Microéconomie, chapter 15 - CEScermsem.univ-paris1.fr/davila/teaching/SBS/Ch15_Pindyck-09.pdf · 1 Solvay Business School – Université Libre de Bruxelles Microéconomie, chapter

1 Solvay Business School – Université Libre de Bruxelles

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Microéconomie, chapter 15

Investment and capital markets


2

Points to be discussed   Discounted present value

  The value of bonds

  The net present value as a criterion for investment decisions

  Adjusting for risks

  Consumers’ investment decisions

  Investing in education

  How are interest rates determined?


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Introduction

 What characterizes capital markets?  investments, financial or physical, take time

to produce a return  They induce a present, sure cost  But they deliver only future, uncertain returns  Present sure costs and future uncertain

returns must be compared


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Net present value NPV

 How do we compute the value of a flow of future returns?  The present value of a future income P is the

investment today necesary to obtain a return P at the same date

 The present value of a flow of future incomes is the sum of the present values of each of them


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 Future value: if the annual return R is constant, then  1 euro invested today delivers 1 + R euros

a year from now x euros invested today deliver (1 + R)x

euros a year from now  Value today of y euros a year from now? (1 + R)-1y euros invested today deliver (1 +

R) (1 + R)-1y=y euros a year from now


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€

(1+ R)n = value n years from now of 1 euro today

1(1+ R)n

= value today of 1 euro n years fom now

 The interest rate R determines the net present value


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Net present value of €1 in the future

R 1 year 5 years 10 years 30 years

1% €0,990 €0,951 €0,905 €0,742

2% €0,980 €0,906 €0,820 €0,552

5% €0,952 €0,784 €0,614 €0,231

10% €0,909 €0,621 €0,386 €0,057


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Assessing flows of future incomes

 Flows of future incomes are compared comparing their net present values

 Their net present values depend on the assumed rate of return R


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Two flows of income

flow A: €100 €100 €0 flow B: €20 €100 €100

today 1 yr 2 yrs


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€

NPV of flow A =100 +100

(1+ R)

NPV of flow B = 20 +100

(1+ R)+

100(1+ R)2

Two flows of income


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Two flows of income

NPV of flow A: €195,24 €190,90 €186,96 €183,33

NPV of flow B: €205,94 €193,54 €182,57 €172,78

R = 0,05 R = 0,10 R = 0,15 R = 0,20

  Which flow of future incomes has a higher neet present value depends of the interest rate R

  For small values of R, B’s net present value is higher   For big values of R, A’s net present value is higher


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The value of lost revenues

 The NPV can be used to compute the value for a household of the lost revenues due to the death of one of the spouses

 Example:  One of the spouses dies at 52 in 2009  wage: €85.000  Retirement age: 60


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 Which is the net present value of the lost revenues for the household?  The wage must be adjusted for foreseable

career advancemts (g%)  say g=8%

 The probability (m) of death at a later date must be taken into account as well   It can be obtained from the statistics of mortality

 Assume the return on Treasury bills is R=9%


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€

NPV = W0 +W0(1+ g)(1−m1)

(1+ R)

+ W0(1+ g)2(1−m2)(1+ R)2 + ...

+W0(1+ g)7(1−m7)

(1+ R)7



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 The sum of column 4 is the NPV of lost revenues: $650.252

 This NPV computations are used to write life insurance contracts, or for litigation for compensation in cases of accidental death


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The value of a bond

 A bond is a contract by which the issuer commits to make a flow of payments to the holder, either indefinitely or until a final payment at a fixed date

 Example: a corporate bond guarantees a payment of €100 per year (the coupon) for the next 10 years and a final payment of €1000  How much is this bond worth?


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The value of a bond

 Payments of the bond:  Payments from coupons = €100 per year for

10 years  Final payment = €1000 within 10 years

€

NPV = 100(1+ R)

+100

(1+ R)2 +

...+ 100(1+ R)10 +

1000(1+ R)10


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The value of a bond

Interest rate 0,05 0,10 0,15 0,20

NPV

of t

he fl

ow o

f pa

ymen

ts (th

ousa

nds €)

0 0,5

1

1,5

2

The higher the interest rate, the smaller the value

of the bond


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The value of a bond

 An perpetual bond pays a fixed income every period indefinitely

 The net present value of a perpetual bond is an infinite sum

 If the interest rate is R, the net present value of a perpetual bond of €100 is

NPV = €100/R


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The value of a bond

€

NPVn = 100(1+ R)

+100

(1+ R)2 + ...+ 100(1+ R)n

€

NPVn1

(1+ R) = 100

(1+ R)2 +100

(1+ R)3 + ...+ 100(1+ R)n+1

€

NPVn 1− 1(1+ R)

=100 1

(1+ R)−

1(1+ R)n+1

€

NPVn =100

1(1+ R)

−1

(1+ R)n+1

1− 1(1+ R)

€

→NPV =100 1R


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The actuarial return of a bond

 Treasury bills and corporate bonds are trade in secondary markets

 The price of a bond is then determined by its demand and supply

 The market price of a bond determines its implicit return


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 If P is the market price of a perpetual bond paying C

€

since P =CR

, then R =CP


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 If the price of a perpetual bond paying €100 is €1000, its actuarial return is

R = €100/ €1000 = 0,10 = 10%


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 Computing the return of a bond:

€

if P equals the NPV = 100(1+ R)

+100

(1+ R)2 +

...+ 100(1+ R)10 +

1000(1+ R)10

then R can be computed as a function of P


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Interest rate 0,05 0,10 0,15 0,20 0

0,5

1

1,5

2 N

PV o

f pay

men

ts (th

ousa

nds €)

The actuarial return is the interest rate that equalizes the NPV of the flow of payments from the bond and its

market price

The actuarial return of a bond is inversely related to its price


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  The return can vary across bonds  Corporate bonds have typically a higher return than

Treasury bills  This is a consequence of the different levels of risk of

different bonds   Riskier bonds yield higher returns   Governments very rarely do not pay   Some firms are much less sure bets


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The return of corporate bonds

 The return of a bond depends on its nominal value and its coupon

 Assume

Nominal value

Annual coupon maturity

firm A €100 €7,5 10 years

firm B €100 €5,5 5 years


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 If the price of a bond A is 120,25, then its return is:


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 If the price of a bond B is 73,50, then its return is:


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The Net Present Value as a guide for investment decisions

 Investors compare the present value of the flow of returns from an investment with its cost

 The investment is profitable if the present value of the flow of returns exceeds this cost


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€

C = cost of capital π n = profits of year n (n =10)

NPV = -C +π1

(1+ R)+

π 2

(1+ R)2 + ...+ π10

(1+ R)10

R = return of an alternative investment (discount rate, oportunity cost) with a similar risk

the investment is profitable is NPV > 0


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 Choice of the discount rate  The choice matters  It should be the return to a similar investment

  It must have the same risk  In absence of risk, the opportunity cost is the

return of Treasury bills


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The Net Present Value as a guide to investment decisions  Example:

 Build a plant costs €10 millions  Its output will be 8000 engines each month

for 20 years  Cost of producing 1 engine = €42,50  Sale price of 1 engine = €52,50  Profit = €10 per engine, i.e. €80.000 per month  Useful life of the plant: 20 years  Scrap value of the plant €1 million

 Is this investment profitable?


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The Net Present Value for investments

  The Net Present Value of the investment is

 R*=7,5% is the discount factor such that NPV=0  If the return of Treasury bills is below 7,5%, the NPV

is positive  If the return of Treasury bills is above 7,5%, the NPV

is negative

€

NPV = -10 + 0,96(1+ R)

+0,96

(1+ R)2 +

...+ 0,96(1+ R)20 +

1(1+ R)20

R* = 7,5%


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The Net Present Value as a guide to investment decisions

discount factor R 0 0,05 0,10 0,15 0,20

-6

Net

Pre

sent

Val

ue

(€ m

illio

ns)

-4

-2

0

2

4

6

8

10 • The investement is not

profitable if Treasury bills yield more than 7,5%

• The investement is not profitable if Treasury bills yield

less than 7,5%

R*=7,5%


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 For investment decisions one must distinguish between real and nominal interest rates  The real interest rate discounts the impact of

inflation


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Real and nominal interest rates

 Assume prices and costs are given in real terms and that the inflation rate is 5%  then nominal prices and costs are

  today, P = 52,50 1 year, P = 1,05 x 52,50 = 55,13, 2 years, P = 1,05 x 55,13 = 57,88…   today, C = 42,50 1 year, C = 1,05 x 42,50 = 44,63, 2 years, C = 1,05 x 44,63 = 46,86…  Profits are 960.000 per year in real terms


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Real and nominal interest rates

 If the flow of profits is in real terms, then the discount rate must also be in real terms, for instance  R real = R nominal - inflation = 9% - 5% =

4%


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discount rate R 0.10 0.15 0.20 0

-6

Net

Pre

sent

Val

ue

(€ m

illio

ns)

-4

-2

0

2

4

6

8

10

0.04**

If real R = 4%, the NPV is positive. The investment

Is profitable

R*=7,5%


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 The present value of some future values may be negative  Temporary losses may be caused by, for

instance  The time to build a plant  High initial costs that decrease progressively

 The investment decision must take into account the possibility of temporary losses


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 The engine production plant  Time to build the plant: 1 year

 Cost today €5 millions  Cost within a year €5 millions

 Expected losses first year €1 million, and €0,5 million the next two years

 Profits of €0,96 million per year starting at year 3 for the next 20 years

 Scrap value of the plant €1 million


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€

NPV = - 5 - 5(1+ R)

−1

(1+ R)2 −0,5

(1+ R)3

+0,96

(1+ R)4 +0,96

(1+ R)5 + ...

+0,96

(1+ R)20 +1

(1+ R)20


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Adjusting for risk

 Under uncertainty any risk can be taken care of adding a risk premium to the riskless interest rate

 Recall: the risk premium is the amount that a risk averse individual would be willing to pay to get rid of the risk


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Diversifiable and non diversifiable risks

 Diversifiable risks can be eliminated investing in different projects or firms

 Non diversifiable risks cannot be aliminated and must be included in the risk premium


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 Diversification spreads the risk thin over several uncorrelated options  Investments funds invest in different

unrelated sectors  Firms invest in several different unrelated

projects  Assets with diversifiable risks do not have a

risk premium – their return is close to the riskless rate


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 Some risks cannot be eliminated  Firms profits depend on the business cycle  Future growth is uncertain

  Investor seek a risk premium for nondiversifiable risks

 The discount factor must include a risk premium


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 Capital Asset Pricing Model (CAPM)  Explains the risk premium as a function of the

correlation of the return of an asset with the average return in the stock market


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Capital asset pricing model

 Consider investing in the stock market through an investmet fund  if rm is the expected average return in the market  and rf is the riskless return  then rm - rf is the risk premium for non diversifiable

risks  It is the excess return needed to accept the non

diversifiable risks of the stock market


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 The return of different assets is correlated with that of the stock market  The CAPM explains the excess return of

each stock by its correlation with the excess return of the market

€

ri − rf = β(rm − rf ) ri = expected return of the stockβ = beta of the stock


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  β mesures the sensitivity of the asset to non diversifiable risks  A beta close to 1 means the stock has the same non

diversifiable risks than the market  A beta close to 0 means the only risks of the stock are

diversifiable and hence its return does not exceed the riskless rate

 The bigger its beta, the higher the expected return of an asset in excess of the riskless rate


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 Once its beta i known, the appropriate discount factor to compute the Net Present Value of an investement in the stock is

€

discount factor = rf + β(rm − rf )

 i.e. the riskless rate plus a risk premium for the non diversifiable risks


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Estimation of beta

  For stocks, beta can be estimated statistically   For direct investments, beta is more difficult to

estimate   Firms typically use the cost of capital for the

frim as nominal discount rate  The weighted average of the expected return of its

stocks and the interest rate the firm pays for loans


54

Consumer’s investment decisions

 Consumers face an investment problem when they buy durable goods  They have to compare a flow of future

services to a present purchase price


55


 Example: costs and benefits of buying a car  A car provides transportation services for 5

or 6 years  One needs to compare the flow of future

services (transportation less insurance, maintenance and gas) and the purchase price


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 Example: costs and benefits of buying a car  Let S be the value in euros of the transportation

services provided by the car  Let E be the annual operating cost

 insurance, maintenance, and gas  Let €15.000 be the purchase price  Let €3.000 be the scrap value after 6 years

  The purchasing decision can be made considering the Net Present Value of the car


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 Example: costs and benefits of buying a car

€

NPV = −15.000 + (S − E) +(S − E)(1+ R)

+

(S − E)(1+ R)2 + ...+ (S − E)

(1+ R)6 +3.000

(1+ R)6


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 The decisions to buy or not depends on the discount rate  If the purchase is financed by a loan, the

interest rate of the loan is the relevant discount factor


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Investing in education

 Individuals have to make decisions about their education  Going to college or not?  Pursuing graduate education, MBA, PhD?

 Education improves the individual’s productivity and therefore improves his of her expected longlife income


60


 Consider the choice between going to college or looking for a job after high school  What is the net present value of going to

college?


61


 Cost of going to college  Opportunity cost of wages non earned – say €20.000 per year

 Registration fees, housing, etc. – say another €20.000 per year

 i.e. €40.000 per year during 4 years


62


 Benefits from going to college  A higher starting wage and better career

prospects  On average, €20.000 per year over the wage

earned with a high school degree only  Assume the active life is 20 years (!)

 What is the net present value of going to college?


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€

NPV = −40 − 40(1+ R)

−40

(1+ R)2−

40(1+ R)3

+20

(1+ R)4+ ...+ 20

(1+ R)23


64


 Which is the relevant discount rate?  Assume there is no inflation, so that we can

use the real discount rate  A rate of 5% is a typical opportunity cost for

most households   It is the return of alternative investments

 The net present value of going to college is approximately €66.000


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 On average, the wage is $30.000 higher per year

 Assume this difference persists for 20 years  An MBA lasts 2 years and costs around

$45.000   The opportunity cost of the non earned pre-

MBA wage is of $45.000 per year  All in all an MBA costs $90.000 per year during

two years


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 The net present value of an MBA is

€

NPV = −90 − 90(1+ R)

+30

(1+ R)2+ ...+ 30

(1+ R)21

 With a discount rate of 5%, it amounts to $158.000


68

How interest rates are determined?

 An interest rate is the price payed by a borrower to a lender in exchange for a loan  It is determined by the supply and demand of funds to

lend  Supply of funds comes from household savings  Demand of funds comes from

 Households wishing to comsume beyond their current income

 Firms intending to make some investments


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How interest rates are determined?   For households, the higher the interest rate, the

costlier is consumption  They borrow less  Households demand decreases with the interest rate

  For firms investments become profitable if the corresponding NPV> 0  Higher interest rates imply lower NPVs  Firms demand is also decreasing in the interest rate

  Total demand, from both households and firms, is therefore decreasing


70


Funds to lend

R Interest

rate

DT

DM

DE

DM and DE , households demad (M) and firms demand (E) are decreasing in the

interest rate


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S


Funds to lend

R Interest

rate

R*

Q*

The equilibrium interest rate is R*

DT

DM

DE


72


 Supply and demand for funds to lend determine the equilibrium interest rate  In a recession, the NPV of investments

decrease, firms invest less and demand for funds decreases. Interest rates therefore decrease

 On the other hand, government debt increases the demand for funds, pushing interest rates upwards


73

Changes in the equilibrium interest rate

S

DT

R*

Q*

During recessions the demand for funds decreases, and so do

interest rates

D’T

Q1

R1

Funds to lend

R Interest

rate


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S

DT

R*

Q*

The government debt pushes interest rates upwards

Q2

R2

D’T

Funds to lend

R Interest

rate


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S

DT

R*

Q*

When the central bank increases the money supply, the supply of

funds to lend increases S’

R1

Q1

Funds to lend

R Interest

rate

Documents

Microéconomie, chapter 15 - CEScermsem.univ-paris1.fr/davila/teaching/SBS/Ch15_Pindyck-09.pdf · 1 Solvay Business School – Université Libre de Bruxelles Microéconomie, chapter