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This Brunette Prefers Bonds: Finding Love in Mathematical Certainty

This Brunette Prefers Bonds: Finding Love in Mathematical Certainty

June 05, 2023

This Brunette Prefers Bonds

While stocks may be sexy talk in social gatherings, I prefer bonds. And not because of their “safety” or income streams, or because they tend to be great diversifiers in a portfolio.  I love bonds because I love math. Yes, I am a math nerd and bonds are math.  Like math, they have an answer: before the bond matures you receive the coupon payment and at maturity, you get the full-face value of the bond back (which might be more or less than you invested assuming it does not default). If something changes in the bond equation, the answer changes.  Easy peasy.

Among bond investors there is often confusion when trying to understand how bonds behave before maturity. This is understandable, for example, when rates rise that SOUNDS like it is a good thing if you hold bonds. Higher rates mean higher returns, don’t they? NO, at least not for current bond investors who bought when rates were lower. This makes sense from a math perspective as well as from a common-sense perspective.

Let’s look at a simple 1-year bond sold at full face value (typically $1000):

Bond price = $ coupon/(1+r)1 + (Face Value- Purchase Price)/(1+r)1

So, you can see from a math perspective if interest rates, r ,go up (denominator gets bigger), bond prices come down, and the reverse is true too (if rates go down, bond prices go up).

And this makes intuitive sense too.  If person A buys a bond when interest rates are, say, 4%, and pays $1000 for it, and then interest rates go to 5%, to entice someone to buy the 4% bond its price needs to come down to make it just as attractive (all other things being equal) as the 5%. And the reverse makes sense too: if rates go to 3%, the 4% bond would be more attractive so its price would go up to equalize the return to the investor (or yield to maturity which is the interest rate in the above equations).  

So, we can see mathematically and common sense-wise that interest rates and bond prices move in opposite directions (not considering creditworthiness and other factors).

But what IS the return to the investor? This is where more confusion comes in as there are many different “returns” that are often cited when describing a bond. No, it is not the coupon.  The coupon is one component of the return, it is the periodic % amount you receive over the life of the bond.  The other component to return is price change, and this is relevant when bonds are purchased at a price other than face value

There’s another term to describe the return of a bond, and that is current yield which is simply the coupon payment/current price.  So, if you buy a bond below its face (maturity) value, the current yield will be higher than its coupon (since the denominator is lower). But this too is not the actual return to the bond investor.

The actual return to the bond investor is the yield to maturity, r, which is arrived at by summing up the coupon payments together with the change in value of the bond (from the purchase date to the date sold or matured) and expressing this sum in todays $ terms as an annualized % of the purchase price. 

To recap:

Coupon = periodic interest payment

Current yield = coupon/bond price (PV), and

Yield to maturity (or return) = rate the bond returns if it pays all coupon payments and face value (FV) at maturity (in n periods):

Now you too can love bonds!

Studied, researched and written by Carla Devlin

The information contained is for educational purposes only and it is not intended to cover all aspects of a particular matter. Neither the information presented, nor any opinion expressed constitutes a representation by us or a solicitation of the purchase or sale of any securities. The information provided is not written or intended as tax or legal advice. Investments are not FDIC-insured, nor are they deposits of or guaranteed by a bank or any other entity, so they may lose value.