Description
different types of yield curves, normal yield curve, flat yield curve, inverted yield curve, humped yield curve and yoeld curve for bonds of different quality. It also explains the concept of Stripping and various theories of term strucutre like expectation theory, segmented market theory, liquidity premium theory.
YIELD CURVE AND TERM STRUCTURE OF INTEREST RATES
YIELD AND TERM-TO-MATURITY
Yield of a Bond depends on its it’s term to maturity. ? Generally, longer the term-to-maturity higher is the yield. ? But some times this relation may invert.
?
Estimated Yield on Government Securities of Different Maturities
Date
Yield on 6 month Bill 2.34% 8.55%
Yield on 5 Year ZCB 7.41% 7.94%
30-June 2010 25 -April 2008
Source : NSE
TERM STRUCTURE OF INTEREST RATES: US
TERM STRUCTURE OF INTEREST RATES: INDIA
YIELD CURVE
?A
graph showing relationship between yield-to-maturity and term-tomaturity for the bonds of a given quality class.
: Spot Rate Curve
? Also
? Yield
Curve : Sources
?
India
?
NSE : http://www.nse-india.com/content/debt/debt_zcyc.htm ? CCIL http://www.ccilindia.com/GSecZCYC.aspx.
?
US
?
US Treasury: http://www.ustreas.gov/offices/domestic-finance/debtmanagement/interest-rate/yield.shtml
Spot interest rates 10 0 0.019178082 4.72 0.536984082 1.054790082 1.572596082 0.61 2.090402082 2.608208082 3.126014082 3.643820082 4.161626082 4.679432082 5.197238082 5.715044082 6.232850082 6.750656082 7.268462082 7.786268082 8.304074082 1 2 3 4 5 6 7 8
9
8.821880082
9.339686082 9.857492082 10.37529808 10.89310408 11.41091008 11.92871608 12.44652208 12.96432808 13.48213408 13.99994008 14.51774608 15.03555208 15.55335808 16.07116408 16.58897008 17.10677608 17.62458208 18.14238808 18.66019408 19.17800008 8.47 19.69580608 8.45 Maturity (years)
NSE ZCYC 30 June 2010
Plot of the Estimated ZCYC
29-Jun-10 30-Jun-10
CCIL ZCYC 30 June 2010
Movement of Zero Coupon Yields
9.00 8.75 8.50 8.25 8.00 7.75 7.50 Zero Coupon Rate (%) 7.25 7.00 6.75 6.50 6.25 6.00 5.75 5.50 5.25 5.00 4.75 4.50
6/29/2010 6/30/2010 1 5.28 5.05 2 5.64 5.55 3 6.10 6.06 4 6.52 6.50 5 6.87 6.87 6 7.15 7.15 7 7.38 7.38 8 7.56 7.56 9 7.70 7.70 10 7.82 7.82 11 7.91 7.91 12 7.99 7.99 13 8.06 8.06 14 8.12 8.12 15 8.16 8.16 16 8.21 8.21 17 8.24 8.24 18 8.28 8.28 19 8.30 8.30 20 8.33 8.33 21 8.35 8.35 22 8.37 8.37 23 8.39 8.39 24 8.41 8.41 25 8.43 8.43 26 8.44 8.44 27 8.45 8.45 28 8.47 8.47 29 8.48 8.48 30 8.49 8.49 31 8.50 8.50
Tenor (In Yrs)
TYPES OF THE YIELD CURVE
Normal Yield Curve
Flat Yield Curve
Inverted Yield Curve
Humped Yield Curve
TERM PREMIUM
?
The difference between short-term and longterm yield rate is known as ‘term-premium’ or ‘term-spread’. The term premium is reflected by the slop of the yield curve.
?
YIELD CURVE FOR BONDS OF DIFFERENT QUALITY
Credit Spread
ZCYC
?
The yield curve for Zero Coupon Treasury Bonds (ZCYC) is used as a benchmark. A coupon paying bond can be viewed as a portfolio of multiple zero coupon bonds (STRIPING). A corporate Bond can be valued using ZCYC after making adjustment for Credit Spread
?
?
STRIPING
STRIPS = Separate Trading of Registered Interest and Principal of Securities Striping= Separating Coupons and Principals
A coupon paying bond can be viewed as a portfolio of zero coupon bonds of different maturities.
USE OF YIELD CURVE
Why it is so important?
VALUATION OF BOND
VALUATION OF BOND
Face Value Redemption Value (%) Coupon Rate Maturity (Years) Zero Coupen Yield Curve Year 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 1000 100 0.08 6 YTM 6.0 6.2 6.5 6.8 7.2 7.7 8.3 9.0 9.7 10.5 11.5 12.7
P??
t ?1
Year 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Bond Price Half-years 1 2 3 4 5 6 7 8 9 10 11 12
n
Ct t (1 ? yt )
PV 38.83 37.63 36.34 34.99 33.52 31.89 30.09 28.13 26.12 23.98 21.63 496.80 839.95
YTM(%) Half-yearly rates(yt) Cashflow (Ct) 6.0 3.0 40 6.2 3.1 40 6.5 3.3 40 6.8 3.4 40 7.2 3.6 40 7.7 3.9 40 8.3 4.2 40 9.0 4.5 40 9.7 4.9 40 10.5 5.3 40 11.5 5.8 40 12.7 6.4 1040
RIDING ON THE YIELD CURVE
CONNECTING THEORY TO PRACTICE ...
Mr. Anil Sharma has Rs. 1,00,000 which he wants to invest in any of the following two annual coupon paying bonds for two years. After that, he has to make an installment for a flat he booked in Surya Apartments. Since he is busy with his job, he wants to follow buy-&-hold strategy. Face Value Maturity Coupon YCYC is as under: Year 1 2 3 4 5 6 Yield 3.0% 4.0% 4.8% 5.5% 6.0% 6.5% Assuming that the yield curve will not change, suggest which bond Mr. Sharma should buy. Bond X Rs. 1,000 2 years 10% Bond Y Rs. 1,000 6 years 18%
Consider and Think!!!!!
REALIZED YIELD
?
Bond X: Bond Y:
3.91% 6.99%
?
What is the source of higher yield for Bond Y. What is the risk associated with this.
Think about it!!!!!
Yield
0
t1
t2
Time
SPOT RATE AND FORWARD RATE
? Spot
rates y t are the interest rates for different durations
? Forward
fis the interest rate for the rate t money to be borrowed between two dates in the future, but under the terms agreed upon today.
RELATIONSHIP BETWEEN SPOT RATE AND FORWARD RATE
(1 ? yt ) (1 ? f t ) ? t ?1 (1 ? yt ?1 )
t
RELATIONSHIP BETWEEN SHORT RATE AND
SPOT RATE
(1 ? yt ) ? {(1 ? f1 )(1 ? f 2 )...( 1 ? f t )}
1/ t
1 f1=8% y1=8%
2 f1=9%
3 f1=10%
4 f1=11%
y2= 8.50
y2=9.00% y2= 9.49%
SHORT RATES
?
A short rate ( t ) is the rate of interest for one period (say year) Expected Short Rate
r
?
E ( rt )
RELATIONSHIP BETWEEN EXPECTED FUTURE
SHORT RATE AND FORWARD RATE
f t ? E ( rt )
Why?
The difference between these two is called liquidity premium
f t ? E (rt ) ? Liquidity Premium
Remember, E(rt) is not observable.
THEORIES OF TERM STRUCTURE
What decides the shape of the yield curve?
TERM STRUCTURE : BASIC FACTS
? Fact
1: Interest rates for different maturities tend
to move together over time.
? Fact
2: Yields on short-term bond more volatile
than yields on long-term bonds.
? Fact
3: Long-term yield tends to be higher than
short term yields (i.e. yield curves usually are upward sloping).
EXPECTATION THEORY
?
Forward rate equals the market consensus expectations of future short rate.
f t ? E (rt )
? ? ?
Bonds of different maturity are perfect substitute of each other. No liquidity premium ; no inflation and interest rate risk. Expectations theory cannot explain why long-term yields are normally higher than short-term yields, in other words, why the yield curve is usually upward sloping.
SEGMENTED MARKETS THEORY
Markets for different-maturity bonds are completely segmented with different sets of investors. ? The interest rate for each bond with a different maturity is determined by the supply of and demand for the bond ? Because bonds of shorter holding periods have lower inflation and interest rate risks, they have lower yield; that’s why the yield curve is usually upward sloping. ? If markets for different-maturity bonds are completely segmented, there is not reason why the short and long yields should move together
?
LIQUIDITY PREMIUM THEORY
Bond investment is subject to inflation and interest rate risk. ? The further we look into the future, the greater the uncertainly about the level of inflation, which implies that a bond’s inflation risk increases with its time to maturity. ? Interest-rate risk arises from a mismatch between investor’s investment horizon and a bond’s time to maturity. ? The longer the term of the bond, the greater the price changes for a given change in interest rates and the larger the potential for capital losses.
?
…LIQUIDITY PREMIUM THEORY
The liquidity premium theory views bonds of different maturities as substitutes, but not perfect substitutes. ? Investors prefer short rather than long bonds because they are free of inflation and interest rate risks. ? Therefore, they must be paid positive liquidity (term) premium to hold long-term bonds.
?
…LIQUIDITY PREMIUM THEORY: IMPLICATIONS
Interest rates of different maturities will move together because the long-term rates are essentially tied to the short-term rates. ? Long rates will also be less volatile because part of the long rate, which is just an average of the short rates, will smoothen out the volatility in the short rates. ? Since the risk premium increases with time to maturity, the liquidity premium theory tells us that the yield curve will normally slope upwards, only rarely will it lied flat or slope downwards.
?
f t ? E (rt ) ? Liquidity Premium
Yield Curve
Yield
Liquidity Premium
Expected Interest Rates
Term-to-Maturity
Yield Curve Liquidity Premium
Yield
Expected Interest Rates
Term-to-Maturity
Yield
Yield Curve
Liquidity Premium Expected Interest Rates
Term-to-Maturity
YIELD CURVE AS A PREDICTOR OF
FUTURE ECONOMIC ENVIRONMENT
THE YIELD CURVE AS A PREDICTOR OF FUTURE ECONOMIC ENVIRONMENT
?
It is believed that the yield curve can be helpful in predicting: ? Future directions of the Interest Rates ? Inflationary expectations ? Change in Business Cycles and Real Activities ? Stock market trends
YIELD CURVE AND FUTURE EXPECTED INTEREST RATES
?
The slope of the yield curve provides an important clue to the direction of future short-term interest rates;
upward sloping curve higher future interest rates downward sloping curve lower rates in the future. humped curve short-term rates (over the next year) are expected to rise, but that over the long-run (several years) rates are expected to fall.
? ? ?
PROBLEMS IN ESTIMATING INTERPRETATION
f t ? E (rt ) ? Liquidity Premium
1. It can not be assumed constant 2. It is unobservable/ can not be estimated
YIELD CURVE AND INFLATIONARY EXPECTATIONS
?
The overall level of the yield curve also may shift up or down—at least in part because of changes in inflationary expectations over time.
Nominal Rate of Interest ? Real Rate of Interest? Inflation
YIELD CURVE AND BUSINESS CYCLES
Both short and long rates tend to rise during business cycle expansions and to decline during subsequent downturns. ? Yield Curve becomes flat during boom. ? Inverted yield curve is observed around business cycle peaks (at the top of a Boom) and predicts the recession/slowdown. ? It becomes steep during recession. ? During early stage of economic expansion it tends to be low and upward sloping.
?
CONSTRUCTION OF A THEORETICAL YIELD CURVE
BOOTSTRAPPING: ESTIMATING YIELD CURVE USING COUPON PAYING BONDS
Zero Coupon Bonds may not be available for all maturities. ? Generally, treasury bills are non-coupon paying but treasury bonds are coupon paying bonds. ? In this situation, theoretical spot rates for zero coupon bonds can be determined using bootstrapping.
?
CURVE FITTING: POLYNOMIAL FUNCTION
y t ? a 0 ? a1t ? a 2 t ? a3 t
2
3
5.000
4.000
3.000
2.000
1.000
0.000 0.000
5.000
10.000
15.000
20.000
25.000
30.000
35.000
-1.000
NELSON SIEGEL MODEL
yt ? a0 ? (a1 ? a2 )
?
t
(1 ? e
?t / ?
) ? a2e
?t / ?
5.000
4.500
4.000
3.500
3.000
2.500
2.000
1.500
1.000
0.500
0.000 0.000
5.000
10.000
15.000
20.000
25.000
30.000
35.000
doc_309840560.pptx
different types of yield curves, normal yield curve, flat yield curve, inverted yield curve, humped yield curve and yoeld curve for bonds of different quality. It also explains the concept of Stripping and various theories of term strucutre like expectation theory, segmented market theory, liquidity premium theory.
YIELD CURVE AND TERM STRUCTURE OF INTEREST RATES
YIELD AND TERM-TO-MATURITY
Yield of a Bond depends on its it’s term to maturity. ? Generally, longer the term-to-maturity higher is the yield. ? But some times this relation may invert.
?
Estimated Yield on Government Securities of Different Maturities
Date
Yield on 6 month Bill 2.34% 8.55%
Yield on 5 Year ZCB 7.41% 7.94%
30-June 2010 25 -April 2008
Source : NSE
TERM STRUCTURE OF INTEREST RATES: US
TERM STRUCTURE OF INTEREST RATES: INDIA
YIELD CURVE
?A
graph showing relationship between yield-to-maturity and term-tomaturity for the bonds of a given quality class.
: Spot Rate Curve
? Also
? Yield
Curve : Sources
?
India
?
NSE : http://www.nse-india.com/content/debt/debt_zcyc.htm ? CCIL http://www.ccilindia.com/GSecZCYC.aspx.
?
US
?
US Treasury: http://www.ustreas.gov/offices/domestic-finance/debtmanagement/interest-rate/yield.shtml
Spot interest rates 10 0 0.019178082 4.72 0.536984082 1.054790082 1.572596082 0.61 2.090402082 2.608208082 3.126014082 3.643820082 4.161626082 4.679432082 5.197238082 5.715044082 6.232850082 6.750656082 7.268462082 7.786268082 8.304074082 1 2 3 4 5 6 7 8
9
8.821880082
9.339686082 9.857492082 10.37529808 10.89310408 11.41091008 11.92871608 12.44652208 12.96432808 13.48213408 13.99994008 14.51774608 15.03555208 15.55335808 16.07116408 16.58897008 17.10677608 17.62458208 18.14238808 18.66019408 19.17800008 8.47 19.69580608 8.45 Maturity (years)
NSE ZCYC 30 June 2010
Plot of the Estimated ZCYC
29-Jun-10 30-Jun-10
CCIL ZCYC 30 June 2010
Movement of Zero Coupon Yields
9.00 8.75 8.50 8.25 8.00 7.75 7.50 Zero Coupon Rate (%) 7.25 7.00 6.75 6.50 6.25 6.00 5.75 5.50 5.25 5.00 4.75 4.50
6/29/2010 6/30/2010 1 5.28 5.05 2 5.64 5.55 3 6.10 6.06 4 6.52 6.50 5 6.87 6.87 6 7.15 7.15 7 7.38 7.38 8 7.56 7.56 9 7.70 7.70 10 7.82 7.82 11 7.91 7.91 12 7.99 7.99 13 8.06 8.06 14 8.12 8.12 15 8.16 8.16 16 8.21 8.21 17 8.24 8.24 18 8.28 8.28 19 8.30 8.30 20 8.33 8.33 21 8.35 8.35 22 8.37 8.37 23 8.39 8.39 24 8.41 8.41 25 8.43 8.43 26 8.44 8.44 27 8.45 8.45 28 8.47 8.47 29 8.48 8.48 30 8.49 8.49 31 8.50 8.50
Tenor (In Yrs)
TYPES OF THE YIELD CURVE
Normal Yield Curve
Flat Yield Curve
Inverted Yield Curve
Humped Yield Curve
TERM PREMIUM
?
The difference between short-term and longterm yield rate is known as ‘term-premium’ or ‘term-spread’. The term premium is reflected by the slop of the yield curve.
?
YIELD CURVE FOR BONDS OF DIFFERENT QUALITY
Credit Spread
ZCYC
?
The yield curve for Zero Coupon Treasury Bonds (ZCYC) is used as a benchmark. A coupon paying bond can be viewed as a portfolio of multiple zero coupon bonds (STRIPING). A corporate Bond can be valued using ZCYC after making adjustment for Credit Spread
?
?
STRIPING
STRIPS = Separate Trading of Registered Interest and Principal of Securities Striping= Separating Coupons and Principals
A coupon paying bond can be viewed as a portfolio of zero coupon bonds of different maturities.
USE OF YIELD CURVE
Why it is so important?
VALUATION OF BOND
VALUATION OF BOND
Face Value Redemption Value (%) Coupon Rate Maturity (Years) Zero Coupen Yield Curve Year 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 1000 100 0.08 6 YTM 6.0 6.2 6.5 6.8 7.2 7.7 8.3 9.0 9.7 10.5 11.5 12.7
P??
t ?1
Year 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Bond Price Half-years 1 2 3 4 5 6 7 8 9 10 11 12
n
Ct t (1 ? yt )
PV 38.83 37.63 36.34 34.99 33.52 31.89 30.09 28.13 26.12 23.98 21.63 496.80 839.95
YTM(%) Half-yearly rates(yt) Cashflow (Ct) 6.0 3.0 40 6.2 3.1 40 6.5 3.3 40 6.8 3.4 40 7.2 3.6 40 7.7 3.9 40 8.3 4.2 40 9.0 4.5 40 9.7 4.9 40 10.5 5.3 40 11.5 5.8 40 12.7 6.4 1040
RIDING ON THE YIELD CURVE
CONNECTING THEORY TO PRACTICE ...
Mr. Anil Sharma has Rs. 1,00,000 which he wants to invest in any of the following two annual coupon paying bonds for two years. After that, he has to make an installment for a flat he booked in Surya Apartments. Since he is busy with his job, he wants to follow buy-&-hold strategy. Face Value Maturity Coupon YCYC is as under: Year 1 2 3 4 5 6 Yield 3.0% 4.0% 4.8% 5.5% 6.0% 6.5% Assuming that the yield curve will not change, suggest which bond Mr. Sharma should buy. Bond X Rs. 1,000 2 years 10% Bond Y Rs. 1,000 6 years 18%
Consider and Think!!!!!
REALIZED YIELD
?
Bond X: Bond Y:
3.91% 6.99%
?
What is the source of higher yield for Bond Y. What is the risk associated with this.
Think about it!!!!!
Yield
0
t1
t2
Time
SPOT RATE AND FORWARD RATE
? Spot
rates y t are the interest rates for different durations
? Forward
fis the interest rate for the rate t money to be borrowed between two dates in the future, but under the terms agreed upon today.
RELATIONSHIP BETWEEN SPOT RATE AND FORWARD RATE
(1 ? yt ) (1 ? f t ) ? t ?1 (1 ? yt ?1 )
t
RELATIONSHIP BETWEEN SHORT RATE AND
SPOT RATE
(1 ? yt ) ? {(1 ? f1 )(1 ? f 2 )...( 1 ? f t )}
1/ t
1 f1=8% y1=8%
2 f1=9%
3 f1=10%
4 f1=11%
y2= 8.50
y2=9.00% y2= 9.49%
SHORT RATES
?
A short rate ( t ) is the rate of interest for one period (say year) Expected Short Rate
r
?
E ( rt )
RELATIONSHIP BETWEEN EXPECTED FUTURE
SHORT RATE AND FORWARD RATE
f t ? E ( rt )
Why?
The difference between these two is called liquidity premium
f t ? E (rt ) ? Liquidity Premium
Remember, E(rt) is not observable.
THEORIES OF TERM STRUCTURE
What decides the shape of the yield curve?
TERM STRUCTURE : BASIC FACTS
? Fact
1: Interest rates for different maturities tend
to move together over time.
? Fact
2: Yields on short-term bond more volatile
than yields on long-term bonds.
? Fact
3: Long-term yield tends to be higher than
short term yields (i.e. yield curves usually are upward sloping).
EXPECTATION THEORY
?
Forward rate equals the market consensus expectations of future short rate.
f t ? E (rt )
? ? ?
Bonds of different maturity are perfect substitute of each other. No liquidity premium ; no inflation and interest rate risk. Expectations theory cannot explain why long-term yields are normally higher than short-term yields, in other words, why the yield curve is usually upward sloping.
SEGMENTED MARKETS THEORY
Markets for different-maturity bonds are completely segmented with different sets of investors. ? The interest rate for each bond with a different maturity is determined by the supply of and demand for the bond ? Because bonds of shorter holding periods have lower inflation and interest rate risks, they have lower yield; that’s why the yield curve is usually upward sloping. ? If markets for different-maturity bonds are completely segmented, there is not reason why the short and long yields should move together
?
LIQUIDITY PREMIUM THEORY
Bond investment is subject to inflation and interest rate risk. ? The further we look into the future, the greater the uncertainly about the level of inflation, which implies that a bond’s inflation risk increases with its time to maturity. ? Interest-rate risk arises from a mismatch between investor’s investment horizon and a bond’s time to maturity. ? The longer the term of the bond, the greater the price changes for a given change in interest rates and the larger the potential for capital losses.
?
…LIQUIDITY PREMIUM THEORY
The liquidity premium theory views bonds of different maturities as substitutes, but not perfect substitutes. ? Investors prefer short rather than long bonds because they are free of inflation and interest rate risks. ? Therefore, they must be paid positive liquidity (term) premium to hold long-term bonds.
?
…LIQUIDITY PREMIUM THEORY: IMPLICATIONS
Interest rates of different maturities will move together because the long-term rates are essentially tied to the short-term rates. ? Long rates will also be less volatile because part of the long rate, which is just an average of the short rates, will smoothen out the volatility in the short rates. ? Since the risk premium increases with time to maturity, the liquidity premium theory tells us that the yield curve will normally slope upwards, only rarely will it lied flat or slope downwards.
?
f t ? E (rt ) ? Liquidity Premium
Yield Curve
Yield
Liquidity Premium
Expected Interest Rates
Term-to-Maturity
Yield Curve Liquidity Premium
Yield
Expected Interest Rates
Term-to-Maturity
Yield
Yield Curve
Liquidity Premium Expected Interest Rates
Term-to-Maturity
YIELD CURVE AS A PREDICTOR OF
FUTURE ECONOMIC ENVIRONMENT
THE YIELD CURVE AS A PREDICTOR OF FUTURE ECONOMIC ENVIRONMENT
?
It is believed that the yield curve can be helpful in predicting: ? Future directions of the Interest Rates ? Inflationary expectations ? Change in Business Cycles and Real Activities ? Stock market trends
YIELD CURVE AND FUTURE EXPECTED INTEREST RATES
?
The slope of the yield curve provides an important clue to the direction of future short-term interest rates;
upward sloping curve higher future interest rates downward sloping curve lower rates in the future. humped curve short-term rates (over the next year) are expected to rise, but that over the long-run (several years) rates are expected to fall.
? ? ?
PROBLEMS IN ESTIMATING INTERPRETATION
f t ? E (rt ) ? Liquidity Premium
1. It can not be assumed constant 2. It is unobservable/ can not be estimated
YIELD CURVE AND INFLATIONARY EXPECTATIONS
?
The overall level of the yield curve also may shift up or down—at least in part because of changes in inflationary expectations over time.
Nominal Rate of Interest ? Real Rate of Interest? Inflation
YIELD CURVE AND BUSINESS CYCLES
Both short and long rates tend to rise during business cycle expansions and to decline during subsequent downturns. ? Yield Curve becomes flat during boom. ? Inverted yield curve is observed around business cycle peaks (at the top of a Boom) and predicts the recession/slowdown. ? It becomes steep during recession. ? During early stage of economic expansion it tends to be low and upward sloping.
?
CONSTRUCTION OF A THEORETICAL YIELD CURVE
BOOTSTRAPPING: ESTIMATING YIELD CURVE USING COUPON PAYING BONDS
Zero Coupon Bonds may not be available for all maturities. ? Generally, treasury bills are non-coupon paying but treasury bonds are coupon paying bonds. ? In this situation, theoretical spot rates for zero coupon bonds can be determined using bootstrapping.
?
CURVE FITTING: POLYNOMIAL FUNCTION
y t ? a 0 ? a1t ? a 2 t ? a3 t
2
3
5.000
4.000
3.000
2.000
1.000
0.000 0.000
5.000
10.000
15.000
20.000
25.000
30.000
35.000
-1.000
NELSON SIEGEL MODEL
yt ? a0 ? (a1 ? a2 )
?
t
(1 ? e
?t / ?
) ? a2e
?t / ?
5.000
4.500
4.000
3.500
3.000
2.500
2.000
1.500
1.000
0.500
0.000 0.000
5.000
10.000
15.000
20.000
25.000
30.000
35.000
doc_309840560.pptx