Bond Risks and Duration Management
Bond Risks
Interest Rate Risk
Interest rate risk (price risk) is the change in the market prices of bonds due to varying interest rates.
- Bond prices and interest rates are inversely related.
Reinvestment Rate Risk
Reinvestment rate risk refers to the uncertainty surrounding the rate at which interim cash flows (e.g., coupon proceeds) can be invested.
- The higher the coupon or holding period, the higher the reinvestment rate risk.
- If interest rates go down, your interest on reinvested interest will decline.
Default Risk
Default risk (credit risk) occurs when the bond issuer is unable or unwilling to pay the interest and principal on a bond.
Credit Rating
Credit rating assesses credit risk.
- Investment grade = BBB or above.
- If below BBB = Junk bonds (lower credit rating, higher yield).
No credit risk for US Treasury Bonds but they do have interest rate risk. Closest to risk-free investment = T-Bills as they have no default risk and little interest rate risk.
Call Risk
Call risk refers to the risk that a bond issuer will redeem its bonds before they mature.
- Call provision = clause that allows the issuer to buy back bonds at a given price before maturity.
- Bond refunding occurs if interest rates fall and prices of bonds rise, allowing issuers to issue new bonds with a lower coupon.
Inflation Risk
Inflation risk arises because the value of a security’s cash flows vary over time due to changes in purchasing power.
- Real interest rate = nominal interest rate – inflation.
- TIPS protect from inflation as they are adjusted to the CPI, with a fixed coupon rate but an adjustable principal.
Liquidity Risk
Liquidity risk or marketability risk depends on the ease with which a bond can be sold at or near its value.
- The size of the spread between the bid and ask price measures liquidity risk; a wider spread indicates higher risk.
Duration
Duration is a direct measure of interest rate risk:
- The higher a bond’s duration, the higher its exposure to interest rate risk and price volatility.
- While maturity is the time until the last payment, duration is the time until the bond price is repaid by cash flows.
Macaulay Duration
Macaulay duration is the weighted average time until cash flows are received and is measured in years.
- It is measured in units of years.
- It is between 0 and the maturity of the bond.
- It is equal to the maturity if and only if the bond is a zero-coupon bond.
Longer duration implies that prices are more sensitive to interest rate changes.
Modified Duration
We sometimes use “modified duration” to rewrite the approximate price change equation.
Modified Duration = Macaulay Duration / (1 + yield)
Modified Duration is the relative change in the value of a bond when the yield changes by a small amount.
Example: A bond’s modified duration is 12 years. What is the approximate percentage price change when the yield increases by 10 basis points? The price changes by -12 x 0.1% = -1.2%.
DV01
DV01: “dollar value of a basis point” is the dollar value change given a one basis point change in the “interest rate”.
- It is just another measure of duration.
- A positive DV01 implies that the bond value is negatively associated with the yield.
- If the yield rises by 1 bp, the bond value decreases by DV01.
- Also known as Basis Point Value (BPV).
- Used to estimate gain and loss or monitor risk.
Linear Approximation
We can calculate the approximate change in a bond price when rates change using linear approximation.
- This is a good approximation when the yield change is small.
Key Rate Durations
Key rate durations measure bond price sensitivity to shifts at “key” points along the yield curve.
- This is difficult and requires a yield curve.
Duration and Duration Matching
Pros and Cons of Duration
Pros: Easy to compute, has closed-form solutions.
Cons: Assumes small parallel shifts of the yield curve.
Duration Matching
Duration matching is an approximation that assumes:
- A flat yield curve.
- Only parallel shifts of the yield curve, not changes in the slope of the curve or other types of shape changes.
- A linear approximation.
Convexity improves the accuracy by using a refined approximation.
Duration Matching Example
We want to buy a hedge position worth P of 10-year zero-coupon bonds so that the sum of the price changes is zero.
That is, the “market value” of the hedge position has to equal that of the old position, scaled by the ratio of the durations.
Example:
- Market value of the existing position is 100 x $915.75 = $91,575.
- Durations are 4.611 (5-year coupon bond) and 10 (10-year zero-coupon bond).
- The formula tells us to sell 10-year zeros worth (4.611/10) x $91,575 = $42,222.
- Zero-coupon bonds have face values of $1,000, so prices are $1,000/1.0610 = $558.39.
- So we need to sell $42,222 / $558.39 = 75.6 units of these bonds.
DV01 Matching Example
Alternatively, we can match DV01.
DV01 on one coupon bond (only accounting for duration):
DV01 = Price x Duration / (1 + yield) / 10000
Example: (915.75) x 4.61 / 1.06 / 10000 = 0.398
DV01 on one 10-year zero-coupon bond:
(558.39) x 10 / 1.06 / 10000 = 0.527
We want 100 coupon bonds and “n” 10-year zero-coupon bonds, such that the total DV01 = 0.
100 x (0.398) + n x (0.527) = 0, n = -75.61
However, it is important to note that immunization based on duration matching is never perfect as it is based on a linear approximation.