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Topic 6. DV01/PVBP , Duration, and Convexity. DV01/PVBP, Duration and Convexity. Measures of risk of debt securities. Dollar value of an 01 or DV01 (also known as the price value of a basis point (PVBP). Duration measures. Convexity . DV01/PVBP, Duration and Convexity.
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Topic 6 DV01/PVBP, Duration, and Convexity
DV01/PVBP, Duration and Convexity • Measures of risk of debt securities. • Dollar value of an 01 or DV01 (also known • as the price value of a basis point (PVBP). • Duration measures. • Convexity.
DV01/PVBP, Duration and Convexity Consider a bond with one year to maturity. It pays a semi-annual coupon of 4% on a par value of 100. It is priced to yield 6%. What is its risk? Price at 6%: Price at 6.01%:
DV01/PVBP, Duration and Convexity The slope can be approximated to get DV01 as follows: Instead of increasing yield by a basis point, we can perturb the yield by half a basis point each way and compute an estimate of DV01.
DV01/PVBP, Duration and Convexity Price at 5.995%: Price at 6.005%:
DV01/PVBP, Duration and Convexity The slope can be approximated to get DV01 as follows:
DV01/PVBP, Duration and Convexity • Macaulay duration is the discounted-cash-flow-weighted • time to receipt of all promised cash flows divided by the • price of the security. Duration is a linear measure or 1st derivative of how the price of a bond changes in response to interest rate changes. It measures the elasticity of price to rate changes – the % change. • This implies that Macaulay duration is the average time • taken by the security, on a discounted basis, to pay back • the original investment. • 3. Hence, longer the duration, greater is the risk. Duration is shorter than maturity for all bonds except zero coupon bonds. Duration of a zero-coupon bond is equal to its maturity
Qualifiers • First-order approximation • Accurate for small changes in yield • Limitation: Depends on parallel shifts in a flat yield curve • Multifactor duration models try to address this • Strictly applicable only to option-free (e.g., non-convertible) bonds
Check your intuition • How does each of these changes affect duration? • Having no coupon payments. • Decreasing the coupon rate. • Increasing the time to maturity. • Decreasing the yield-to-maturity.
DV01/PVBP, Duration and Convexity Modified duration is the percent change in the price divided by the change in yield.
DV01/PVBP, Duration and Convexity This is Table 7.8 in the text with errors corrected.
Convexity • Convexity is the second order derivative of the price equation w.r.t. yield. It’s a measure of the degree to which a bond’s price-yield curve departs from a straight line i.e. how the duration of a bond changes as the interest rate changes. • Convexity is a risk management figure, used similarly to the way 'gamma' is used in options risks management; it is a number used to manage the market risk a bond portfolio is exposed to. If the combined convexity of a trading book is high, so is the risk. However, if the combined convexity and duration are low, the book is hedged, and little money will be lost even if fairly substantial interest movements occur. (Parallel in the yield curve.)
Numerical example with Convexity • Consider a 20-year 9% coupon bond selling at $134.6722 to yield 6%. Coupon payments are made semiannually. • Dmac= 10.98: D-mod =? • The convexity of the bond is given as 164.106 (annual), 81.95 seminannual.
Numerical example • If yields increase instantaneously from 6% to 8%, the percentage price change of this bond is given by: • First approximation (Duration): –10.66 .02 100 = –21.32 • Second approximation (Convexity) 81.95 (.02)2 100 = +3.28 Total predicted % price change: –21.32 + 3.28 = –18.04% (Actual price change = –18.40%.)
Numerical example • What if yields fall by 2%? • If yields decrease instantaneously from 6% to 4%, the percentage price change of this bond is given by: • First approximation (Duration): –10.66 –.02 100 = 21.32 • Second approximation (Convexity) 81.95 (–.02)2 100 = +3.28 Total predicted price change: 21.32 + 3.28 = 24.60% Note that predicted change is NOT SYMMETRIC.
Takeaways: Duration and Convexity • Price approximation using only duration: New Bond Price ($) = P + [P (Duration)] • Price approximation using both duration and convexity: New Bond Price ($) = P + [P (Duration)] + [P (Convexity)] (P +Eq. 7.16):
Convexity Enhancement Swaps: Barbell-Bullet Trades Barbell: • Combination of short and long tenor bonds Bullet: • Single bond of intermediate maturity Barbell-Bullet trade: • Arbitrage spread trade • Long one position, short the other • Equal duration risk and market values
Bullet – Barbell Example Worksheet: • Barbell: purchase bonds A&B • Bullet: sell bond C • What weights to assign to bonds A&B? Use Solver
Bullet – Barbell Example • Net Yield Gain: 0.06%
Session 7 - Conclusions/Main insights • Price-yield relationship is convex. As a first approximation, we can use DV01, Macaulay Duration and Modified Duration as measures of risk of debt securities. • Modified duration is the percentage change in price for a change in yield, and is used in practice along with DV01. • These ideas can be extended to duration of portfolio of securities. • Convexity can be quantified under some assumptions. • Barbell portfolios of same duration tend to have higher convexity than bullet securities with same duration.
