Fixed Income: Duration
Kevin Crotty
BUSI 448: Investments
Where are we?
Last time:
- Equity asset pricing models
- Multifactor models
Today:
- Interest rate risk
- Duration
Fixed Income Topics
- Interest rate risk
- Duration
- Convexity
- Duration
- Credit risk
- Leverage
- Ratings
- Credit default swaps
- Reinvestment risk
Bond pricing refresher
\[ P = \frac{CF_1}{(1+y/m)}+\frac{CF_2}{(1+y/m)^2}+...+\frac{CF_T}{(1+y/m)^T} \]
- \(m\): number of payments per year
- \(y/m\): per period yield (i.e., the discount rate)
For bonds, the cash flows are usually fixed coupon payments, so this reduces to:
\[ P = \frac{C}{(1+y/m)}+\frac{C}{(1+y/m)^2}+...+\frac{C+FACE}{(1+y/m)^T}\]
where \(C\) is the coupon payment of the bond.
Interest Rate Risk
Duration defined
\[ P = \frac{C}{(1+DR)} + \frac{C}{(1+DR)^2}+ \frac{C}{(1+DR)^3}+ ... + \frac{C+FACE}{(1+DR)^T} \]
We can rewrite this as:
\[ P = PV(CF_{t_1}) + PV(CF_{t_2})+ PV(CF_{t_3})+ ... + PV(CF_{t_T}) \]
where \(t_1\) is the time of the first cash-flow in years.
Now divide both sides by \(P\):
\[ 1 = \frac{PV(CF_{t_1})}{P} + \frac{PV(CF_{t_2})}{P}+ \frac{PV(CF_{t_3})}{P}+ ... + \frac{PV(CF_{t_T})}{P} \]
Each term on the RHS is a weight!
Duration defined
- Duration is a weighted-average time to cash flows.
- The weights are the fraction of the total PV (the price) that is due to the cash flows at each time.
\[\text{duration}=\left[\frac{PV(CF_{t_1})}{P} \right] \cdot t_1 + \left[\frac{PV(CF_{t_2})}{P} \right] \cdot t_2 + ... + \left[\frac{PV(CF_{t_T})}{P} \right] \cdot t_T \]
Duration visualized
What happens to duration as:
- Maturity increases?
- Coupon rate increases?
Duration is related to interest rate risk!
For a change in yield \(y\) of \(\Delta y\), the percent change in price is:
\[\frac{\Delta P}{P} \approx - \left[ \frac{\text{duration}}{1+DR} \right] \cdot \Delta y.\]
The term in brackets is modified duration.
Alternatively, we can work in prices rather than returns:
\[ P_{\text{new}} \approx P_0 - P_0 \cdot \left[ \frac{\text{duration}}{1+DR} \right] \cdot \Delta y \]
Let’s go to today’s notebook and calculate duration
Duration and the bond pricing function
How good is this approximation?
Consider two bonds with
- Same YTM of 10%
- Same coupon rate of 5%
- Different maturities of 5 and 10 years
Let’s look at how well the duration approximation works for different yield change magnitudes.
Drawbacks of duration
- Duration is a linear approximation.
- It can be improved using curvature of pricing function (convexity).
- Also, price risk is not the only risk associated with rate changes.
- reinvestment risk!
Duration and reinvestment risk
Consider a rate decline from 10% to 9%
Reinvestment risk
The risk that interest payments cannot be reinvested at the same rate.
- If rates fall
- bond prices rise
- but the value of reinvested coupons falls.
When investment horizon matches duration, reinvestment risk and interest rate risk cancel out!
An example
Suppose you need to pay out $X at year 5 (think of a pension company).
What is your investment strategy, using bonds, that ensures that you can meet your obligation?
Best bet is to buy a zero-coupon bond maturing in 5 years
If unavailable, buy a bond with duration of 5 years
Duration tells us three very useful things
- Effective maturity of a bond
- Interest rate risk (sensitivity of bond prices to rate changes)
- Horizon at which interest rate risk and reinvestment risk cancel out
For next time: Convexity
BUSI 448