 4.24% 5.03% 4.59% 5.14% 5.60% 5.60% 5.94% 6.14% –
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2

Suppose you have an annual yield of 5% on bonds (or a return of 5% on stocks), and annual inflation is running say 7%.
What’s the the “real” yield after adjusting for inflation?

INCORRECT ANSWER:
AdjustedYield = Yield - Inflation

CORRECT ANSWER:
AdjustedYield = (1+Yield)/(1+Inflation) - 1

CORRECT ANSWER DERIVATION:
The definition of ‘yield’ (or ‘return’), evaluated at some date in the future is:

Yield = (FuturePrice - TodaysPrice)/TodaysPrice = FuturePrice/TodaysPrice - 1

When there’s inflation then both prices in the above equation for Yield (or Return) need to be put on equal terms. Let’s adjust TodaysPrice to FuturePrice based on an annual inflation rate, “Inflation”. For simplicity, we’ll assume the future date is one year from today:

TodaysPriceAdjusted = (1+Inflation)*TodaysPrice

We need to use TodaysPriceAdjusted instead of TodaysPrice in the above formula in order to get AdjustedYield:

AdjustedYield = FuturePrice/TodaysPriceAdjusted - 1

Substituing in TodaysPriceAdjusted:

AdjustedYield = FuturePrice/[(1+Inflation)*TodaysPrice] - 1

Rearrange to isolate FuturePrice/TodaysPrice
AdjustedYield = [1/(1+Inflation)] * FuturePrice/TodaysPrice - 1

We know from above that
Yield = FuturePrice/TodaysPrice - 1
so
FuturePrice/TodaysPrice = 1 + Yield

Substitute this into the formula for AdjustedYield to get:

AdjustedYield = [1/(1+Inflation)]*(1 + Yield) - 1
i.e.

AdjustedYield = (1+Yield)/(1+Inflation) - 1

Example 1:
If the nominal yield is 5% and inflation is 8%, then the adjusted yield is
(1+0.05)/(1+0.08) -1 = -0.0277 or a loss of 2.77%
The loss is not 5-8= -3%

Example 2:
If the nominal yield is 10% and inflation is 8%, then the adjusted yield is
(1+0.10)/(1+0.08) -1 = 0.0185 or a gain of 1.85%
The gain is not 10-8= 2%

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3

What you say above is literally correct, but not the entire story, right?

Formula 1: AdjustedYield = Yield - Inflation

is an approximation of

Formula 2: AdjustedYield = (1+Yield)/(1+Inflation) - 1

where the approximation is increasingly accurate as the value of the [Inflation] term is small (for those interested, the approximation is calculated by a Taylor Series of the exact formula). That is why in your examples 1 and 2

-2.77% is not too different from -3%
and
1.85% is not too different from 2%

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4

Yes, for suitable limits on inflation and return the Taylor expansion shows that simple subtraction is close to correct.
I just wanted to point out that simple subtraction is not the entire story, as many people might think.

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