Mini Chapter Three
Duration & Convexity
Duration
For instance assuming the same tenor of 5 years:
a. A zero coupon bond that has no intermediate cash flows would take exactly the tenor of the bond i.e. 5 years to realise the instrumentâ€™s cash flows
b. A 5% couponbearing bond with intermediate coupon earnings would have a shorter time period/duration to realise all cash flows
c. And a 7% couponbearing bond would have a still shorter time period and hence the lowest duration among allÂ
Duration as described above can be mathematically denoted as below:Â
, where:

 T is the time period in years and Táµ¢ = i^{th} time period

 C denotes the cash flow and Cáµ¢ is the cash flow at time period i

 y is the annualised yield to maturity (YTM) of the bond payable annually

 B is the current bond price which if you recall is the present value of all bond cash flows discounted at y,Â Â i.e.
Mathematically this is explained applying the power and chain rule of differentiation on the bond price notation B to arrive at:
Where Î”B is the dollar change in the bondâ€™s present value for a marginal change in its yield. To standardize this concept across bond tenors/types we assess present value sensitivity for a 1 basis point change in yield also termed as price value per basis point (PVBP) or dollar value per basis point (DV01). Elaborating further,
For example a long bond position for a face value of USD 5 mio, with a modified duration of 3.55 and market price at 101.75 would have a DV01 (assuming 1bp move up in yield) of:
3.55 x 101.75% x 5,000,000 x 0.01% = ~ – USD 1806
Or in case of a 1bp move down in yield the dollar value change in the bond price would be ~ +USD 1806.
It follows then higher the duration higher the price sensitivity of the bond.
Now letâ€™s get to the second order impact of the change in bond yields on its prices i.e. the change in duration with a marginal change in yield. We already know from the duration formulae above that a higher/lower yield corresponds with a lower/higher duration. From the approximate notation above we can see that:
Interestingly as yields fall not only does the price of the bond rise but even the percentage change in the bond price is higher given the same movement in yields. This is the same as saying that Duration (both Macaulay and Modified) also rises as yields fall and vice versa (notation above proves it). Conversely when yields rise the bond prices fall but the rate of fall also decreases (as duration decreases) and hence the bond price fall is not as much as the bond price rise for the same rise vs fall in yields. This leads to a convex (or normally referred to as positively convex) behaviour of prices for a long bond position.
Graph 1  Bond price vs yield relationship to show convexity
Source: Pandemonium.Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
Table 1 – Duration across different coupon tenor bonds vs IRRs/YTM
IRR  5% 2y  5% 5y  5% 10y  0% 2y  0% 5y  0% 10y  10% 2y  10% 5y  10% 10y 

1.0%  1.9  4.5  8.3  2.0  4.9  9.8  1.9  4.3  7.5 
2.0%  1.9  4.5  8.1  2.0  4.9  9.7  1.9  4.2  7.3 
3.0%  1.9  4.4  8.0  1.9  4.8  9.6  1.9  4.1  7.2 
4.0%  1.9  4.4  7.8  1.9  4.8  9.5  1.8  4.1  7.0 
5.0%  1.9  4.3  7.7  1.9  4.7  9.4  1.8  4.0  6.9 
6.0%  1.8  4.3  7.5  1.9  4.7  9.3  1.8  4.0  6.7 
7.0%  1.8  4.2  7.4  1.9  4.7  9.3  1.8  3.9  6.6 
8.0%  1.8  4.2  7.2  1.9  4.6  9.2  1.8  3.9  6.4 
9.0%  1.8  4.1  7.1  1.8  4.6  9.1  1.8  3.8  6.2 
Source: Pandemonium.Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
1. Group 1 – Same coupon for different tenors – longer duration bond i.e. 10y is more
convex then 5y is more convex than 2y. Hence longer the tenor for the same coupon
more convex it is.
2. Group 2 – Different coupons for same tenors – thereâ€™s more duration in lower
coupons but lower convexity in them and vice versa.
3. Group 3 – Higher coupons (consider 10% above) that tend to annuities would be
lowest on duration for any specific tenor but highest on convexity.Â
Thus intuitively portfolio managers who are uncertain about direction of rates tend to buy higher coupon (more convex) bonds as those would give them more protection in adverse moves. However the tradeoff is that when yields fall they would make less money because of its lower duration. If one is convinced that yields would fall then they would tend towards the longest duration bond – aka a zero coupon bond – to maximise gains.
 Trivia
For instance in a 5% interest rate regime, a 30y or 50 year coupon bearing bonds would tend to have duration in a fairly tight range peaking around 1619 years whereas in a 10% regime the similar peak is observed around the 10 year duration mark. Low single digit coupons do create more dispersion of duration even for the longer tenors as – firstly the max duration level is higher than that seen on higher coupons, secondly the incremental exponential impact of lower coupons is larger than the higher coupons which causes more dispersion.
Positive vs Negative Convexity
The regular usage of convexity is to denote positive convexity which comes into play as duration goes lower with higher interest rates and vice versa. All vanilla fixed income instruments whether bonds or swaps are positively convex in that regard. Negative convexity is just the reverse i.e. an increase in duration as interest rates increase. The magnitude of convexity for a fixed income portfolio is the sum of convexity across all its holdings – long/short positions would have positive/negative convexity and portfolio managers would tend to construct portfolios with positive convexity for obvious reasons.
Being exposed to Bonds with an embedded call option in them are a classic example ofÂ negative convexity. Holders of mortgage loans (banks) are effectively short an option whereby the borrower could prepay the loan and refinance it at a much lower rate. (i.e. Â the borrower is long a put option on the higher rate loan) thereby denying the banks to earn a higher interest rate for a longer duration. The loan prepayment not only reduces duration in the books of banks as rates go lower, it also delivers them a new longduration exposure at a much lower rate. The opposite would happen when borrowers choose to run the loan for longer as rates go higher, delivering the banks a longer duration exposure in a rising rates environment implying higher marktomarket losses.
 Connecting the Dots
It would be easier to identify a negatively convex situation for a portfolio as being short gamma (that comes with being short an option) and positively convex as being long gamma. We would discuss that in detail in later sections on option greeks.