Théorie Financière 2004-2005 2. Valeur actuelle Professeur André Farber.

Théorie Financière2004-20052. Valeur actuelle

Professeur André Farber

Tfin 2004 02 Present Value |2August 23, 2004

Present Value

• Objectives for this session :

1. Review present value calculation in a simple 1-period setting

2. Extend present value calculation to several periods

3.Analyse the impact of the compounding periods

4.Introduce shortcut formulas for PV calculations

5. Apply present value for pricing bonds


Interest rates and present value: 1-period

• Review: 1-period

• Future value of C0 : FV1(C0) = C0 ×(1+r1) = C0 / DF1

• Present value of C1: PV(C1) = C1 / (1+r1) = C1 × DF1

• Data: r1 → DF1 = 1/(1+r1) or Data: DF1 → r1 = 1/DF1 - 1


Using Present Value

• Consider simple investment project: C0 = -I = -100 C1 = +125

• Interest rate r1 = 5%, DF1 = 0.9523

05.19

9523.0125100

1 111

1

DFCIr

CINPV


Internal Rate of Return

• Alternative rule: compare the internal rate of return for the project to the opportunity cost of capital

• Definition of the Internal Rate of Return IRR : (1-period)

IRR = Profit/Investment = (C1 - I)/I

• In our example: IRR = (125 - 100)/100 = 25%

• The Rate of Return Rule: Invest if IRR > r

• In this simple setting, the NPV rule and the Rate of Return Rule lead to the same decision:

• NPV = -I+C1/(1+r) >0 C1>I(1+r) (C1-I)/I>r IRR>r


IRR: a general definition

• The Internal Rate of Return is the discount rate such that the NPV is equal to zero.

• -I + C1/(1+IRR) 0

• In our example:

• -100 + 125/(1+IRR)=0

• IRR=25%

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

25.00

30.00

0.0% 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% 17.5% 20.0% 22.5% 25.0% 27.5% 30.0%

Discount Rate

Net P

rese

nt V

alue

IRR


Present Value Calculation with Uncertainty

• Consider the following project:

• C0 = -I = -100 Cash flow year 1:

• The expected future cash flow is C1 = 0.5 * 50 + 0.5 * 200 = 125

• The discount rate to use is the expected return of a stock with similar risk

• r = Risk-free rate + Risk premium

• = 5% + 6% (this is an example)

• NPV = -100 + 125 / (1.11) = 12.6

+50 with probability ½

+200 with probability ½


Several periods: a simple investment problem

• Consider the following project:

• Cash flow t = 0 C0 = - I = - 100

• Cash flow t = 5 C5 = + 150 (risk-free)

• How to calculate the economic profit?

• Compare initial investment with the market value of the future cash flow.

• Market value of C5 = Present value of C5

• = C5 * Present value of $1 in year 5

• = C5 * 5-year discount factor

• = C5 * DF5

• Profit = Net Present Value = - I + C5 * DF5


Using prices of U.S. Treasury STRIPS

• Separate Trading of Registered Interest and Principal of Securities

• Prices of zero-coupons

• Example: Suppose you observe the following prices

Maturity Price for $100 face value

1 98.03

2 94.65

3 90.44

4 86.48

5 80.00

• The market price of $1 in 5 years is DF5 = 0.80

• NPV = - 100 + 150 * 0.80 = - 100 + 120 = +20


Present Value: general formula

• Cash flows: C1, C2, C3, … ,Ct, … CT

• Discount factors: DF1, DF2, … ,DFt, … , DFT

• Present value: PV = C1 × DF1 + C2 × DF2 + … + CT × DFT

• An example:

• Year 0 1 2 3

• Cash flow -100 40 60 30

• Discount factor 1.000 0.9803 0.9465 0.9044

• Present value -100 39.21 56.79 27.13

• NPV = - 100 + 123.13 = 23.13


Several periods: future value and compounding

• Invests for €1,000 two years (r = 8%) with annual compounding

• After one year FV1 = C0 × (1+r) = 1,080

• After two years FV2 = FV1 × (1+r) = C0 × (1+r) × (1+r)

• = C0 × (1+r)² = 1,166.40• Decomposition of FV2

• C0 Principal amount 1,000

• C0 × 2 × r Simple interest 160

• C0 × r² Interest on interest 6.40

• Investing for t years FVt = C0 (1+r)t

• Example: Invest €1,000 for 10 years with annual compounding

• FV10 = 1,000 (1.08)10 = 2,158.82

Principal amount 1,000Simple interest 800Interest on interest 358.82


Present value and discounting

• How much would an investor pay today to receive €Ct in t years given market interest rate rt?

• We know that 1 €0 => (1+rt)t €t

• Hence PV (1+rt)t = Ct => PV = Ct/(1+rt)t = Ct DFt

• The process of calculating the present value of future cash flows is called discounting.

• The present value of a future cash flow is obtained by multiplying this cash flow by a discount factor (or present value factor) DFt

• The general formula for the t-year discount factor is: t

tt r

DF)1(

1


Discount factors

Interest rate per year

# years1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091

2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 0.8573 0.8417 0.8264

3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 0.7938 0.7722 0.7513

4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 0.7350 0.7084 0.6830

5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209

6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 0.6302 0.5963 0.5645

7 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 0.5835 0.5470 0.5132

8 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 0.5403 0.5019 0.4665

9 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 0.5002 0.4604 0.4241

10 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 0.4632 0.4224 0.3855


Spot interest rates

• Back to STRIPS. Suppose that the price of a 5-year zero-coupon with face value equal to 100 is 75.

• What is the underlying interest rate?

• The yield-to-maturity on a zero-coupon is the discount rate such that the market value is equal to the present value of future cash flows.

• We know that 75 = 100 * DF5 and DF5 = 1/(1+r5)5

• The YTM r5 is the solution of:

• The solution is:

• This is the 5-year spot interest rate

55 )1(

10075

r

%92.5175

100 51

5

r


Term structure of interest rate

• Relationship between spot interest rate and maturity.

• Example:

• Maturity Price for €100 face value YTM (Spot rate)

• 1 98.03 r1 = 2.00%

• 2 94.65 r2 = 2.79%

• 3 90.44 r3 = 3.41%

• 4 86.48 r4 = 3.70%

• 5 80.00 r5 = 4.56%

• Term structure is:

• Upward sloping if rt > rt-1 for all t

• Flat if rt = rt-1 for all t

• Downward sloping (or inverted) if rt < rt-1 for all t


The Euro yield curve


Using one single discount rate

• When analyzing risk-free cash flows, it is important to capture the current term structure of interest rates: discount rates should vary with maturity.

• When dealing with risky cash flows, the term structure is often ignored.

• Present value are calculated using a single discount rate r, the same for all maturities.

• Remember: this discount rate represents the expected return.

• = Risk-free interest rate + Risk premium

• This simplifying assumption leads to a few useful formulas for:

• Perpetuities (constant or growing at a constant rate)

• Annuities (constant or growing at a constant rate)


Constant perpetuity

• Ct =C for t =1, 2, 3, .....

• Examples: Preferred stock (Stock paying a fixed dividend)

• Suppose r =10% Yearly dividend = 50

• Market value P0?

• Note: expected price next year =

• Expected return =

50010.

501 P

r

CPV

Proof:PV = C d + C d² + C d3 + …PV(1+r) = C + C d + C d² + …PV(1+r)– PV = CPV = C/r

50010.

500 P

%10500

)500500(50)(

0

011

P

PPdiv


Growing perpetuity

• Ct =C1 (1+g)t-1 for t=1, 2, 3, ..... r>g

• Example: Stock valuation based on:

• Next dividend div1, long term growth of dividend g

• If r = 10%, div1 = 50, g = 5%

• Note: expected price next year =

• Expected return =

gr

CPV

1

000,105.10.

500

P

050,105.10.

5.521

P

%10000,1

)000,1050,1(50)(

0

011

P

PPdiv


Constant annuity

• A level stream of cash flows for a fixed numbers of periods

• C1 = C2 = … = CT = C

• Examples:

• Equal-payment house mortgage

• Installment credit agreements

• PV = C * DF1 + C * DF2 + … + C * DFT +

• = C * [DF1 + DF2 + … + DFT]

• = C * Annuity Factor

• Annuity Factor = present value of €1 paid at the end of each T periods.


Constant Annuity

• Ct = C for t = 1, 2, …,T

• Difference between two annuities:

– Starting at t = 1 PV=C/r

– Starting at t = T+1 PV = C/r ×[1/(1+r)T]

• Example: 20-year mortgage

Annual payment = €25,000

Borrowing rate = 10%

PV =( 25,000/0.10)[1-1/(1.10)20] = 25,000 * 10 *(1 – 0.1486)

= 25,000 * 8.5136

= € 212,839

])1(

11[

Trr

CPV


Annuity Factors

Interest rate per year

# years1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091

2 1.9704 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355

3 2.9410 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869

4 3.9020 3.8077 3.7171 3.6299 3.5460 3.4651 3.3872 3.3121 3.2397 3.1699

5 4.8534 4.7135 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908

6 5.7955 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553

7 6.7282 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684

8 7.6517 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5348 5.3349

9 8.5660 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590

10 9.4713 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446


Growing annuity

• Ct = C1 (1+g)t-1 for t = 1, 2, …, T r ≠ g

• This is again the difference between two growing annuities:

– Starting at t = 1, first cash flow = C1

– Starting at t = T+1 with first cash flow = C1 (1+g)T

• Example: What is the NPV of the following project if r = 10%?

Initial investment = 100, C1 = 20, g = 8%, T = 10

NPV= – 100 + [20/(10% - 8%)]*[1 – (1.08/1.10)10]

= – 100 + 167.64

= + 67.64

T

r

g

gr

CPV

1

111


Present Value: general formula

• Cash flows: C1, C2, C3, … ,Ct, … CT

• Discount factors: DF1, DF2, … ,DFt, … , DFT

• Present value: PV = C1 × DF1 + C2 × DF2 + … + CT × DFT

TT

Tt

t

t

r

C

r

C

r

C

r

CPV

)1(...

)1(...

)1()1( 22

2

1

1

TT

tt

r

C

r

C

r

C

r

CPV

)1(...

)1(...

)1()1( 221

If r1 = r2 = ...=r


Useful formulas: summary

• Constant perpetuity: Ct = C for all t

• Growing perpetuity: Ct = Ct-1(1+g)

r>g t = 1 to ∞

• Constant annuity: Ct=C t=1 to T

• Growing annuity: Ct = Ct-1(1+g)

t = 1 to T

r

CPV

gr

CPV

1

))1(

11(

Trr

CPV

))1(

)1(1(1

T

T

r

g

gr

CPV


Compounding interval

• Up to now, interest paid annually

• If n payments per year, compounded value after 1 year :

• Example: Monthly payment :

• r = 12%, n = 12

• Compounded value after 1 year : (1 + 0.12/12)12= 1.1268

• Effective Annual Interest Rate: 12.68%

• Continuous compounding:

• [1+(r/n)]n→er (e= 2.7183)

• Example : r = 12% e12 = 1.1275

• Effective Annual Interest Rate : 12.75%

n

n

r)1(


Juggling with compounding intervals

• The effective annual interest rate is 10%

• Consider a perpetuity with annual cash flow C = 12

– If this cash flow is paid once a year: PV = 12 / 0.10 = 120

• Suppose know that the cash flow is paid once a month (the monthly cash flow is 12/12 = 1 each month). What is the present value?

• Solution 1:

1. Calculate the monthly interest rate (keeping EAR constant)

(1+rmonthly)12 = 1.10 → rmonthly = 0.7974%

2. Use perpetuity formula:

PV = 1 / 0.007974 = 125.40

• Solution 2:

1. Calculate stated annual interest rate = 0.7974% * 12 = 9.568%

2. Use perpetuity formula: PV = 12 / 0.09568 = 125.40


Interest rates and inflation: real interest rate

• Nominal interest rate = 10% Date 0 Date 1

• Individual invests $ 1,000

• Individual receives $ 1,100

• Hamburger sells for $1 $1.06

• Inflation rate = 6%

• Purchasing power (# hamburgers) H1,000 H1,038

Real interest rate = 3.8%

• (1+Nominal interest rate) = (1+Real interest rate)×(1+Inflation rate)

• Approximation:

Real interest rate ≈ Nominal interest rate - Inflation rate


Bond Valuation

• Objectives for this session :

– 1.Introduce the main categories of bonds

– 2.Understand bond valuation

– 3.Analyse the link between interest rates and bond prices

– 4.Introduce the term structure of interest rates

– 5.Examine why interest rates might vary according to maturity


Zero-coupon bond

• Pure discount bond - Bullet bond

• The bondholder has a right to receive:

• one future payment (the face value) F

• at a future date (the maturity) T

• Example : a 10-year zero-coupon bond with face value $1,000

•

• Value of a zero-coupon bond:

• Example :

• If the 1-year interest rate is 5% and is assumed to remain constant

• the zero of the previous example would sell for

TrPV

)1(

1

91.613)05.1(

000,110

PV


Level-coupon bond

• Periodic interest payments (coupons)

• Europe : most often once a year

• US : every 6 months

• Coupon usually expressed as % of principal

• At maturity, repayment of principal

• Example : Government bond issued on March 31,2000

• Coupon 6.50%

• Face value 100

• Final maturity 2005

• 2000 2001 2002 2003 2004 2005

• 6.50 6.50 6.50 6.50 106.50


Valuing a level coupon bond

• Example: If r = 5%

• Note: If P0 > F: the bond is sold at a premium

• If P0 <F: the bond is sold at a discount

• Expected price one year later P1 = 105.32

• Expected return: [6.50 + (105.32 – 106.49)]/106.49 = 5%

49.1067835.01003295.45.61005.6 5505.0 dAP

TTrTT

dACrr

C

r

C

r

CP

100

)1(

100

)1(...

)1(1 20


When does a bond sell at a premium?

• Notations: C = coupon, F = face value, P = price

• Suppose C / F > r

• 1-year to maturity:

• 2-years to maturity:

• As: P1 > F

FPrF

C

Fr

FCP

00 1

1

1

r

PCP

1

10 with

r

FCP

11

FrF

C

Fr

FCP

1

1

10


A level coupon bond as a portfolio of zero-coupons

• « Cut » level coupon bond into 5 zero-coupon

• Face value Maturity Value

• Zero 1 6.50 1 6.19

• Zero 2 6.50 2 5.89

• Zero 3 6.50 3 5.61

• Zero 4 6.50 4 5.35

• Zero 5 106.50 5 83.44

• Total 106.49


Law of one price

• Suppose that you observe the following data:

Bond Price 1 2 3

A 100.97 104 0 0B 105.72 7 107 0C 101.56 5.5 5.5 105.5

Maturity

What are the underlying discount factors? Bootstrap method

100.97 = DF1 104105.72 = DF1 7 + DF2 107101.56 = DF1 5.5 + DF2 5.5 + DF3 105.5

1 2 3Discount factors 0.971 0.925 0.864Spot rates 3% 4% 5%


Bond prices and interest rates

Bond prices fall with arise in interest rates and rise with a fall ininterest rates

0,00

20,00

40,00

60,00

80,00

100,00

120,00

140,00

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%

Interest rate

Bo

nd

pri

ce


Sensitivity of zero-coupons to interest rate

0,00

50,00

100,00

150,00

200,00

250,00

300,00

350,00

400,00

450,00

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%

Interest rate

Bo

nd

pri

ce

5-Year

10-Year

15-Year


Duration for Zero-coupons

• Consider a zero-coupon with t years to maturity:

• What happens if r changes?

• For given P, the change is proportional to the maturity.

• As a first approximation (for small change of r):

trP

)1(

100

Pr

t

rr

t

rt

dr

dPtt

1)1(

100

1)1(

1001

rr

t

P

P

1

Duration = Maturity


Duration for coupon bonds

• Consider now a bond with cash flows: C1, ...,CT

• View as a portfolio of T zero-coupons.

• The value of the bond is: P = PV(C1) + PV(C2) + ...+ PV(CT)

• Fraction invested in zero-coupon t: wt = PV(Ct) / P

• •

• Duration : weighted average maturity of zero-coupons

D= w1 × 1 + w2 × 2 + w3 × 3+…+wt × t +…+ wT ×T


Duration - example

• Back to our 5-year 6.50% coupon bond.

Face value Value wt

Zero 1 6.50 6.19 5.81%

Zero 2 6.50 5.89 5.53%

Zero 3 6.50 5.61 5.27%

Zero 4 6.50 5.35 5.02%

Zero 5 106.50 83.44 78.35%

Total 106.49

• Duration D = .0581×1 + 0.0553×2 + .0527 ×3 + .0502 ×4 + .7835 ×5

• = 4.44

• For coupon bonds, duration < maturity


Price change calculation based on duration

• General formula:

• In example: Duration = 4.44 (when r=5%)

• If Δr =+1% : Δ ×4.44 × 1% = - 4.23%

• Check: If r = 6%, P = 102.11

• ΔP/P = (102.11 – 106.49)/106.49 = - 4.11%

rr

Duration

P

P

1

Difference due to convexity


Duration -mathematics

• If the interest rate changes:

• Divide both terms by P to calculate a percentage change:

• As:

• we get:

)(1

...)(1

2)(

1

1

)(...

)()(

21

21

T

T

CPVr

TCPV

rCPV

r

dr

CdPV

dr

CdPV

dr

CdPV

dr

dP

))(

...)(

2)(

1(1

11 21

P

CPVT

P

CPV

P

CPV

rPdr

dP T

P

CPVT

P

CPV

P

CPVDuration T )(

...)(

2)(

1 21

r

Duration

Pdr

dP

1

1


Yield to maturity

• Suppose that the bond price is known.

• Yield to maturity = implicit discount rate

• Solution of following equation:Ty

FC

y

C

y

CP

)1(...

)1(1 20

0,00

20,00

40,00

60,00

80,00

100,00

120,00

140,00

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%

Interest rate

Bo

nd

pri

ce


Spot rates

• Consider the following prices for zero-coupons (Face value = 100):

Maturity Price

1-year 95.24

2-year 89.85

• The one-year spot rate is obtained by solving:

• The two-year spot rate is calculated as follow:

• Buying a 2-year zero coupon means that you invest for two years at an average rate of 5.5%

%51

10024.95 1

1

rr

%5.5)1(

10085.89 22

2

rr


Forward rates

• You know that the 1-year rate is 5%.

• What rate do you lock in for the second year ?

• This rate is called the forward rate

• It is calculated as follow:

• 89.85 × (1.05) × (1+f2) = 100 → f2 = 6%

• In general:

(1+r1)(1+f2) = (1+r2)²

• Solving for f2:

• The general formula is:

111

)1(

2

1

1

22

2

d

d

r

rf

11)1(

)1( 11

1

t

tt

t

tt

t d

d

r

rf


Forward rates :example

• Maturity Discount factor Spot rates Forward rates

• 1 0.9500 5.26

• 2 0.8968 5.60 5.93

• 3 0.8444 5.80 6.21

• 4 0.7951 5.90 6.20

• 5 0.7473 6.00 6.40

• Details of calculation:

• 3-year spot rate :

• 1-year forward rate from 3 to 4

%80.51)8444.0

1(

)1(

18444.0 3

1

333

rr

%21.618444.0

8968.011

)1(

)1(

3

22

2

33

3

d

d

r

rf


Term structure of interest rates

• Why do spot rates for different maturities differ ?

• As

• r1 < r2 if f2 > r1

• r1 = r2 if f2 = r1

• r1 > r2 if f2 < r1

• The relationship of spot rates with different maturities is known as the term structure of interest rates

Time to maturity

Spotrate

Upward sloping

Flat

Downward sloping


Forward rates and expected future spot rates

• Assume risk neutrality

• 1-year spot rate r1 = 5%, 2-year spot rate r2 = 5.5%

• Suppose that the expected 1-year spot rate in 1 year E(r1) = 6%

• STRATEGY 1 : ROLLOVER

• Expected future value of rollover strategy:

• ($100) invested for 2 years :

• 111.3 = 100 × 1.05 × 1.06 = 100 × (1+r1) × (1+E(r1))

• STRATEGY 2 : Buy 1.113 2-year zero coupon, face value = 100


Equilibrium forward rate

• Both strategies lead to the same future expected cash flow

• → their costs should be identical

• In this simple setting, the foward rate is equal to the expected future spot rate

f2 =E(r1)

• Forward rates contain information about the evolution of future spot rates

)1)(1(

100))(1)(1(

)1(

1113.1100

21112

2 frrEr

r

Théorie Financière 2004-2005 2. Valeur actuelle Professeur André Farber.

Documents

Transcript of Théorie Financière 2004-2005 2. Valeur actuelle Professeur André Farber.