Interest-Rate Risk II

Interest-Rate Risk II

Duration Rules

Rule 1: Zero Coupon Bonds What is the duration of a zero-coupon bond? Cash is received at one time t=maturity weight = 1 So the duration of a zero coupon bond is just

its time to maturity in terms of how we have defined “one period” (usually six months)

Duration Rules

Rule 2: Coupon Rates Coupons early in the bond’s life reduce the

average time to get payments. Weights on early “times” are higher Holding time to maturity constant, a bond’s

duration is lower when the coupon rate is higher.

Duration Rules

Rule 3: Time to MaturityHolding the coupon rate constant, a bond’s duration generally increases with time to maturity.

– If yield is outrageously high, then higher maturity decreases duration.

–

Rule 4: Yield to MaturityFor coupon bonds, as YTM increase, duration decreases.

Rule 5:The duration of a level perpetuity is (1+y)/y

Modified Duration of a Portfolio

Banks hold several assets on their balance sheets. Let vi be the fraction of total asset PV attributed to asset i. Suppose the bank holds 3 assets Duration of total assets:

*33

*22

*11

* DvDvDvD

Example

Bank Assets:– Asset 1: PV=$ 8M D*=12.5– Asset 2: PV=$38M D*=18.0– Asset 3: PV=$ 2M D*= 1.75

Total PV = $48M– v1=8/48=0.17, v2=38/48=0.79 v3=2/48=0.04

Modified Duration of Portfolio:D*=(0.17)(12.5) + (0.79)(18) + (0.04)(1.75)=16.42

Review

For zero coupon bonds:– YTM=effective annual return

For annual bonds:– Effective annual return = YTM assuming we can

reinvest all coupons at the coupon rate

For semi-annual bonds– Effective six-month return =YTM/2 assuming we

can reinvest all coupons at the coupon rate

Effective Annual Return of a Portfolio

Example: Portfolio Value: $110 Annual Bond 1: PV=$65, EAR=5% Annual Bond 2: PV=$45, EAR=3% What is effective return on portfolio? (get/pay-1) Get=65*1.05+45*1.03=114.6 Pay=110 Return=114.6/110-1=4.18% But (65/110)*.05+(45/110)*.03=4.18%

Bottom line: the EAR of a portfolio is the weighted sum of the EARs of the individual assets in the portfolio where weights are the fraction of each asset of total portfolio value.

Back to Building a Bank

From previous example (Building a Bank) Assets: D*=23.02, PV=100M (YTM=1.8%) Liabilities: D*=0.99, PV=75M (YTM=1%) Equity: 25M Currently a 10 bp increase in rates causes:

A = -23.02*.001*100M = -2.30M

L = -0.99*.001*75M = -0.074M

E =-2.30M-(-0.074M) = -2.23M (drop of 8.8%)

Building a Bank

Suppose you want a 10bp increase in rates to cause equity to drop by only 4% (1M).

Options:A: Hold D* of assets constant and raise D*of liabilities

B: Hold D* of liabilities constant and lower D* assets

C: Raise D* of liabilities and lower D* of assets

Building a Bank: Option A

Hold D* of assets at 23.02 For any given D* of liabilities, a 10bp increase in

rates will cause equity to change as follows:

E = -2.30M- (-D*75M*.001) Given that you want a 10bp increase in rates to

cause equity to drop by 1M:

-1M= -2.30M- (-D*75M*.001) solve for D*

D*=17.333


How to get D* of liabilities to 17.33? Issue a bond or CD with duration greater than

17.33. Example: Issue a zero-coupon bond that

matures in 25 years. Assume YTM=1.5%.– Duration=25– D* = 25/1.015 = 24.63

How much should you issue?


You want the D* of your “liability portfolio” to be 17.33.

Let v=fraction of liability portfolio in the 25yr zero-coupon bond. The rest of your liabilities will come from short-term deposits.

17.33 = v(24.63)+(1-v)(0.99) solve for vv = .6912


So make the 25yr bond 69.12% of your liability portfolio.

Total liabilities = 75M Issue .6912*75M = $51.84M in 25yr zero-

coupon bonds with D*=24.63 Raise $23.16M in short-term deposits with

D*=0.99


Checking the approximation: Liabilities:

– 51.84 in 25yr zero-coupon bonds (YTM=.015)– 23.16 in deposits (YTM=.01)

We use the duration approximation to set the target. How do we know if the approximation works? Let’s find the exact change in equity for a 10bp

increase in rates. First, we need to find future values


Future value of Liabilities:– 51.84 in 25yr zero-coupon bonds (YTM=.015)

Future value at expiration (face value) = 51.84*(1.015)^25=75.22

– 23.16M in deposits (YTM=.01) Future value at expiration = 23.16*1.01 = 23.39

Present value if rates jump by 10bp:– Zero-coupon bonds: 75.22/1.016^25=50.58– Deposits: 23.39/1.011 = 23.14

Change in PV of liabilities if rates jump by 10bp:(50.58M + 23.14) – 75M = -1.28M


We know (slides last Wed) that if rates jump by 10bp, assets will drop by exactly 2.27M (PV of bonds drops from 100M to 97.73M)

Change in equity, given a 10bp increase in rates, will be -2.27M-(-1.28M)= -0.99M

Our objective was to have it drop by 1M. So we are very close.


By switching away from short-term deposits we’ve lowered interest-rate risk.

Cost (before rates change):

Before we tailored the balance sheet: – Liabilities (75M) YTM=1%– Assets (100M) YTM=1.8%– Profits=1.8M-.75M=1.05M

After tailoring the balance sheet– Liabilities: 0.6912*.015+0.3088*.01 = 1.3%– Assets (100M) YTM=1.8%– Profits=1.8M-1.3M=0.50M

Building a Bank: Option B

Hold D* of liabilities at 0.99 For any given D* of assets, a 10bp increase in

rates will cause equity to change as follows:

E = -D*100M*.001-(-0.074M) Given that you want a 10bp increase in rates to

cause equity to drop by only 1M:

-1 = -D*100*.001-(-0.074) solve for D*

D*=10.74


How to get D* of liabilities to 10.74? Buy a bond duration less than 10.74 Example: zero-coupon bond than matures in 5

years. Assume YTM=1.2%.– Duration=5– D* = 5/1.012 = 4.94

How much should you purchase?


You want the D* of your asset portfolio to be 10.74. Let v=fraction of asset portfolio in the 5yr zero-

coupon bond (D*=4.94). The rest of your assets will be in the 30-yr coupon bonds (D*=23.02).

10.74 = v(4.94)+(1-v)(23.02) solve for v

v = 0.679


So make the 5yr zero 67.9% of your assets Total assets = 100M Buy .679*100M = $67.9M in 5yr zeros Purchase $32.1M in the 30-year coupon

paying bond


Checking the effect: Assets:

– 67.9 in 5yr zero-coupon bonds (YTM=.012)– 32.1M in 30-year coupon bonds (YTM=.018)

We want to see how the PV of these assets change as we observe a parallel shift in the yield curve. To do this, we need to find future values.


Future value of Assets:– 67.9 in 5yr zero-coupon bonds (YTM=.012)

Future value at expiration (face value) = 67.9*(1.012)^5 = 72.07

– 32.1 in 30-year bonds (YTM=.018, coupon rate=0.18) Future value at expiration (face value)=32.1

Present value if rates jump by 10bp:– 5yr zeros: 72.07/1.013^5=67.56– 30-yr bonds: N=30, FV=32.1, pmt=.018*32.1, ytm=0.019

PV=31.37

Change in PV of assets if rates jump by 10bp:(67.56+31.37) – 100 = -1.07 (million)


We know (from class last Wed) that if rates jump by 10bp, liabilities will drop by exactly 0.074M

So, given new structure of assets, given a 10bp increase in rates, equity will change as follows:

-1.07M-(-0.074M)= -0.996M Our objective was to have it drop by 1M. So we are

very close.


By switching away from short-term deposits we’ve lowered interest-rate risk.

Cost (before rates change):

Before we tailored the balance sheet: – Liabilities (75M) YTM=1%– Assets (100M) YTM=1.8%– Profits=1.8M-.75M=1.05M

After tailoring the balance sheet– Liabilities (75M) YTM=1%– Assets (100M) YTM=.679*.012+.321*.018=1.4%– Profits=1.4M-.75M=0.65M

Important Facts

We hedged only at the present time. As time changes and yields change, modified

durations will change. Need to periodically rebalance hedging

portfolio, even if yields remain constant, or hedge will become useless.

Building a Bank: Option C

You can choose several different combinations of the modified durations of assets and liabilities to accomplish the same objective.

Next slide: The possible combinations

Building a Bank: Option C

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35

D* of Assets

D*

of

Lia

bili

tie

s

D* of Assets=23.02D* of Liabilities=17.33

D* of Assets=10.74D* of Liabilities=0.99

Duration

Using only duration can introduce approximation error.

Duration matching works best for small changes in yields.

Duration allows us to match the slope of the price-curve at a given point.

As you move away from this point, the slope will change – the source of approximation error.

Duration

10000000

12000000

14000000

16000000

18000000

20000000

22000000

24000000

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05

Change in Yield

Bo

nd

Pri

ce

Convexity

Convexity is a measure of how fast the slope is changing at a given point.

Not very convex. More convex.

Convexity

Bond investors like convexity– When yields go down, the prices of bonds with more

convexity increase more.

– When yields go up, the prices of bonds with more convexity drop less

The more convex a bond is, the worse the duration approximation will do.

– Possible to incorporate convexity into analysis above.

Appendix:

Modified Duration of a Portfolio

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Documents

Interest-Rate Risk II