valuing financial assets using spot and forward rates
DESCRIPTION
More About Present Values. Valuing Financial Assets Using Spot and Forward Rates. Valuing a Bond - Simple Approach. Bond Prices and Yields. Price. Yield. YTM (r). 1981. 1987 & Normal. 1976. Year. 1 5 10 20 30. Term Structure of Interest Rates. - PowerPoint PPT PresentationTRANSCRIPT
Berlin, 04.01.2006 Fußzeile 1
Valuing Financial Assets
Using Spot and Forward
Rates
More AboutPresent Values
Berlin, 04.01.2006 Fußzeile 2
Valuing a Bond - Simple Approach
NN
r
C
r
C
r
CPV
)1(
000,1...
)1()1( 22
11
Berlin, 04.01.2006 Fußzeile 3
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.
Berlin, 04.01.2006 Fußzeile 6
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 -×
Berlin, 04.01.2006 Fußzeile 8
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)
Berlin, 04.01.2006 Fußzeile 9
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
Berlin, 04.01.2006 Fußzeile 10
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 €
Berlin, 04.01.2006 Fußzeile 11
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 :
Berlin, 04.01.2006 Fußzeile 12
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
Berlin, 04.01.2006 Fußzeile 13
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
Berlin, 04.01.2006 Fußzeile 14
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
Berlin, 04.01.2006 Fußzeile 15
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%
Berlin, 04.01.2006 Fußzeile 16
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 17
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%)