FINANCE3 . Present Value

Professor André Farber

Solvay Business SchoolUniversité Libre de BruxellesFall 2004

MBA 2004 |2

Present Value

• Objectives for this session :

• 1. Introduce 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

MBA 2004 |3

Present Value: teaching strategy

• 1-period:

• FV, PV, 1-year discount factor DF1

• NPV, IRR

• Several periods: start from Strips

– Zero-coupons & disc. Factors

• General PV formula

– From prices to interest rates: spot rates

– Shortcut formulas: perpetuities and annuities

– Compounding interval

MBA 2004 |4

Interest rates and present value: 1 period

• Suppose that the 1-year interest rate r1 = 5%

• €1 at time 0 → €1.05 at time 1

• €1/1.05 = 0.9523 at time 0 → €1 at time 1

• 1-year discount factor: DF1 = 1 / (1+r1)

• Suppose that the 1-year discount factor DF1 = 0.95

• €0.95 at time 0 → €1 at time 1

• € 1 at time 0 → € 1/0.95 = 1.0526 at time 1

• The 1-year interest rate r1 = 5.26%

• 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

MBA 2004 |5

Using Present Value

• Consider simple investment project:

• Interest rate r = 5%, DF1 = 0.9523

125

-100

0 1

MBA 2004 |6

Net Future Value

• NFV = +125 - 100 1.05 = 20

• = + C1 - I (1+r)

• Decision rule: invest if NFV>0

• Justification: takes into cost of capital

– cost of financing

– opportunity cost

-100

+100+125

-105

0 1

MBA 2004 |7

Net Present Value

• NPV = - 100 + 125/1.05 = + 19

• = - I + C1/(1+r)

• = - I + C1 DF1

• = - 100+125 0.9524

• = +19

• DF1 = 1-year discount factor

• a market price

• C1 DF1 =PV(C1)

• Decision rule: invest if NPV>0

• NPV>0 NFV>0

-100

+125

-125

+119

MBA 2004 |8

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

MBA 2004 |9

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

MBA 2004 |10

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

MBA 2004 |11

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

MBA 2004 |12

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

MBA 2004 |13

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

MBA 2004 |14

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

MBA 2004 |15

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

MBA 2004 |16

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

MBA 2004 |17

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

MBA 2004 |18

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

MBA 2004 |19

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)

MBA 2004 |20

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

MBA 2004 |21

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

MBA 2004 |22

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.

MBA 2004 |23

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

MBA 2004 |24

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

MBA 2004 |25

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

MBA 2004 |26

Review: 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

MBA 2004 |27

Review: Shortcut formulas

• 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

MBA 2004 |28

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(

MBA 2004 |29

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

MBA 2004 |30

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