Part 3' Interest Rates and Interest Rate Related Risks

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Part 3.Interest Rates and Interest Rate Related
Risks
Derivative Instruments Introduction To
Mechanisms, Applications and Valuation
1
2
Topics Covered

• Term Structure of Interest Rates
• Bond Valuation Using Average Returns
• Spot and Forward Rates
• Valuations with Spot and Forward Rates
• Risk and McCauley Duration
• Modified Duration and Convexity
• A Different Approach F.R.A.

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Term Structure of Interest Rates Germany
(selected years since 2000)
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Term Structure of Interest Rates

• Interest Rate - the interest rate according to
  the term structureSpot Rate implied rate to
  valuate future cash flows
• Forward Rate - The interest rate, fixed today for
  a future period
• Current Yield Coupon payments on a security as
  a percentage of the securitys market price
  (gross of accrued interest)
• Yield To Maturity (YTM) - The IRR on an interest
  bearing instrument

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Term Structure (Theory)

• What Determines the Shape of the Term Structure ?
• 1 Unbiased Expectations Theory Short term rates
  reflects the current monetary policy of the
  central bank. In the long range interest rates
  reflect the future expected short term rates.
• Liquidity Premium Theory. Investors tend to
  prefer short term investments. This is to prevent
  from being exposed to interest rate related risk.
  To invest money in the long range, investors ask
  for higher premiums.
• Market Segmentation Theory There is no linkage
  between short- medium and long term interest
  rates. Different groups of investors have
  different motivations to invest money.

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Valuing a Bond - Simple Approach
Straight Bond, 5yrs. To Maturity, 5,5 Coupon
Rate, annual payment, market rate 5
or

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Valuing a Bond - Simple Approach
Semi-annual Coupon Payments
Example n 10 C 5,5 r 5 FV 1000 t 5

Floating Interest Rates (1/year)
Example n 5 C 5,5 r 5-9 FV 1000 t 5
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Valuing a Bond - Simple Approach
Zero Bonds
Example r 5 FV 1000 t 5
Example r ? FV 1000 P 783.53 t 5
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Excursus Yield To Maturity

• All interest bearing instruments are priced to
  fit the term structure
• This is accomplished by modifying the asset price
• The modified price creates a New Yield, which
  fits the Term Structure
• The new yield is called the Yield To Maturity
  (YTM)

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Excursus Yield to Maturity

• Example
• A 1,000 treasury bond expires in 5 years. It
  pays a coupon rate of 10.5. If the market price
  of this bond is 1,079 what is the YTM ?

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Excurs Yield to Maturity

• Example
• A 1,000 treasury bond expires in 5 years. It
  pays a coupon rate of 10.5. If the market price
  of this bond is 1,079 what is the YTM?

C0 C1 C2 C3 C4 C5____
-1079 105 105 105 105 1105 Calculate
IRR 8.5
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Excursus Yield to Maturity

• A 1000 treasury bond expires in 5 years. It pays
  a coupon rate of 10.5. If the market price of
  this bond is 1079 what is the YTM?

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Valuation - Spot Rates (At Flat 3-y-Rates)
t
t
t
t
1
0
2
3
40.000,00
40.000,00
1.040.000,00
Market Value
-
1

1,07
40.000
37.383,18
34.937,55
848.949,79
921.270,52
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Valuation Spot RatesAverage Returns
 
 
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Valuation - Spot RatesDuplicating Cash Flows




40.000,00
40.000,00
1.040.000,00
Market Value ?





-
971.962,62
Loan




interest 7
Interest 7
interest 7



971962,62
- 68.037,38
- 68.037,38
- 68.037,38

Difference 0
Difference
- 28.037,38



26.450,36
Investment


interest 6
interest 6


- 26.450,36
1.587,02
1.587,02

Difference 0
Difference
- 26.450,36



25.190,82
Investment

Interest 5
1.259,54

- 25.190,82

Difference 0
920.321,44

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Which is the Right Value ?
Three approaches lead to three results
But which is the right one ??????
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Use Spot Rates to Valuate the Bond
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Interest Rates (YTM) and Related Spot Rates
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Forward Rates
An interest rate that is not agreed upon a
defined period of time starting right now, but
for a defined time period, which begins (and
ends) in the future, is called a forward
rate. Example Due to an expected future
business development your corporate needs a
1-year loan of 10 Mio . The loan should be
available 1 year from now. Youd like to fix the
future interest rate now.
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Spot Rates and Related Forward Rates(Basic Idea)
You can either borrow now, or fix a rate using a
Forward. The rate, that can be locked in today,
results from this model The cost of borrowing
now for two years must equal the cost of
borrowing now for one year with an obligation to
extend the loan for a second year at a forward
rate not known now.
Using the spot rates from the example above and
solving the equation for rf,1,1 results in
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Spot Rates and Related Forward Rates
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Interest Rate Risk
What does this tell you about the relationship
between bond prices and yields for bonds with
different maturities?
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Bond Premiums and Discounts

• What happens to bond values if required return is
  not equal to the coupon rate?

The bond's price will differ from its par value
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The Duration of A Zerobond Equals its Time to
Maturity
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Excursus Calculation of a Portfolios Value
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The Duration of a 8 Bond Does Not Depend On
Different YTM
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Example Calculation of Duration
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Modified Duration
In a next step, the Duration can be transformed
into the Modified Duration. Technically then the
Duration must be divided by the expression (1r),
where r represents the current average market
yield.
As a result, MD informs roughly about the changes
in portfolio value to be expected when the market
yield changes by 1 . Hereby MD serves as a
prognostic tool to check the interest rate
sensitivity of each asset or the whole portfolio
An expected interest rate increase of 2.5 will
change the Value of an interest bearing portfolio
with a Duration of 7.04 relativ to a current
market rate of 10 by 16 ....
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Modified Duration and Convexity
Unfortunately, the MD assumes a linear relation
between a bonds price and the market yield. As
this is not the case, the usage of MD as a tool
to forecast value changes must lead to a
systematic error, called the Convexity error or
the tracking error (see the left figure)
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Forward Rates (F.R.A. - Application)
To contract a F.R.A. means to lock in an interest
rate concerning a future period. Referring to
our term structure, your bank might use an F.R.A.
to make sure, that her future costs of financing
a 1-year 10 Mio loan will not exceed 3,30 .
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Forward Rates (F.R.A. - Application)
Scenario 1Short rate in t1 is at 5. Financing
costs will be 500 T. Compensations on F.R.A.
will be (5-3,3)x10 Mio 170 T. Total costs
at end of year 2 (500-170)330 T ( 3,3)
Long F.R.A.
Scenario 2Short rate in t1 is at 2. Financing
costs will be 200 T. Payments on F.R.A. will be
(2-3,3)x10 Mio -130 T. Total costs at end of
year 2 (200 130)330 T ( 3,3)
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Lessons Learnt

• The Term Structure of Interest Rates Determines
  Asset Values
• To Value Correctly, Spot Rates should be used
• Spot and Forward Rates directly result from a
  given Term Structure of Rates
• RRRRRR
• The McCauley Duration informs about a point in
  time, where an assets total value is immune
  against interest rate changes.
• Modified Duration and Convexity are used in the
  context of the McCauley Duration
• A Different Approach F.R.A.bbbbbbbbbö
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