Duration and Convexity
Locking in a NOMINAL Rate of Return
I’m Back…For Awhile
It’s hard to describe chemo week other than feeling like I’m taking sleeping pills for the week (plus a couple of days after). The good news is that it is over in a week or so and life returns to (somewhat) normal. The bad news is it looks like there is another 10 months to go. Just a head’s up though that we’ll probably be on a 3 week on, 1 week off schedule for a bit.
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Duration and Convexity
One issue that I’ve stayed away from since starting this update is the whole concept of bonds. A bond typically pays a fixed coupon stream every 6 months (for simplicity, we’re going to assume annually as we go through this). Therefore, you buy a 10-year 5% coupon bond, you are going to receive $25 every 6-months and then at the end of the 10 years, get your $1000 back. Granted, this is a bit of a simplification as it ignores options (puts, calls, convertibles, etc.) or abnormal bonds (catastrophe bond, variable rate bonds, etc.). If anyone wants to dive into these issues, let me know and I can write another column or two. Unfortunately, over the past couple of years, interest rates were so low that bonds were guaranteed to offer a low return (TINA — There Is No Alternative). Granted, it was positive, but very low on a nominal basis (unless you are assuming negative inflation rates…not exactly what we’ve seen for the last few months). If you look at the graph below, you’ll see the yellow line marking a 2% yield on a 10-year Treasury. This means that if you buy a 10-year bond and plan to hold it for 9.16 years (more on this in a minute) to lock in your 2% return (on a nominal basis). Assuming you bought in at a 2% rate…things were even worse in 2020 when the yield was below 1%. Fortunately rates are rising (slowly), so bonds may be better investments going forward.
Duration refers to the holding period where your reinvestment rate risk is exactly offset by your price risk. If I buy a bond, there are two major sources of risk. First, if rates rise, you will earn a higher return on your reinvested coupons (which is great). Unfortunately, it is offset by the price declining (which is less great). The opposite is also true. If rates fall, your reinvested coupons are worth less, but you offset that by a higher price.
Let’s walk through the example using the 2% coupon, 10-year bond. Again, I’ll simplify by making it annual…the difference is more detail than needed as convexity — another topic we’ll introduce in a minute — and assumptions about your reinvestment rate and future interest rate changes will cause FAR more distortion than your actual calculation.
Let’s assume that you buy the bond today and then (in a severe case of bad fortune), interest rates rise overnight to 4%. ARGHH, WHY ME!!!! As we know, the bond price will drop because now new bonds are paying 4% and our “old” bonds are paying a mere 2%. We can calculate the new bond price as follows:
PMT = $20 FV = $1000 N = 10 I/Y = 4 SOLVE FOR PV ==> $837.78.
Remember that this is not what we were hoping for as our bond price dropped. However, remember that if you hold until maturity, you will get the $1000 value of the bond so this is more of a loss in “value” (an opportunity cost). Now, imagine that rates stay exactly where they are, what will happen to the value of your bond? Each year as it gets closer to maturity, the price will rise slightly so that in year 10, you get the full $1000 face value (assuming no default). What happens then is a loss in value (what the bond is worth to you today when you COULD be earning a 4% return and you are only earning a 2% return).
Let’s look at that. If you have a cash flow stream of $20 per year (but COULD BE earning $40 per year), you are “losing” $20 per year for the remaining life of the bond.
PMT = -$20 FV = $0 (this is $1000 in either case) N = 10 I/Y = 4
SOLVE FOR PV ==> $162.22
Note that the difference in value between $1000 (what your bond was worth when you bought it) and the $837.78 that it is worth today is exactly $162.22.
So, we’ve figured out why the bond is worth less — it pays us less than we could earn on an equivalent new investment (opportunity loss).
One thing in finance is that we never assume we are spending the return, we are instead going to reinvest over the life of the asset, which in this case is 10 years. Thus, if interest rates stayed at 2%, I would get my $20 and reinvest it for 9 years at 2% and would have $23.90. However, now that rates are at 4%, I can reinvest it for the next 9 years and get $28.47. A “bonus” $4.56. My total reinvestment rate “profit” over the life of the bond is $21.13.1 Thus, I actually will not only get my money back, but will get more reinvestment income over the 10 years, which will allow me to earn a slightly higher return than my original 2%.
So, did this sudden rise in interest rates hurt me or help me? The answer lies in the calculation of duration. A sudden rise in interest rates makes the bond worth less today (a drop in the relative price of the bond) but is offset by a slightly better reinvestment rate. In other words, it is both bad (price risk) and good (reinvestment risk). The key then is to not hold your bond to maturity, but to hold it until duration…which in the 2% 10-year Treasury example is 9.16 years and will lock in a 2% rate of return.
Immunization — Rate Changes are Irrelevant!
If we start with a hypothetical example (and you can modify the time to maturity or coupon rate to whatever you want) of a 7% coupon, 30-year bond when the current market rate of interest is 8.5%. This gives us a duration of almost exactly twelve years (11.9941 if you want to be precise). We can see that it doesn’t matter whether rates rise from 8.5% to 9.5% or fall from 8.5% to 7.5%. In either case, after twelve years you will have a total of cash flow (rounded to the nearest dollar…a slight difference as the duration is not exactly 12) of $2241. In the first case, your bond is worth less (it drops in value from $838.80 to $788.82…a loss of $49.98), but that was offset by the higher coupon reinvestment rate ($1452.65). In the second case, your bond is worth more (It increase in value from $838.80 to $951.47…a gain of $112.67), but that is offset by the lower coupon reinvestment rate ($1289.66). By holding your bonds NOT for the 30-year holding period, but instead until the 12-year duration, you no longer care about whether or not rates go up or go down. In scenario 1, you end up with $2241 in total cash flows and in scenario 2, you end up with $2241 in total cash flows. You’ve locked in your rate of return!
You might have noticed a few problems with duration. First, it is an approximation. Yes, you can calculate the exact duration, but it also assumes that interest rates move once. They don’t. What happens if interest rise and then fall. You can take your 10-year Treasury yielding 2% and have rates climb to 4% over the first year which drops your duration to 9.07 years. Then have rates fall back to 2% over the next 3 years. Now you have a bond with 7 years to maturity and the duration is 6.60 years (3 + 6.60 = 9.60 years). So, do I hold for 9.07 years or 9.60 years? What about changes in interest rate sensitivity? For Treasury bonds that should be minimal, but what about financial institutions in 2007? What about the reinvestment rate? If I own a 10-year bond, do I need to anticipate what the spot rates will be at the various 1-year increments? Can I reinvest my coupons at a 9-year, 8-year, 7-year, etc. time frames. I can…but there is a cost to adding the flexibility that is miniscule for a larger account but significantly larger for a $20,000 bond holding where my $20 per year in coupon interest will net me $400 in interest.
In addition to a bond’s duration (the holding period at which we have immunized our risk from interest rate changes), we need to look at the convexity of the bonds. Duration works best for small changes in interest rates (what happens when interest rates move by 1% or so) rather than for larger changes. So, what happens if interest rate changes move by 3%? You end up needing to account for convexity as well (the non-linear aspect of bond prices). For example, if you think of a bond trading at a yield of 4% and rates fall or rise by 1%, you are looking at a range of 3% - 5%. However, what if rates fall or rise by 3%? Now you have a range of 1% - 7% and the relationship changes significantly. You can see the issue here with a 5-year 6% coupon bond and a 30-year 6% coupon bond. Clearly the longer the time to maturity, the greater the convexity (the purple line) distorts from the duration. Also, while not shown, the higher the coupon rate the greater the convexity (a 14% coupon bond will have more convexity than a 6% coupon bond).
Let’s recap a few issues with duration and convexity.
Tells us how (approximately) how much a given security's price will change for a given change in interest rates.
Tells us the (approximate) holding period where reinvestment rate risk and price risk offset one another (immunized portfolio risk).
Is more accurate for small changes in interest rates.
Is more accurate for shorter-term bonds.
Convexity is biggest for bonds with (a) higher coupons and (b) higher times to maturity.
Duration is an approximation (as with all financial calculations, it assumes only one thing changes) and should be used with a bit of skepticism.
Note that this does not make duration and convexity worthless. They are still incredibly valuable tools (and there is a lot not covered here). However, they are an approximation that can help you manage your interest rate risk in terms of nominal rates. It will not help you manage your real interest rate risk unless you happen to know exactly what inflation WILL average over the remaining life of your bonds (seeing into the future is a handy little skill in which we all apparently assume we have despite large evidence to the contrary).
The total is $4.56 + $3.94 + $3.34 + $2.78 + $2.25 + $1.75 + $1.27 + $0.82 + $0.40 + $0.00 = $21.13