1. Interest Rates Chapter 4 (part 2)
Geng Niu

2. Duration
Suppose a bond provides the holder with cash flows ci at time ti (i=1,2,…n), the bond yield is y, then bond price B is

3. Duration Duration of a bond that provides cash flow ci at time ti is
where B is its price and y is its yield (continuously compounded)
Duration is a weighted average of the times when payments are made.

4. Duration Taylor expansion:
∆𝐵= 𝑑𝐵 𝑑𝑦 ∆𝑦+ 1 2! 𝑑 2 𝐵 𝑑 𝑦 2 ∆𝑦 ! 𝑑 3 𝐵 𝑑 𝑦 3 ∆𝑦 3 +⋯
With a small change Δy in the yield, it is approximately true that:

5. Duration

6. Key Duration Relationship
Duration is important because it leads to the following key relationship between (small) changes in a yield on the bond and the percentage changes in its price

7. Duration
Duration: an estimate of economic life of a bond measured by the weighted average time to receipt of cash flows
The shorter the duration, the less sensitive is a bond’s price to fluctuations in interest rates

8. Key Duration Relationship
When the yield y is expressed with compounding m times per year
The expression
is referred to as the “modified duration”
How to prove?

9. Managing interest rate risk
Interest rate risk is measured by comparing the weighted average duration of assets with the weighted average duration of liabilities.
If Asset duration > Liability duration
interest rates Market value of equity will fall.

10. Managing interest rate risk
Interest rate immunization,  is a strategy that ensures that a change in interest rates will not affect the value of a portfolio.
To ensure that the value of a firm's assets will increase or decrease in exactly the opposite amount of their liabilities, thus leaving the value of the firm's equity unchanged, regardless of changes in the interest rate.

11. Convexity

12. Convexity The convexity, C, of a bond is defined as
This leads to a more accurate relationship
When used for bond portfolios it allows larger shifts in the yield curve to be considered, but the shifts still have to be parallel

13. Convexity
∆𝐵= 𝑑𝐵 𝑑𝑦 ∆𝑦+ 1 2! 𝑑 2 𝐵 𝑑 𝑦 2 ∆𝑦 2 𝑑 2 𝐵 𝑑 𝑦 2 =− 𝑖=1 𝑛 𝑡 𝑖 𝑐 𝑖 𝑒 𝑖 −𝑦 𝑡 𝑖 𝑑𝑦 = 𝑖=1 𝑛 𝑡 𝑖 2 𝑐 𝑖 𝑒 𝑖 −𝑦 𝑡 𝑖 𝐶= 1 𝐵 𝑑 2 𝐵 𝑑 𝑦 2 = 1 𝐵 𝑖=1 𝑛 𝑡 𝑖 2 𝑐 𝑖 𝑒 𝑖 −𝑦 𝑡 𝑖 ∆𝐵 𝐵 =−𝐷∆𝑦+ 1 2 𝐶 ∆𝑦 2

14. Theories of the Term Structure
Expectations Theory: forward rates equal expected future zero rates
Market Segmentation: short, medium and long rates determined independently of each other. Yield curve reflects the hedging and maturity needs of institutional investors
Liquidity Preference Theory: forward rates higher than expected future zero rates. Long term yield is greater because investors prefer the liquidity in short term issues.

15. Liquidity Preference Theory
Suppose that the outlook for rates is flat and you have been offered the following choices
Which would you choose as a depositor? Which for your mortgage?
Maturity
Deposit rate
Mortgage rate
1 year 3% 6%
5 year

16. Liquidity Preference Theory
The majority of a bank’s customers will opt for one—year deposits and five-year mortgages.
This creates an asset/liability mismatch for the bank.
No problem if interest rates fall.
If rates rise, the deposits will cost more in the future.

17. Liquidity Preference Theory cont
To match the maturities of borrowers and lenders a bank has to increase long rates above expected future short rates
In our example the bank might offer
Maturity
Deposit rate
Mortgage rate
1 year 3% 6%
5 year 4% 7%