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Finance in a nutshell

Fundamentals of FinanceFinance in a nutshell

Jukka Perttunen

University of Oulu - Department of Finance

Fall 2018

Jukka Perttunen Fundamentals of Finance


Valuation of a future cash-flow

Finance is about valuation of future cash-flows is terms of today’s euros.

Future cash-flow can be considered as a random variable with a probability distribution.

Probability distribution describes the

probabilities p(x̃) of the different values

of a random variable x̃ .

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6

�x̃

p(x̃)

The present value x0 of the future cash-flow

equals to the discounted value of the

expected future cash-flow E(x̃).

E(x̃)��

��

��

��x0 = dxE(x̃)

The value of the discount factor dx depends on

– the time-value of money

t = T

t = 0

, and

– the uncertainty in the future cash flow x̃ .

� -

In terms of an annual discount rate kx , it is about

– the annual risk-free rate r ,= e−kx T E(x̃)

= e−(r+px )T E(x̃)

– the annual cash-flow-specific risk premium px, and

– the time in years to the cash-flow, i.e. T .



Valuation of the future cash-flow

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6

�x̃

p(x̃)

E(x̃)��

��

��

��x0

T = 2

t = 0

At this point, let us assume that we expect to receive a cash-flow of e5000 two years from now.

= 5000

To be able to calculate the present value x0 of the cash-flow, we need to identify the risk-free rate r ,

as well as the cash-flow-specific risk-premium px .

= e−(r+px )T E(x̃) = e−(r+px )×2 × 5000

Let us start with the risk-free rate r , which is

applicable over the two-year period in question.



Estimation of the risk-free rate

The annual risk-free rate r is typically estimated

in terms of the annual percentage yield of risk-free

(government-issued) zero-coupon bonds.

The holder of the bond receives back

the e1000 principal at time T = 2.

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6

x̃

p(x̃)

x = 1000

T = 2

There is no uncertainty in the payments

of a government-issued bond. There is

no need for any probability distribution.

The bond currently trades at e950.

��

��

��x0 = 950

t = 0

The value of the discount factor

is df = 0.95000.

= df × 1000

⇒ df = 0.95000

We may express the risk-free discount factor in terms of the

(two-year) continuously compounded annual risk-free rate r :

e−r×T = df ⇒ e−r×2 = 0.95000

⇒ r = −1

2ln 0.950000 = 0.02565 = 2.565%




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6

�x̃

p(x̃)

E(x̃)��

��

��

��x0

T = 2

t = 0

Two years from now, we expect to receive a cash-flow of e5000,

= 5000

The two-year risk-free rate is r = 2.565%.

= e−(r+px )T E(x̃) = e−(0.02565+px )×2 × 5000

Next we need to determine the cash-flow-specific

risk-premium px , which depends on the uncertainty

of the cash-flow.

� -



Uncertainty in the future cash-flow

The risk-premium px is about, how much reward the market is ready to pay to the uncertainty in the cash-flow x̃ .

The uncertainty in the cash-flow is a bit difficult issue to analyze.

The primary measurement of the uncertainty

or variability of a random variable is variance

s2 = E[(

x̃ − E(x̃))2]. s2

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6

�x̃

p(x̃)

t = T� -

However, from the point of view of the financial market, it is essential, how much the cash-flow x̃ contributes the

riskiness of a well-diversified portfolio, i.e. the riskiness of the market portfolio.

In other words, a part of the variability in the cash-flow is ignored, because diversification of investments enables us

to get rid of it, if we just want to.

The remaining part of the variability – the undiversifiable systematic risk – the risk we are not able to get rid of by

diversification, determines the size of the risk-premium px .



Uncertainty in the return provided by the cash-flow

The riskiness of a portfolio is measured in terms of the return variance of the portfolio, and thus we turn to analyze

the return R̃x provided by the cash-flow x̃ over the time-period from t = 0 to t = T :

R̃x =x̃ − x0

x0

.

Yes, we do not know the current value x0 of the cash-flow, but let us postpone the issue for a while. Anyway, it is

quite obvious, that the variability of return R̃x mirrors the variability the cash-flow x̃ . Thus we are able to measure

the variability of the cash-flow by the variance σ2x of the return provided by the cash-flow:

σ2x = E

[(R̃x − E(R̃x )

)2].

However, if the cash-flow x̃ is combined as a part of a diversified portfolio m, the most part of the variance risk σ2x

disappears, and what remains left is the covariance-risk σxm of the cash-flow with all the other assets in the portfolio:

σxm = E[(

R̃x − E(R̃x ))(

R̃m − E(R̃m))].

At this point, it is clear that we want to be able to express the discount rate kx in terms of the risk-free rate r , and a

risk-premium px , the size of which depends on the amount of the covariance-risk σxm involved with the cash-flow x̃ :

kx = r + px = r + f (σxm).



Uncertainty in the return provided by the cash-flow

We decide to build the functional form of the risk-premium in a way that it contains a common measurement of the

unit-price of risk – the expected excess return µm − r of the market portfolio over the risk-free rate.

The chosen measurement enables us to express the risk-premium in terms of a rate of return, and, furthermore, as

the rate of return of such a benchmark portfolio, which is common to all market participants.

At the same time we need to adjust our measurement σxm of the amount of risk to correspond to the above chosen

measurement of the price of risk. It is, we multiply the covariance-risk σxm by a constant c:

kx = r + px = r +(µm − r

)cσxm .

To find the value of the adjustment factor c, we apply the model to the market-portfolio, the expected return, the

required return, and the discount rate of which we know to be equal to km = µm . Correspondingly, the amount

of the covariance-risk of the market-portfolio equals to σmm = σ2m , and we get

µm = r +(µm − r

)cσ2

m ⇒ cσ2m = 1 ⇒ c =

1

σ2m

.

The general form of the models appears now as

kx = r +(µm − r

)σxmσ2m

.



Capital asset pricing model

The model is typically written as

kx = r +(µm − r

)βx , βx =

σxm

σ2m

.

In the above Capital Asset Pricing Model

– the risk-free rate, r , represents the time-value of money,

– the term µm − r represents the unit-price of market-related undiversifiable risk,

– the beta, βx , represents the investment-specific amount of market-related systematic risk.

When it comes to the value of the beta-coefficient,

– values near to one indicate a tendency to mirror the market, on average,

– values greater than one indicate a tendency to follow the market, but with a stronger variability, on average,

– values lower than one indicate a tendency to follow the market, but with a weaker variability, on average.

– values near to zero indicate independecy from the market,

– negative values indicate a tendency to move to the directon opposite to that of the market, on average.

In order to be able to apply the model, we need to

– estimate the unit-price µm − r of the market-related risk,

– estimate the amount of market-related systematic risk, i.e. the beta-coefficient βx of the investment.



Unit-price of market-related risk

The unit-price of market-related risk is typically estimated in terms of a long-term historical average excess return of

the stock market over the return of a risk-free bond portfolio.

The Procedure:

Every year, over a long period (say 30 years) of yearly historical data, calculate:

– the yearly logarithmic return, Rm , of the stock market,

– the yearly logarithmic return, Rf , of a portfolio of government-issued risk-free bonds,

– the yearly excess return, Re = Rm − Rf , of the stock market over the return of the risk-free bond portfolio.

The time-series average, R̄e , of the excess returns serves as an estimate of the expected price of the market risk in

terms of an annual percentage rate.

In the following numerical examples, let us assume the value of R̄e = 0.04000 = 4.00%.

Arithmetic return:

Rt =Pt − Pt−1

Pt−1

=11− 10

10= 0.10

Logarithmic return:

Rt = ln Pt − ln Pt−1 = ln 11− ln 10 = 0.0953

e0.0953 − 1 = 0.10ln(1 + 0.10) = 0.0953



Amount of market-related risk

When it comes to the estimation of the amount of systematic risk, in our case of an investment project, we do not

necessarily have any historical data available for the direct estimation of the beta-coefficient.

To get an estimate of the amount of the market-related systematic risk, we

– use an educated guess of the value of the beta-coefficient , or

– choose an appropriate benchmark-asset, for which we are able to estimate the beta by available empirical data.

Having empirical data available, the beta-coefficient is typically estimated by regressing the asset (excess) returns

against the market (excess) returns over a period of historical data:

Rxt = α̂x + β̂xRmt + εxt , β̂x =σ̂xm

σ̂2m

.

The Procedure:

Over a period of an appropriate length (say 5 years) of monthly historical data,

– calculate the monthly logarithmic (excess) returns, Ri , of the asset,

– calculate the monthly logarithmic (excess) returns, Rm , of the stock market,

– regress the asset (excess) returns against the market returns,

– use the estimated regression coefficient as the estimate of the systematic risk.

In our numerical example, let us assume the value of β̂x = 1.20.




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6

�x̃

p(x̃)

E(x̃)��

��

��

��x0

T = 2

t = 0

Two years from now, we expect to receive a cash-flow of e5000,

= 5000


kx = r + R̄eβx = 0.02565 +

The historical average market excess return is R̄e = 4.000%.

0.04000×

The estimated beta of the cash-flow is β̂x = 1.20.

1.20 =

The applicable discount rate is kx = 7.365%.

0.07365 = 7.365%

The present value of the cash-flow is e4315.

= e−kx T E(x̃) = e−0.07365×2 × 5000 ≈ 4315



Multiple cash-flows

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6

�T3 = 3

x̃3

p(x̃3)

E(x̃3) = 3300

r3 = ?

kx3= ?

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6

�T2 = 2

x̃2

p(x̃2)

E(x̃2) = 300

r2 = 2.565%

kx2= ?

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6

�T1 = 1

x̃1

p(x̃1)

E(x̃1) = 300

r1 = ?

kx1= ?

��

��

��

��

��


The estimated beta of the multiple cash-flow project is β̂x = 1.10.



Estimation of the one-year risk-free rate

The holder of a one-year government-issued bond receives back


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6

x̃1

p(x̃1)

x1 = 1000

T = 1

��

��


x0 = 975

t = 0

The value of the discount factor is df = 0.97500.

= df × 1000

⇒ df = 0.97500

The one-year risk-free rate is r1 = 2.532%.

e−r1×1 = 0.97500

⇒ r = − ln 0.97500 = 0.02532 = 2.532%



Estimation of the three-year risk-free rate

The holder of a three-year government-issued bond receives back


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6

x̃3

p(x̃3)

x3 = 1000

T = 3

��

��

��

��

��


x0 = 925

t = 0

The value of the discount factor is df = 0.92500.

= df × 1000

⇒ df = 0.92500

The three-year risk-free rate is r3 = 2.599%.

e−r3×3 = 0.92500

⇒ r = −1

3ln 0.92500 = 0.02599 = 2.599%



Multiple cash-flows



The one-year risk-free rate is r = 2.532%.


The three-year risk-free rate is r = 2.599%. -

6

�T3 = 3

x̃3

p(x̃3)

E(x̃3) = 3300

r3 = 2.599%

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6

�T2 = 2

x̃2

p(x̃2)

E(x̃2) = 300

r2 = 2.565%

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6

�T1 = 1

x̃1

p(x̃1)

E(x̃1) = 300

r1 = 2.532%

��

��

��

��

��

Market yield curve

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6

xx

x



Multiple cash-flows

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6

�T3 = 3

x̃3

p(x̃3)

E(x̃3) = 3300

r3 = 2.599%

kx3=

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6

�T2 = 2

x̃2

p(x̃2)

E(x̃2) = 300

r2 = 2.565%

kx2=

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6

�T1 = 1

x̃1

p(x̃1)

E(x̃1) = 300

r1 = 2.532%

kx1=

��

��

��

��

��



The one-year discount rate is r = 6.932%.

kx1= r1 + R̄eβx = 0.02532 + 0.04000× 1.10 = 0.06932 = 6.932%

6.932%

The two-year discount rate is r = 6.965%.

kx2= r2 + R̄eβx = 0.02565 + 0.04000× 1.10 = 0.06965 = 6.965%

6.965%

The three-year discount rate is r = 6.999%.

kx3= r3 + R̄eβx = 0.02599 + 0.04000× 1.10 = 0.06999 = 6.999%

6.999%



Multiple cash-flows

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6

�T3 = 3

x̃3

p(x̃3)

E(x̃3) = 3300

r3 = 2.599%

kx3=

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6

�T2 = 2

x̃2

p(x̃2)

E(x̃2) = 300

r2 = 2.565%

kx2=

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6

�T1 = 1

x̃1

p(x̃1)

E(x̃1) = 300

r1 = 2.532%

kx1=

��

��

��

��

��



6.932%

6.965%

6.999%

The present value of the cash-flows is e3216.

x0 = e−0.06932×1 × 300 + e−0.06965×2 × 300 + e−0.06999×3 × 3300 ≈ 3216



Infinite cash-flow

We may have a case of an infinite cash-flow.

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6

�T3 = 3

x̃3

p(x̃3)

E(x̃3)

-

6

�T2 = 2

x̃2

p(x̃2)

E(x̃2)

-

6

�T1 = 1

x̃1

p(x̃1)

E(x̃1)

��

��

��

��

��

��

��*

Continues ”forever”...

Here we assume that the cash-flow continues infinitely

with a constant growth rate of g = 5%.

= 110.25

= 105.00

= 100.00

= (1 + g)E(x̃2)

= (1 + g)E(x̃1)

= (1 + g)2E(x̃1)

= 1.05× 100.00

= 1.052 × 100.00

We are able to express the present value of the cash-flows in

terms of an infinite geometric series, which converges if g ≤ akx .

x0 =E(x̃1)

1 + akx+

(1 + g)E(x̃1)

(1 + akx )2+

(1 + g)2E(x̃1)

(1 + akx )3+

(1 + g)3E(x̃1)

(1 + akx )4+ . . . =

E(x̃1)

akx − g

Notice: The discount rate akx is here in terms of annual compounding!



Infinite cash-flow

An infinite cash-flow with a constant growth-rate of g = 5%.

-

6

�T3 = 3

x̃3

p(x̃3)

E(x̃3)

-

6

�T2 = 2

x̃2

p(x̃2)

E(x̃2)

-

6

�T1 = 1

x̃1

p(x̃1)

E(x̃1)

��

��

��

��

��

��

��*


= 110.25

= 105.00

= 100.00

= (1 + g)2E(x̃1)

= (1 + g)E(x̃1)

The long-term government-issued bond yield is r = 2.700%.

Market yield curve

-

62.700%



Infinite cash-flow

An infinite cash-flow with a constant growth-rate of g = 5%.

-

6

�T3 = 3

x̃3

p(x̃3)

E(x̃3)

-

6

�T2 = 2

x̃2

p(x̃2)

E(x̃2)

-

6

�T1 = 1

x̃1

p(x̃1)

E(x̃1)

��

��

��

��

��

��

��*


= 110.25

= 105.00

= 100.00

= (1 + g)2E(x̃1)

= (1 + g)E(x̃1)

x0 =E(x̃1)

1 + akx+

(1 + g)E(x̃1)

(1 + akx )2+

(1 + g)2E(x̃1)

(1 + akx )3+

(1 + g)3E(x̃1)

(1 + akx )4+ . . . =

E(x̃1)

akx − g



The estimated beta of the cash-flow is β̂x = 1.15.

kx = r + R̄eβx

= 0.02700 + 0.04000× 1.15 = 0.07300

akx = e0.07300 − 1 = 0.07573 = 7.573%

The applicable discount rate is akx = 7.573%.

=100.00

0.07573− 0.05

≈ 3886

The present value of the cash-flows is e3886.



Valuation of a future asset price

So far, we have implicitly assumed a cash-flow(s) to follow a normal distribution.

Alternatively, we may assume a variable to follow a lognormal distribution.

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6

0 x̃

p(x̃)

E(x̃)

We are still allowed to value the asset in terms of the discounted expected value.

��

��

��x0

T

t = 0

kx = r + R̄eβx

= e−kx T E(x̃)



Valuation of a future asset price

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6

0 x̃

p(x̃)

E(x̃)��

��

��

��x0

T = 2

t = 0

Two years from now, we expect the asset price to be e25.00

= 25.00


kx = r + R̄eβx = 0.02565 +


0.04000×

The estimated beta of the asset is β̂x = 0.90.

0.90 =

The applicable discount rate is kx = 6.165%.

0.06165 = 6.165%

The current asset price should be e22.10.

= e−kx T E(x̃) = e−0.06165×2 × 25.00 ≈ 22.10



Infinite dividend-stream

An infinite dividend-stream with a constant growth-rate of g = 4%.

-

6

T3 = 3

0 x̃3

p(x̃3)

E(x̃3)

-

6

T2 = 2

0 x̃2

p(x̃2)

E(x̃2)

-

6

T1 = 1

0 x̃1

p(x̃1)

E(x̃1)

��

��

��

��

��

��

��*


= 2.1632

= 2.08

= 2.00

= (1 + g)2E(x̃1)

= (1 + g)E(x̃1)

x0 =E(x̃1)

1 + akx+

(1 + g)E(x̃1)

(1 + akx )2+

(1 + g)2E(x̃1)

(1 + akx )3+

(1 + g)3E(x̃1)

(1 + akx )4+ . . . =

E(x̃1)

akx − g



The estimated beta of the stock is β̂x = 0.90.

kx = r + R̄eβx

= 0.02700 + 0.04000× 0.90 = 0.06300

akx = e0.06300 − 1 = 0.06503 = 6.503%

The applicable discount rate is akx = 6.503%.

=2.00

0.06503− 0.04

≈ 79.90

The present value of the dividends is e79.90.



Corporate bond with default risk

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6

T3 = 3

x̃3

p(x̃3)

x3 = 1040

-

6

T2 = 2

x̃2

p(x̃2)

x2 = 40

-

6

T1 = 1

x̃1

p(x̃1)

x1 = 40

��

��

��

��

��

A 1000-euro, 3-year corporate bond, with an annual 4% coupon.

The risk-profile of the bond corresponds to that of the BB-rated bonds.

Default risk

Default risk

Default risk

1-year BB-bonds trade at the yield of 4.157%.



y3 = 4.234%

y2 = 4.185%

y1 = 4.157%

BB-rated bond yield curve

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6

xx

x

The value of the bond is e991.

x0 = e−0.04157×1 × 40 + e−0.04185×2 × 40

+ e−0.04234×3 × 1040 ≈ 991



Valuation of an option contract*

In derivatives pricing, we assume the risk-neutral price x̃ of an underlying asset to follow a lognormal distribution.

-

6

0 x̃

p(x̃)

A European call option on the asset pays off an amount of max(0, x̃ − X ) at the maturity T of the contract.

X is a contracted strike price.

We draw the payoff max(0, x̃ − X ) in the same

coordinate system, but with a different vertical

axis x̃ − X .

X

��

6̃x − X

p(x̃) max(0, x̃ − X )





-

6

0 x̃

p(x̃)



X

��

6̃x − X

p(x̃) max(0, x̃ − X )

We are interested in the probability-weighted payoff p(x̃)max(0, x̃ − X ) of the contract.

We draw the probability-weighted payoff

p(x̃)max(0, x̃ − X ) in the same coordinate

system, but with a third vertical axis.

-

6p(x̃)max(0, x̃ − X )

p(x̃)max(0, x̃ − X )







-

6

0 x̃

p(x̃)max(0, x̃ − X )

X

The expected risk-neutral call option payoff equals

to the sum of all probability-weighted option payoffs:

E [max(0, x̃ − X )] =

∫ ∞X

p(x̃)max(0, x̃ − X )dx̃

The risk-neutral expected payoff E [max(0, x̃ − X )]

is discounted by the risk-free rate r to

get the value c0 of the option.

��

��

��

T

t = 0

c0 = e−rT E [max(0, x̃ − X )] = e−rT∫ ∞X

p(x̃)max(0, x̃ − X )dx̃


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