Valuing Financial Assets Using Spot and Forward Rates

Berlin, 04.01.2006 Fußzeile 1

Valuing Financial Assets

Using Spot and Forward

Rates

More AboutPresent Values


Valuing a Bond - Simple Approach

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r

C

r

C

r

CPV

)1(

000,1...

)1()1( 22

11


Bond Prices and Yields

0

200

400

600

800

1000

1200

1400

1600

0 2 4 6 8 10 12 14

5 Year 9% Bond 1 Year 9% BondYield

Price

Berlin,

Term Structure of Interest Rates

Interest Rate - the interest rate according to the term structureSpot Rate – implied rate to valuate future cash flowsForward Rate - The interest rate, fixed today for a future periodCurrent 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)

Year

1981

1987 & Normal

1976

1 5 10 20 30

Berlin,

Term Structure of Interest RatesWhat Determines the Shape of the TS?1 - Unbiased Expectations Theory2 - Liquidity Premium Theory

Term Structure & Capital Budgeting CF should be discounted using Term Structure info Since the spot rate incorporates all forward rates,

then you should use the spot rate that equals the term of your project.

If you believe in other theories take advantage of the arbitrage.


Term – Structure of Interest Rates Germany

4,97%4,90% 4,88% 4,89% 4,92% 4,96% 5,00% 5,05% 5,09% 5,14%

3,12% 3,17%

3,39%

3,62%

3,82%

3,99%

4,14%4,27%

4,38%4,47%

2,41%

2,85%

3,23%

3,54%

3,81%

4,21%

4,37%4,50%

4,61%

2,22%

2,41%

2,64%

2,88%

3,10%

3,33%

3,48%

3,64%3,78%

3,90%

3,48%

4,03%

2,79%

2,62%

2,41%

2,93%

3,06%

3,17%3,26%

3,34%3,42%

2,00%

2,50%

3,00%

3,50%

4,00%

4,50%

5,00%

5,50%

1 2 3 4 5 6 7 8 9 10

1. November 2000

1. November 2001

1. November 2003

1. November 2004

1. November 2005

Berlin,

Valuation - Spot Rates (Flat Rate)

t t 1 t t

40.000,00 40.000,00 1.040.000,00

37.383,18

34.937,55

848.949,79

2-×1,0740.000

3-×1,071.040.000

0 2 3

921.270,52

Market Value

11,0740.000 -×


t 0 t 1 t 2 t 3

40.000,00 40.000,00 1.040.000,00

38.095,24

35.599,86

848.949,79

922.644,89

Marktwert ?

11,0540.000

2 1,0640.000

31,071.040.000

ValuationInterest Rates (Yields)


t 0 t 1 t 2 t 3

40.000,00 40.000,00 1.040.000,00

Loan:

971962,62

-971.962,62 interest 7 % Interest 7 % interest 7 %

- 68.037,38

- 68.037,38

- 68.037,38

Difference: 0

Difference: - 28.037,38

Investment:

- 26.450,36

+ 26.450,36 interest 6 % interest 6 %

+ 1.587,02

+ 1.587,02

Difference: 0

Difference: - 26.450,36

Investment:

- 25.190,82

25.190,82

Interest: 5 %

1.259,54

Difference: 0

Market Value ?

920.321,44

Valuation - Spot RatesDuplication-Portfolio


Which Priceis the Right One ?

Three approaches lead to three results:

But which is the right one ??????

Valuation Mode Result (P.V.)

3y Interest Rate flat (7%) 921.270,52 €

Term – Structure of Interest Rates (5,6,7%)

922.644,89 €

Replication of Cash Flows 920.321,44 €


Use Spot Rates to Valuate the Price of a Bond

1 2 3

Yield 5% 6% 7%

Spot Rates 5% 6,03% 7,1%

00010711

0701

06031

70

051

70

40510021071

0701

061

70

051

70

32

32

.,

.

,,

,.,

.

,,

Proof :


t r[t] q[s,t] r[s,t]

1 2,41% 1,0241 2,41%2 2,85% 1,028562975 2,86%3 3,23% 1,032472836 3,25%4 3,54% 1,03571334 3,57%5 3,81% 1,038588645 3,86%6 4,03% 1,040972195 4,10%7 4,21% 1,042955747 4,30%8 4,37% 1,044757588 4,48%9 4,50% 1,046243224 4,62%

10 4,61% 1,047521695 4,75%

t

1

1t

1i

it,st

tt,s

qr1

r1q

Term – Structure of Interest Rates and related Spot Rates (Calculation)

Example:

%,,,,,

,, 5731

03247102856102411035401

03541 4

1

3214

sr


Forward Rates

A financial contract that does not start immediately but at a specified date in the future is called a Foward Contract. 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.

t0 t1 t2


Spot Rates and related Forward Rates

1112

2 111 ,,frrr

To solve the problem you can fix a rate using a Forward Contract. The rate, that can be locked in today, results from a simple 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.

Using the spot – rates from the example above and solving the equation for rf,1,1 results in:

%,

,,

,,

,,

303

1024101028601

11

112

f

f

r

r


Spot Rates and relatedForward Rates

for years 1 2 3 4 5 6 7 8 9

in year

1 3,30% 3,67% 3,96% 4,22% 4,44% 4,61% 4,77% 4,90% 5,02%2 4,03% 4,29% 4,53% 4,72% 4,88% 5,02% 5,14% 5,23%3 4,55% 4,78% 4,95% 5,09% 5,22% 5,32% 5,40%4 5,02% 5,16% 5,27% 5,39% 5,47% 5,55%5 5,30% 5,40% 5,51% 5,59% 5,65%6 5,49% 5,62% 5,69% 5,74%7 5,75% 5,78% 5,83%8 5,82% 5,87%9 5,91%

Maturity Term Spot Ratesstructure

t r[t] r[s,t]1 2,41% 2,4100%2 2,85% 2,8563%3 3,23% 3,2473%4 3,54% 3,5713%5 3,81% 3,8589%6 4,03% 4,0972%7 4,21% 4,2956%8 4,37% 4,4758%9 4,50% 4,6243%10 4,61% 4,7522%


Forward Rates (F.R.A. - Application)

To contract a Forward-Rate means to lock in an interest rate concerning a future period. Your corporation might use an F.R.A. (= Forward Rate Agreement) to make sure, that her future costs of financing a 1-year 10 Mio € loan will not exceed 3,30 %.

Fixed Rate: 3,30%

Maturity of F.R.A.Time to Market


Forward Rates(F.R.A. - Application)

Locked-in Rate: 3,3%

Profit

Loss

Long F.R.A.

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: (500-170)=330 T€ (= 3,3%)

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: (200 +130)=330 T€ (= 3,3%)

Valuing Financial Assets Using Spot and Forward Rates

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