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Part 3. Interest Rates and Interest Rate Related Risks Derivative Instruments Introduction To Mechanisms, Applications and Valuation 1 Term Structure of Interest Rates Bond Valuation Using Average Returns Spot and Forward Rates Valuations with Spot and Forward Rates
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Derivative Instruments
Introduction
To Mechanisms,
Applications and Valuation
1
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.
Topics CoveredForward Rate  The interest rate, fixed today for a future period
Current Yield – Coupon payments on a security as a percentage of the security’s market price (gross of accrued interest)
Yield To Maturity (YTM)  The IRR on an interest bearing instrument
YTM (r)
2000 (flat)
2003 (Normal)
1991 (Invers)
Year
1 2 3 4 5 6 7 8 9 10
Term Structure of Interest Rates1 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.
Term Structure (“Theory”)Valuing a Bond  Simple Approach
Straight Bond, 5yrs. To Maturity, 5,5 Coupon Rate, annual payment, market rate: 5%
or
+
Semiannual 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= 59%
FV= 1000
t= 5
Zero Bonds:
Example:
r= 5%
FV= 1000
t= 5
Example:
r= ?
FV= 1000
P= 783.53
t= 5
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)
Excursus: Yield To MaturityA $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 ?
Excursus:Yield to MaturityA $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?
Excurs:Yield to MaturityC0 C1 C2 C3 C4 C5____
1079 105 105 105 105 1105
Calculate IRR = 8.5%
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?
2
×
40.000
1,07

3
×
1.040.000
1,07
Valuation  Spot Rates(At Flat 3yRates)t
t
t
t
0
1
2
3
40.000,00
40.000,00
1.040.000,00
Market Value

1
×
40.000
1,07
37.383,18
34.937,55
848.949,79
921.270,52
t
t
t
0
1
2
3
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
Result (P.V.)
3y Interest Rate flat (7%)
921.270,52 €
Term – Structure of InterestRates (5,6,7%)
922.644,89 €
Replication of Cash Flows
920.321,44 €
Which is the Right Value ?
Three approaches lead to three results:
But which is the right one ??????
2
3
Yield
5%
6%
7%
Spot Rates
5%
6,03%
7,1%
Proof :
A 3yBond at a 3yYTM of 7% will be issued at par, i.e. $ 1,000.
Use Spot Rates to Valuate the Bond
Interest Rates (YTM) and Related Spot Rates
t1
t2
1y Forward Ratelocked in
1y Forward Rate starting in t1
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 1year loan of 10 Mio €. The loan should be available 1 year from now. You‘d like to fix the future interest rate now.
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:
At a market yield of 6.5 % (flat) the two 6.5% coupon bonds are both priced at $ 1,000.
Interest Rate RiskWhat does this tell you about the relationship
between bond prices and yields for bonds with
different maturities?
r > Coupon Interest Rate
P0 < par value
DISCOUNT
=
r < Coupon Interest Rate
P0> par value
PREMIUM
=
Bond Premiums and Discounts100
100,00
2%
80,00
78,35
75,13
74,30
67,68
67,29
60,00
56,45
55,68
5%
48,10
40,00
42,41
37,69
10%
31,86
20,00
23,94
14,86
0,00
t 20
t
5
The Duration of A Zerobond Equals it‘s Time to Maturity
Total Portfolio Value:141,26 €
Value of ReinvestedPayments : 48,84 €
Present Value of FuturePayments : 92,42 €
Interest payments from the bond: 8% / y.Interest rate in the market: 10%
100
+
8
8
8
8
8
8
8
8
8
8
t 10
t 5
t 0
Present Value of future cash flows generated by the bond: 8% / 5y.
Excursus: Calculation of a Portfolio‘s Value
In a next step, the Duration can be transformed into the Modified Duration. Technically then the Duration must be divided by the expression (1+r), 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 %....
Modified Duration and Convexity
Unfortunately, the MD assumes a linear relation between a bond‘s 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):
Maturity of F.R.A.
Time to Market
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 1year 10 Mio € loan will not exceed 3,30 %.
Lockedin Rate: 3,3%
Loss
Forward Rates (F.R.A.  Application)
Scenario 1:Short 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: (500170)=330 T€ (= 3,3%)
Long F.R.A.
Scenario 2:Short 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%)
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 asset’s 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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Lessons Learnt