Mini Chapter Three

Duration & Convexity

Duration

Duration quite literally is the average time taken to receive the promised cash flows on a financial instrument. We say financial instrument and not merely a fixed income instrument because in theory anything with future cash flows is sensitive to interest rates (time value of money) and therefore would have duration risk in it. Three main drivers of duration are: a) tenor of investment b) coupon rate c) nominal yields (YTM).

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 instrument’s cash flows

b. A 5% coupon-bearing bond with intermediate coupon earnings would have a shorter time period/duration to realise all cash flows

c. And a 7% coupon-bearing bond would have a still shorter time period and hence the lowest duration among all

Duration as described above can be mathematically denoted as below:

\frac{Σ_{i = 1 t o n} {T_{i} \times \frac{C_{i}}{{(1 + y)}^{T_{i}}}}}{B}

, where:

- T is the time period in years and Tᵢ = i^th time period

- C denotes the cash flow and Cᵢ 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, i.e.

B = Σ_{i = 1 t o n} {\frac{C_{i}}{{(1 + y)}^{T_{i}}}}

For the sake of simplicity, we have assumed annual compounding above, but in case of coupon payment frequency of more than once a year do remember to adjust the YTM for that frequency. For instance an 8% annual coupon payable quarterly should have

{(\frac{1 + 8%}{4})}^{0.25}

as the discount factor for the first quarter and so on.

{(\frac{1 + 8%}{4})}^{0.25}

You can think of the formula above as the weighted average time period for receiving all cash flows where the time period T_i is weighted by the proportion of the bond’s total present value received at time Tᵢ. Sum of these weights would naturally be equal to 1. Only in case of a non-zero coupon-bearing bond or an instrument with intermediate cash flows will the duration (measured in years above) be lesser than the tenor/maturity of the bond. This is also called Macaulay’s Duration, named after the Canadian Economist Frederick Macaulay.

As an extension of the above what’s used in practice is the Modified Duration which is a practical representation of the price sensitivity of a bond given a marginal change in its yield i.e.

\frac{Δ B}{Δ y}

.

Mathematically this is explained applying the power and chain rule of differentiation on the bond price notation B to arrive at:

\frac{Δ B}{Δ y}

\frac{Δ B}{Δ y} = Σ_{i = 1 t o n} {- T_{i} \times \frac{C_{i}}{{(1 + y)}^{(T_{i} + 1)}}}

= \frac{- 1}{(1 + y)} \times Σ_{i = 1 t o n} {T_{i} \times \frac{C_{i}}{{(1 + y)}^{T_{i}}}}

= \frac{- 1}{(1 + y)} \times {Σ_{i = 1 t o n} \frac{{T_{i} \times \frac{C_{i}}{{(1 + y)}^{T_{i}}}}}{B}} \times B

= - \frac{1}{(1 + y)} M a c a u l a y' s D u r a t i o n \times B

First term, {\frac{1}{(1 + y)} \times M a c a u l a y' s D u r a t i o n}

First term, {\frac{1}{(1 + y)} \times M a c a u l a y' s D u r a t i o n}

on the right-hand side of the equation above is the modified duration carrying the negative sign to reflect the inverse yield vs price relationship.

First term, {\frac{1}{(1 + y)} \times M a c a u l a y ‘ s D u r a t i o n}

on the right-hand side of the equation above is the modified duration carrying the negative sign to reflect the inverse yield vs price relationship.

The expression can be approximated to read as:

Δ B = - \frac{1}{(1 + y)} \times M a c a u l a y' s D u r a t i o n \times B \times Δ y

Where ΔB is the dollar change in the bond’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,

Δ B = D V 01 = - M o d i f i e d d u r a t i o n \times M a r k e t v a l u e o f t h e B o n d N o t i o n a l \times 0.01%

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.

For dollar based investors trading local currency EM bonds, dollar value per basis point can be calculated from the PVBP of the local currency bond as below:

\frac{P V B P \times L o c a l C u r r e n c y N o t i o n a l}{S p o t F X}

DV01 is sometimes incorrectly confused with duration though latter is a key component in computing the former. The notation and example above clarify that DV01 is the dollar value change in the Bond price in response to a 1bp change in yield while Duration is a percentage metric i.e. change in cents versus a marginal change in yield.

Now let’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:

\frac{Δ B}{B} = \frac{- M a c a u l a y' s D u r a t i o n}{(1 + y)} \times Δ y

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.

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.

I’ve tried to assess several combinations of bond tenors and coupons to exhibit the change in  duration (convexity) at different yields. Segregating them into 3 groups you’ll notice:

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 – there’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.

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 trade-off 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.

For any given level of rates you would notice that duration of coupon bearing bonds tend to cap out irrespective of how long the tenors as the longer maturity cash flows being discounted by higher exponentials tend to have smaller and smaller impact on the duration value.

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 16-19 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 of 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. 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 pre-payment not only reduces duration in the books of banks as rates go lower, it also delivers them a new long-duration 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 mark-to-market losses.

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.

Mini Chapter Three

Duration & Convexity

Duration

Graph 1 - Bond price vs yield relationship to show convexity

Positive vs Negative Convexity

Share

Related Resources

Time Value of Money

1: Time Value of Money & Power of Compounding

2: Bootstrapping of Yield Curves – Par, Zero, Forward

4: Duration and PVBP for a Floating Rate Bond

5: Money Market and Cash Bonds

6: Foreign Exchange & Interest Rate Parity

quick links

Varda Pandey