Duration, Convexity and ALM Flashcards
What is duration?
Duration measures how much a bond’s price changes in response to interest rate movements.
Why do we use duration?
It helps investors understand interest rate risk and how bond prices will move when rates change.
What is the relationship between interest rates and bond prices?
Interest rates and bond prices move in opposite directions. When interest rates rise, bond prices fall and vice versa.
What is Macauley Duration?
Macauley Duration is the weighted average time it takes for an investor to receive a bond’s cash flows.
How to calculate the Macauley Duration?
Note the maturities and cash flows. Extract the discount factors from the interest rates. Calculate the present value for each maturity. Calculate the weights of PVs. Then multiply the weight with the maturity and sum them all up.
What is Modified Duration?
Modified duration converts the Macauley duration into a price sensitivity measure, showing how much a bond’s price changes for a 1% change in interest rates.
How do you estimate a bond’s price change using duration?
DeltaP = -Dmod * P * DeltaR
Which bonds have the highest duration?
Longer-maturity bonds
Low-coupon bonds (zero-coupon bonds have the highest duration)
What happens to duration as interest rates increase?
Duration decreases because future cash flows are discounted more heavily.
How do you calculate the duration of a portfolio?
Weight times each duration
Why does a portfolio’s duration matter?
A portfolio with higher duration is more sensitive to interest rate changes.
What is dollar duration?
Dollar duration measures the dollar impact of interest rate changes: Ddollar = P * Dmod
What is convexity?
Convexity measures the curvature of the price-yield relationship, showing how duration changes with interest rates.
Why is convexity important?
Convexity corrects the error in duration estimates, making bond price predictions more accurate.
What is the formula for convexity?
C = (“value of one move down” + “value of one move up” - “value of current price”^2) / (DeltaR^2 * “value of current price”)
How do you improve the duration price estimate with convexity?
DeltaP = -Dmod * P * DeltaR + 0.5 * C * P * DeltaR^2
What happens when a bond has higher convexity?
It loses less value when rates rise and gains more value when rates fall, so high convexity bonds are good for investors.
What type of bonds have the highest convexity?
Long-term bonds
Low-coupon (zero-coupon) bonds
What is Asset-Liability Management (ALM)?
ALM is how banks manage mismatches between their assets (loans, bonds) and liabilities (deposits, borrowings.
What happens when banks have a duration mismatch?
If assets have longer duration than liabilities, interest rate spikes reduce the value of assets more than liabilities, causing losses.
How can banks hedge duration risk?
Duration matching (aligning asset and liability durations)
Interest rate swaps (convert fixed-rate exposure to floating-rate)
What happens to a bank when interest rates rise?
Bond values drop, creating losses on asset side.
Deposit rates increase, raising bank costs.
Loans may default if borrowers struggle with higher rates.
What caused Silicon Valley Bank (SVB) to collapse?
- SVB invested heavily in long-term bonds.
- Interest rates rose, causing massive bond losses.
- Depositors withdrew funds, forcing SVB to sell bonds at a loss.
- This created a bank run and SVB collapsed.
What key risk did SVB ignore?
Duration risk, their assets had much longer duration than their liabilities.
How could SVB have avoided collapse?
Hedging interest rate risk using swaps and reducing exposure to long-term bonds.
How can you hedge duration risk?
Take a position in another asset with opposite duration exposure.
What is the formula for a duration hedge?
K1 = - (P * D) / P1 * D1)
How can you hedge convexity risk?
Use two hedging securities with different convexities:
D * P + K1 * D1 * P1 + K2 * D2 * P2
What happens to duration as a bond gets closer to maturity?
Duration decreases because the bond’s cash flows are received sooner.
What is effective duration?
Effective duration is used for bonds with embedded options (callable/putable). It measures how much a bond’s price changes when interest rates move, accounting for the fact that some bonds may be called early.
Why is effective duration different from modified duration?
Modified duration assumes a fixed cash flow schedule, but effective duration accounts for changing cash flows due to embedded options.
How does convexity affect risk?
Higher convexity means the bond loses less when rates rise and gains more when rates fall, making it safer.
What are the drawbacks of convexity?
Complexity
Convexity is small for short-term bonds
What is a gap risk in ALM?
Gap risk occurs when a bank’s asset duration is longer than its liability duration, meaning rising interest rates can cause large losses.
What is a duration gap?
Duration gap is the difference between the duration of asset and liabilities. A positive gap means high interest rate risk.
How do banks manage duration gaps?
Hedging with swaps
Shortening bond durations by holding more short-term assets.
What key risk did SVB ignore?
Interest rate risk - SVB held long-term bonds but didn’t hedge against rising rates.
Why did SVB’s depositors panic?
As SVB took large bond losses, depositors feared insolvency and withdrew money rapidly, causing a bank run.
What could SVB have done differently?
Hedged duration risk with swaps.
Maintained a shorter asset duration to avoid large losses.
Raised capital sooner to prevent panic.
