MODULE 12: Yield-Based Bond Convexity and Portfolio Properties

1. Think of duration as your first rough guess. You see yield move and you ask, "Fine, how much should price move?" Convexity enters one step later and says, "Not so fast, the path is curved, so your first guess needs a correction."

2. Convexity matters because the true price-yield relationship is curved, not straight. Why is convexity used: to improve price estimates when duration alone starts missing the real move.

3. For an option-free fixed-rate bond, convexity is always positive. That is a very friendly feature. When yields fall, the bond helps you more than duration promised. When yields rise, it hurts you less than duration threatened.

4. That is why convexity is valuable to investors. Why is convexity valuable: it reduces downside damage and increases upside gain for the same absolute yield move.

1. The percentage price change estimate is: $$ \% \Delta PV_{\text{Full}} \approx (-\text{AnnModDur} \times \Delta \text{Yield}) + \left(\frac{1}{2} \times \text{AnnConvexity} \times (\Delta \text{Yield})^2\right) $$

2. Read that formula like a tired exam candidate. First grab the big move from duration. Then calm down and add the smaller convexity correction. Do not try to feel the whole formula at once.

3. Use full price in these formulas, not flat price. What is full price: the bond price including accrued interest.

4. Approximate convexity can be estimated from bumped prices: $$ \text{ApproxCon} = \frac{PV_- + PV_+ - (2 \times PV_0)}{(\Delta \text{Yield})^2 \times PV_0} $$

5. Here, \(PV_0\) is today's full price, \(PV_+\) is the bruised price after yield goes up, and \(PV_-\) is the happier price after yield goes down. You are basically poking the bond on both sides and seeing how curved its reaction is.

6. For BRWA's five-year 3.2% bond, \(PV_+ = 99.770965\) and \(PV_- = 100.229641\) after a 5 basis point shock. That gives approximate convexity of 24.23896, which is almost identical to the annual convexity from the cash-flow table.

7. This price-bump method is handy when the neat textbook cash-flow route becomes annoying. If you can reprice the bond after a small bump up and bump down, you can still get the bend without begging for a perfect closed-form setup.

1. Convexity rises when maturity is longer, coupon is lower, and yield-to-maturity is lower. Why: those features push more value into later cash flows, and later cash flows react more violently to discount-rate changes.

2. One more driver is cash-flow dispersion. That just means how spread out the bond's payments are across time.

3. If two bonds have the same duration, do not relax and call them identical. The one with cash flows spread farther across the timeline usually has more convexity, so its price path bends more.

4. The Romanian 30-year bond makes this obvious. Its approximate modified duration is 15.90637 and approximate convexity is 369.64203. BRWA's five-year bond has duration 4.58676 and convexity 24.23896.

5. The Romanian bond wins on convexity mainly because 30 years of maturity overwhelms the fact that its coupon and yield are higher. Why: longer maturity increases convexity, while higher coupon and higher yield reduce it.

1. Duration alone pretends the world is polite and symmetric. Real bond prices are not. The same 100 basis point move up and down does not create mirror-image results, because the real price-yield line is curved.

2. For BRWA's five-year bond, a 100 basis point yield increase gives an actual price change of -4.46788%, while a 100 basis point decrease gives 4.71035%.

3. Duration alone estimates both moves at 4.58676% in absolute value. Adding convexity improves the estimate to -4.46556% for the yield rise and 4.70795% for the yield decline.

4. This is the whole practical point of learning convexity. You are not memorizing a fancy extra term for decoration. You are learning how to be less wrong when yields move hard.

1. Money convexity is what happens when the abstract bend finally turns into money. Now you are no longer saying, "This bond has convexity 24." You are saying, "Fine, but how many dollars is that bend worth to me?"

2. The formula is: $$ \text{MoneyCon} = \text{AnnConvexity} \times PV_{\text{Full}} $$

3. Then money price change is estimated with: $$ \Delta PV_{\text{Full}} \approx -(\text{MoneyDur} \times \Delta \text{Yield}) + \left(\frac{1}{2} \times \text{MoneyCon} \times (\Delta \text{Yield})^2\right) $$

4. For a USD 100 million BRWA position, money convexity is USD 2,423,894,503. With a 100 basis point yield increase, the actual position loss is USD 4,467,884.

5. Money duration alone estimates a bigger loss of USD 4,586,759. Adding money convexity improves the estimate to USD 4,465,564, so the miss shrinks to only USD 2,320.

1. At portfolio level, the story gets more practical. You can build duration and convexity from the whole pile of cash flows, or you can take weighted averages of the bonds sitting in the portfolio.

2. The aggregate-cash-flow method is the clean textbook answer. It is also the kind of answer people quietly avoid when the portfolio is large and real life is messy.

3. So what do managers actually do? Usually the weighted-average route. Take each bond's duration and convexity, weight them by market value, and add them. It is fast, usable, and good enough for a lot of decisions.

4. In the EUR 100 million two-bond example, the weights are 40% and 60%. Portfolio duration is 6.169 and portfolio convexity is 52.921.

5. If yields rise 50 basis points, estimated portfolio percentage price change is: $$ \% \Delta PV_{\text{Full}} \approx (-6.169 \times 0.005) + \left(\frac{1}{2} \times 52.921 \times 0.005^2\right) \approx -3.018\% $$

6. In the BRWA-Romania portfolio, weighted-average modified duration is 10.19004 and weighted-average convexity is 195.21581.

7. If all yields in that portfolio rise 100 basis points in parallel, the estimated portfolio decline is 9.21396%.

8. Here is where the shortcut quietly cheats. It assumes a parallel shift, meaning yields across maturities all move together by the same amount and in the same direction.

9. But you and I both know markets are rarely that polite. Curves steepen, flatten, and twist. So weighted-average portfolio measures are useful, but they are still a shortcut, not holy truth.

10. The weighted-average method becomes more accurate when yield differences across bonds are smaller and the yield curve is flatter.

11. In Example 2, adding a USD 50 million par 10-year US Treasury lowers portfolio duration from 10.19004 to 9.87415 and convexity from 195.21581 to 161.62749.

12. If the investor expects a 100 basis point parallel fall in yields, the Treasury should not be added. You want the portfolio that leans harder into the rally, and that means more duration plus more convexity, not less.