More About
Present Values
Valuing Financial Assets
Using Spot and Forward
Rates
Berlin, 04.01.2006
Fußzeile
1
Valuing a Bond - Simple Approach
1,000  C N
C1
C2
PV 

 ... 
1
2
N
(1  r )
(1  r )
(1  r )
Berlin, 04.01.2006
Fußzeile
2
Bond Prices and Yields
1600
1400
1200
Price
1000
800
600
400
200
0
0
2
4
5 Year 9% Bond
Berlin, 04.01.2006
6
8
10
1 Year 9% Bond
Fußzeile
12
14
Yield
3
Term Structure of Interest Rates
YTM (r)
1981
1987 & Normal
1976
Year
1
5
10
20
30
Interest Rate - the interest rate according to the term structure
Spot 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
security’s market price (gross of accrued interest)
Yield To Maturity (YTM) - The IRR on an interest bearing instrument
Berlin,
Term Structure of Interest Rates
What Determines the Shape of the TS?
1 - Unbiased Expectations Theory
2 - 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.
Berlin,
Term – Structure of
Interest Rates Germany
5,50%
4,97%
4,90%
5,00%
4,88%
4,89%
4,92%
4,96%
5,00%
5,05%
5,09%
5,14%
4,61%
4,50%
4,50%
4,37%
4,21%
4,03%
4,00%
3,81%
3,82%
3,78%
3,64%
2,79%
2,62%
3,48%
3,33%
3,10%
2,93%
2,85%
3,90%
3,99%
3,06%
3,17%
4,27%
3,48%
3,39%
3,23%
3,00%3,12%
3,42%
3,34%
3,26%
3,17%
1. November 2000
1. November 2001
2,88%
2,64%
1. November 2003
2,41%
2,50%
2,41%
4,47%
4,14%
3,54%
3,62%
3,50%
4,38%
1. November 2004
1. November 2005
2,41%
2,22%
2,00%
1
Berlin, 04.01.2006
2
3
4
5
6
Fußzeile
7
8
9
10
6
Valuation - Spot Rates
(Flat Rate)
t0
Market Value
t1
40.000,00
t2
40.000,00
t3
1.040.000,00
40.000 ×1,07 1
37.383,18
40.000 ×1,07 2
34.937,55
848.949,79
1.040.000 ×1,07
921.270,52
Berlin,
-3
Valuation
Interest Rates (Yields)
t0
Marktwert ?
t1
t2
t3
40.000,00
40.000,00
1.040.000,00
40.000  1,05 1
38.095,24
40.000
 1,06
-2
35.599,86
1.040.000  1,07
-3
848.949,79
922.644,89
Berlin, 04.01.2006
Fußzeile
8
Valuation - Spot Rates
Duplication-Portfolio
t0
t1
Market Value ?
t2
40.000,00
40.000,00
Loan:
971962,62
interest 7 %
Interest 7 %
- 68.037,38
- 68.037,38
Difference:
- 26.450,36
interest 6 %
+ 1.587,02
+ 1.587,02
Difference: - 26.450,36
Investment:
- 25.190,82
920.321,44
Berlin, 04.01.2006
1.040.000,00
971.962,62
interest 7 %
- 68.037,38
Difference: 0
- 28.037,38
+ 26.450,36
interest 6 %
Investment:
t3
Difference: 0
25.190,82
Interest: 5 % 1.259,54
Difference: 0
Fußzeile
9
Which Price
is the Right One ?
Three approaches lead to three results:
Result
(P.V.)
Valuation Mode
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 €
But which is the right one
Berlin, 04.01.2006
Fußzeile
??????
10
Use Spot Rates to
Valuate the Price of a Bond
1
2
3
Yield
5%
6%
7%
Spot Rates
5%
6,03%
7,1%
Proof :
70
70
1.070


 1.002,4051
2
3
1,05 1,06
1,07
70
70
1.070


 1.000
2
3
1,05 1,0603
1,071
Berlin, 04.01.2006
Fußzeile
11
Term – Structure of Interest Rates
and related Spot Rates (Calculation)


1  rt

qs ,t 
t -1

 1 - rt  qs ,t
i 1


Example:
-i






1
t
t
r[t]
q[s,t]
1
2
3
4
5
6
7
8
9
10
2,41%
2,85%
3,23%
3,54%
3,81%
4,03%
4,21%
4,37%
4,50%
4,61%
1,0241
1,028562975
1,032472836
1,03571334
1,038588645
1,040972195
1,042955747
1,044757588
1,046243224
1,047521695
r[s,t]
2,41%
2,86%
3,25%
3,57%
3,86%
4,10%
4,30%
4,48%
4,62%
4,75%
1
4


1,0354
rs,4  
- 1  3,57%
-1
-2
-3 
1 - 0,0354  1,0241  1,02856  1,03247 

Berlin, 04.01.2006

Fußzeile
12
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
Berlin, 04.01.2006
t1
Fußzeile
t2
13
Spot Rates and related
Forward Rates
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.
1 r2 
2
 1  r1   1  rf ,1,1 
Using the spot – rates from the example above and solving
the equation for rf,1,1 results in:
1  0,0286 
2
 1  0,0241  1  rf ,1,1 
rf ,1,1  3,30%
Berlin, 04.01.2006
Fußzeile
14
Spot Rates and related
Forward Rates
Maturity
Term
Spot Rates
structure
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%
for years
1
2
3
4
5
4,22%
4,72%
5,09%
5,39%
5,59%
5,74%
4,44%
4,88%
5,22%
5,47%
5,65%
6
7
8
9
in year
1
2
3
4
5
6
7
8
9
Berlin, 04.01.2006
3,30%
4,03%
4,55%
5,02%
5,30%
5,49%
5,75%
5,82%
5,91%
3,67%
4,29%
4,78%
5,16%
5,40%
5,62%
5,78%
5,87%
3,96%
4,53%
4,95%
5,27%
5,51%
5,69%
5,83%
Fußzeile
4,61% 4,77% 4,90% 5,02%
5,02% 5,14% 5,23%
5,32% 5,40%
5,55%
15
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
Berlin, 04.01.2006
Fußzeile
16
Forward Rates
(F.R.A. - Application)
Profit
Scenario 1:
Long F.R.A.
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%)
Locked-in Rate: 3,3%
Loss
Berlin, 04.01.2006
Fußzeile
17