Bonds and the Upsides/Downsides of Interest Rate Changes

Scope Of This Post

One discussion I keep having with people centers around the hypothetical scenario of a person buying bonds and then interest rates change.  In particular, people seem to be worried about buying bonds and then interest rates increase (especially an unexpected increase since expected increases are supposedly already priced into bonds).

My assertion is that although an investor would have benefited from delaying purchase of bonds until after the (unexpected) increase in interest rates, an investor holding pre-existing bonds is not necessarily worse off when interest rates increase.

For most of my post, I will be talking about interest rates changing but inflation staying constant.  Also, the characters in my stories like to buy 3-year bonds and only 3-year bonds.  So, when my story involves "interest rates changing from 5% to 6.9%", I'm specifically talking about the interest rates for 3-year bonds only.  I know the world has more than 3-year bonds, but I'm keeping my stories short and simple.

I will also ignore callable bonds, TIPS, taxes, convexity, and default risk; I believe those concepts do not change the conclusions about the benefits/harms of interest rate changes on bond holders.

I welcome comments and especially corrections to the assertions I make in this post.

A Very Brief Review of Bonds And Some Shorthand Notation

I will try to use standard bond terminology, but I will also use some custom notation to tersely describe bonds with different coupon rates and times until maturity.

When I say, "2 of Bond{5/100,3y}", that will mean I am talking about holding 2 bonds that have a $5 annual coupon, and have a $100 redemption payment in 3 years.  A holder of 2 of Bond{5/100,3y} will receive:

• 2*$5 coupons ($10) 1 year from now
• 2*$5 coupons ($10) 2 years from now
• 2*$5 coupons ($10) 3 years from now
• 2*$100 redemption payments ($200) 3 years from now

Once we simplify things by ignoring that bonds have different maturities and credit risks, we can see that bonds compete price-wise so that their yield-to-maturity (YTM) is equal to the market interest rate, regardless of coupon rate and face value.  Some simple example relationships between price and yield-to-maturity...

• If a Bond{5/100,3y} is priced at face value ($100), then the yield-to-maturity (YTM) is 5%.
• If the market interest rate is 5%, then a Bond{5/100,3y} will be priced at $100.
• If a Bond{5/100,3y} is priced at $95, then its YTM is 6.9%.
• If the market interest rate is 6.9%, then a Bond{5/100,3y) will be priced at $95.

Thus, in a interest rate environment of 5%, a Bond{5/100,3y} will be priced at $100, but if the interest rate increases to 6.9%, the Bond{5/100,3y} will drop in price to $95.  If YTM is an interesting new concept to you, or you wonder why price depends on YTM so heavily, I would recommending googling and reading up on: "time value of money", "internal rate of return", "bond pricing", and "bond yield to maturity".

Super-DUPER Fun Fact: When someone accuses you of being a "fat cat, clippin' your coupons!", they are not talking about clipping coupons to save money at the grocery store.  They are referring to many years ago when bond owners would clip physical coupons from their bond certificates so they could exchange the coupons for payment.

Interest Rate Hikes Help Those Who Can Invest In The New Bonds

My basic assertion is that a bond holder benefits from an increase in interest rates as long as that bond holder is able to reinvest at that higher interest rate.  As a rule of thumb, someone maintaining or increasing his bond holdings will benefit, and someone who is heavily decreasing his bond holdings over time will be hurt.

This assertion comes from the fact that a bond holder's future cash flows from a particular bond are unchanged by changes in interest rates.  If you buy and hold a Bond{5/100,3y}, you're going to get the same coupon payments and redemption payment regardless of changes in interest rates.  And if interest rates have increased, that means your coupon and redemption payments are able to buy BETTER bonds with BETTER returns than you could before.  Your cash flows have remained the same, and the returns on investing those cash flows (in new bonds) has increased.

People often object along the lines of, "but let's say that interest rates increase from 5% to 6.9% and my holdings of Bond{5/100,3y} decreases in price from $100 to $95; surely I am poorer after the increase in interest rate".

I will say yes, you are worse off if you immediately need to liquidate your bond holdings, but you are better off if you can use your coupon and redemption payments to purchase bonds at the new 6.9% YTM.

It's kind of like asking "would you rather have $100 and 5% returns on bonds or $95 and 6.9% returns on bonds?".  It depends on when you want to cash in on your bonds.

Example1: Larry, Who Reinvests Everything

Let's do an example with Larry and see how Larry's life is affected by changes in interest rates:

• Larry holds 1 Bond{5/100,3y}.
• The market interest rate is 5%
• Larry's bond holdings are priced at $100
• Larry plans to reinvest all of his coupon and redemption payments (always in 3 year bonds).  Thus, his holdings will grow at an annual rate of 5%.
• As the years go by at this interest rate, his holdings will be valued at: $105.00, $110.25, $115.76, $121.55, ...
• Suddenly interest rates change to 6.9%.
• Larry's bond holdings are now priced at $95
• Larry continues the plan of reinvesting everything, thus his holdings will grow at an annual rate of 6.9%
• As the years go by at this interest rate, his holdings will be valued at: $101.56, $108.56, $116.05, $124.06, ...
• 20 years pass at the 6.9% interest rate, and Larry finally sells his bonds to buy an abandoned coal mine to live in.
• Note that after at least three years pass, Larry is richer in the 6.9% interest rate scenario than the 5% interest rate scenario.  This happens whether he holds onto his Bond{5/100,3y} for a while, or trades it immediately for a fractional 0.95 of a Bond{6.9/100,3y}.

I feel it is uncontroversial to say that Larry benefited from the increase in interest rates.  Larry should welcome an increase in interest rates even after he first bought bonds, as that increase gets him closer to that sweet, sweet abandoned coal mine.  It is only if Larry had to liquidate his bond holding after the interest rate increase but before three years had passed, then Larry would have been hurt by the interest rate increase.

Example2: Moe, Who Was Happy With Beans But Got More

For another example, let's pretend Moe holds 1 Bond{5/100,3y} while interest rates are at 5%.  Moe is different from Larry; Moe needs $5 per year from this bond holding to spend on food and clean drinking water.

• Moe holds 1 Bond{5/100,3y}
• The market interest rate is 5%
• Moe's bond holdings are priced at $100
• Moe plans to spend all his 5% coupons on food/water and to use all redemption payments on repurchasing 3-year bonds.  Thus, his holdings and his spending will stay constant.
• Suddenly interest rates change to 6.9%.
• Moe's bond holdings are now priced at $95.
• Moe continues his plan of spending $5 per year on food and clean drinking water.
• Moe decides to sell his 1 Bond{5/100,3y} at $95 and purchase a fractional 0.95 of Bond{6.9/100,3y} for $95.  This bond trade is not necessary, but it makes the example much simpler.
• Moe is now getting annual $6.56 coupons (0.95 * $6.90 = $6.56), but only needs $5 per year.  Moe can now start getting growth on his bond holdings by reinvesting part of the coupon payments.  Moe also has the option to upgrade his lifestyle by increasing his spending.  Hopefully people won't think Moe is "putting on airs" when he buys a tin cup and spoon.  It's okay to enjoy the luxuries in life.

 I feel it is uncontroversial to say that Moe benefited from the increase in interest rates.  Even if Moe had been slowly eating into his principal, Moe would still benefit from the increase in interest rates.

For instance, imagine an alternate Moe, who has annual expenses of $10 per year, which he finances through coupon payments and selling of pieces of his bond.  If the interest rate stays at 5%, he'll run out of money in ~14.1 years.  The interest rate increase to 6.9% lets his money last ~15.5 years.  Alternate Moe also benefits from the interest rate increase.  If Alternate Moe's annual expenses are greater than ~$23.10, he'll start to get hurt by the interest rate increase.

But Larry/Moe Would Have Been Even Richer If...

A possible objection to my examples is that Larry (and Moe) both suffered an initial $5 loss as their bond holdings went from $100 to $95.  If only Larry had the $100 in cash, and bought the bonds AFTER the interest rate increase, then this alternate Larry would not suffer any loss at all and is always in a better position than the original Larry.

Yes, Original Larry is worse off than Alternate Larry, but both of them benefited from the increase in interest rates.  Larry might feel bad that he could have ended up even richer if he had just been able to foresee the unexpected increase in interest rates, but maybe Larry should focus on his improved situation for 3+ years from now.  I would also urge Larry to not try to "beat the market" in his predictions of interest rates.
 
Again, when Larry is holding his Bond{5/100,3y} at a market interest rate of 5%, he should welcome an increase in interest rates rather than worry about his exposure to interest rate risk.  If interest rates increase, Larry's bond will be worth fewer dollars, but Larry will be better off.

Example3: Groucho, Who Couldn't Afford His Tin Roof

Interest rate increases are not good for everyone. This example has someone who is hurt by an interest rate increase.

• Groucho holds 1 Bond{5/100,3y}
• The market interest rate is 5%
• Groucho's bond holdings are priced at $100
• The weatherman forecasts that it will start raining in one year.  Groucho wants to buy a $105 tin roof in one year so he'll be able to stay dry while sleeping.
• Groucho's plan is in one year to collect one $5 coupon and then sell his bond for $100.
• Suddenly interest rates change to 6.9%.
• Groucho's bond holdings are now priced at $95.
• At this interest rate, in one year Groucho will be able to collect a $5 coupon payment and sell his bond for $96.56 (that's the price that at that time gives his bond a YTM of 6.9%); at that time, Groucho will only be able to buy a roof that costs $101.56.
• Alternatively, Groucho could right now exchange his 1 Bond{5/100,3y} for a fractional 0.95 of Bond{6.9/100,3y}.  Then, in one year, he'll collect a $6.56 coupon and be able to sell his fractional bond for $95.  He'll still end up with $101.56.
• A year passes at the 6.9% rate, and Groucho buys a $101.56 roof.  But, as we know, roofs below $102 are a bit leaky, and Groucho is not able to stay dry at night.  As a direct result, he dies from several diseases.

I feel it is uncontroversial to say that Groucho was hurt by the increase in interest rates.

Example 3 Addendum: Groucho Was Not Foolish

I've gotten feedback that Groucho comes off as foolish, since he gambled on a 3-year bond, and died as a result of the gamble.  I will flesh out Groucho's story some more, to show that Groucho was not necessarily foolish in his purchase of a 3-year bond.

Let us go back in time to just before Groucho buys 1 Bond{5/100,3y}, and let's imagine that Groucho has $100 in cash.  Also, Groucho's labor income perfectly matches his ongoing expenses, so this $100 is his only way to prepare for a purchase of a roof in one year.  Imagine that Groucho has the following investment opportunities available to him:

• Groucho could keep the $100 in cash for the whole year, but that would only leave him with $100 to buy a leaky roof.  This strategy seems guaranteed to end badly (leaky roof).
• The market is offering Bond{1/100,1y} for $100.  Groucho could buy one of these 1-year bonds and have $101 in one year, regardless of changes in interest rates.  This strategy also seems guaranteed to end badly.
• The market is also offering Bond{5/100,3y} for $100.  Groucho could buy one of these 3-year bonds and have $105 or more in one year as long as 3-year interest rates don't increase.  Also, interest rates can rise up to 1.65 percentage points, and Groucho will still end up with a $102 leakproof roof or better.  This strategy has a chance of working, but Groucho is exposed some to interest rate risk.
• The market is also offering Bond{6/100,30y} for $100.  Groucho could buy one of these 30-year bonds and have $106 or more in one year as long as 30-year interest rates don't increase.  A $106 roof sounds really nice, but if 30-year interest rates increase by more than 0.3 percentage points, Groucho will end up with a <$102 leaky roof.  This strategy has a chance of working, but Groucho is heavily exposed to interest rate risk.
• Groucho could keep the $100 in cash, and maybe later buy some bonds once interest rates increase.  Unfortunately, the longer Groucho waits, the bigger the interest rate increase would have to be in order to end up with $102 or more dollars at year's end.  This strategy has a chance of working that depends on Groucho being smarter than the market (ugh).  Groucho would be exposed to a "reverse" interest rate risk of sorts.

So, what should Groucho do if he wants to maximize his chances of a leakproof roof?  Among this simplified list of opportunities, buying a Bond{5/100,3y} for $100 seems like Groucho's safest option.  Yes, Groucho would be "gambling" by buying a 3-year bond, but it would be silly to prefer guaranteed failure over a "gamble" that might succeed.

But What About Inflation?

The previous examples all assumed no inflation.  If the previous examples were altered to include a constant 1% inflation (that was already expected by the market), that would change the expenses for Moe and Groucho, but the conclusions of who was helped and hurt by the change in interest rates would remain the same.

If interest rates increased unexpectedly from 5% to 6.9% only because of an unexpected increase in inflation from 0% to 1.9%, that would hurt all bond holders.  But it's the unexpected increase in inflation that is the villain, not the increase in interest rates.  Increases in inflation always hurt holders of fixed-rate securities.

But What About Bond Funds?

The previous examples had characters that held individual (or even fractional) bonds, and sometimes chose to reinvest coupon payments and/or redemption payments in new bonds.  The stories do not change if the characters instead held bond funds that initially held 3-year bonds.

In each of the examples, the fate of the characters were always unchanged when they swapped out a bond for another bond of the same maturity.  The only thing that mattered was whether the characters decided to buy or sell bonds and at what times they did it.  It's the same thing with an ever-replenishing bond fund: the constant swapping out of bonds doesn't matter; the timing of your buying or selling that bond fund matters.

But What If Interest Rates Go Down?

If interest rates decrease, then the prices of current bonds will go up.  Pre-existing bond holders will be able to sell their bonds at a profit (and perhaps reinvest in something else) or stay in bonds if they wish.  As we've previously shown, (ignoring transaction costs and taxes) it does not matter if the bond holders keep their old bonds or swap them for new bonds of same maturity.

The downside of an interest rate decrease is that further investment in bonds will now give a lower return than before.  It might be proper to increase your allocation of other asset classes to best suit your desired risk-and-return profile.

The possibility of interest rates unexpectedly going down is not an argument against holding bonds now.  A decrease in interest rates helps people currently invested in bonds and hurts people who want to invest in bonds in the future; you can be both types of people at the same time.

It Sounds Like Interest Rate Risk Isn't Really A Risk?

No, there are definitely bad possibilities that can arise from changes in interest rate.  The longer the maturity of a bond, the worse it will be hit by an interest rate increase.  This interest rate risk is part of why long term bonds are often priced to have higher YTM than short term bonds.  So, the increased YTM creates a temptation to buy a bond that you'll need to sell before the redemption date.

Imagine a Bond{4/100,1y} and a Bond{5/100,30y}, both priced at $100.  Like Groucho, you'd love to have $105 in one year for a tin roof.  Unfortunately you only have $100 dollars right now.  The 1-year bond is guaranteed to leave you $1 short.  The 30-year bond will let you have your beloved $105 tin roof as long as interest rates don't go up before you sell the bond.

Are you nervous?  You can live with a $104 roof...but you'd really love to have a $105 tin roof, BUT if interest rates go down even a little bit, then the 30-year bond could let you have a ridiculously luxurious $110 roof, BUT BUT that 30-year bond could easily lead to a roof below $102 WHICH MIGHT KILL YOU.  Will interest rates go up or down?  Why does this 30-year bond have to be so sensitive to changes in the interest rate?!?!?

That's interest rate risk.  Quite the roller coaster if you're hoping to sell some bonds way before their redemption date.

SO INTEREST RATE RISK IS LIFE AND DEATH?!

Well, if you're Groucho.  Re-reading the Larry and Moe examples might make you feel better.  Even Alternate Moe, who was aggressively liquidating ever increasing portions of his bond holdings, benefited from an interest rate increase.  But, thank goodness Alternate Moe was holding 3-year bonds and not 100-year bonds.

In general, if you're holding and buying N-year bonds, it takes N years or less to start to benefit from an interest rate increase.

Also, the word "risk" is heavily associated with "possible losses", but if you hold onto a bond until maturity, that bonds is going to give you the same cash flows no matter what interest rates do.  So yes, there's risk in terms of uncertainty of how much you can sell your bonds for in the future, but a change in interest rates doesn't cause a loss of any future coupon or redemption payment.

Perhaps it is better for Larry and Moe to think of interest rate risk as mostly just a source of uncertainty.  Groucho can definitely think of interest rate risk as a source of potential losses.

Reminder: I'm ignoring default risk and inflation risk.  The possibilities of very real losses on bonds mostly come from those two risks.

Does Anyone Smart/Experienced Agree With Your Thesis?

I think so.  Part of the purpose of this post is to test that I have properly understood what I have read about bonds.  (Reading something and seeing if you feel like you understood it is not a good enough test.)

Here are some pieces by smart/experienced people that I believe are consistent with what I've been saying:

• Why rising rates are good for long-term investors
• "While bond prices take a hit initially when rates rise, income reinvested at higher yields not only helps to recover bond losses, but can also compound into a significant portion of a bond fund's long-term total return."
• "Market expectations of a rate rise are already priced into the yield curve and, thus, reflected in bond prices. It's the surprise to expectations that moves rates and bond values."
• For bondholders, rising interest rates can have an upside 
• "Conventional wisdom holds that interest rate increases are bad for bond portfolios. But in fact, depending on your time horizon, you can benefit from rising rates."
• "A helpful rule of thumb is that if your time horizon is longer than the duration of your bond fund, you stand to benefit"
• Fear of interest rate risk creates opportunity costs 

Appendix A: Reminder That There Are Three Types of Bond Duration

 This bond duration Wikipedia article is decent (and so is the more advanced bond convexity article).  Also, this Investopedia article is a bit shorter but the formulas are missing some math notation.

Effective Duration:

• How much a bond's market price will change per percentage point change in market interest rate; note that effective duration is not a time frame; it is a price sensitivity to interest rate changes.
• Formula: EffectiveDuration = (PriceDown - PriceUp) / (2 * PriceOrig * InterestRateDelta);
• PriceDown: new market price if interest rate decreases by InterestRateDelta.
• PriceUp: new market price if interest rate increases by InterestRateDelta.
• PriceOrig: old market price before any interest rate changes.
• InterestRateDelta: hypothetical change in interest rate, in natural units (one percentage point interest rate change will be 0.01)
• Note that the effective duration calculation changes depending on the chosen interest rate change, which is getting into bond convexity.

Modified Duration:

• TODO

Macaulay Duration:

• TODO