1

The Calculus of Finance

Portfolio Diversification, Life Time Savings, and Bankruptcy

Bill Scottscottw@saic.com

858-826-6586SAIC

May 2003, caveat added May 2005, page 1 notes

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Calculus of Finance

Goal -- Understand several Nobel prize winning finance theories using a scientific understanding of basic calculus to derive the fundamental finance equations with less effort.

Focus on portfolio selection as an issue of most importance to scientists and engineers.

Later talks can discuss lifetime savings and bankruptcy.

3

Mean-Variance Utility

U =α P −12γσ P

2Adjust your portfolio to maximize the utility

where expected portfolio growth αP is desired, but portfolio volatility σP

2is to be avoided.

γ is your personal risk aversion. It represents your risk-return tradeoff, how much risk you are willing to bear in order to hope for higher returns.

eq 1

4

Growth is a Weighted Average

The expected portfolio growth is the weighted-average expected growth over all assets

αP = wii

∑ α i eq 2

where wi is the fraction of wealth held in the ith investment.This assumes frequent rebalance instead of buy-and-hold.

5

Uncertainty Adds like Vectors

r1

r2

r3

w1σ1

w2σ2

w3σ3

σP

σP2 = w1

2σ 12 + w2

2σ 22 + w3

2σ 32

Assuming no covariance.

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Portfolio Volatility

σP2 = wi

2

i∑ σ i

2

Volatility adds over a portfolio like vectors

when there is no covariance.

σP2 = wi

2

i∑ σ i

2 + wij ≠i∑

i∑ σ ijw j

If the performance of i and j are not independent, then

where σij is the covariance between i and j.

eq 3

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Volatility and CovariancePast volatility and past covariance are thought to be the best measures of future volatility and covariance; however, past growth is not thought to be a good measure of future growth.

rt =1Δt

(ln St /St−1)σ 2 =

Δt

(n −1)(rt − r )2

n∑

σ ij =Δt

(n − 1)(rit

n∑ − r i)(rjt − r j)

Δt is in years so that the rates and variances are annualized.

8

Tobin Two Asset PortfolioAssume a risk free investment that returns f, and a riskyasset that you expect to return α with volatility σ2. What fraction of wealth w should be invested in the risky asset? Maximize the mean-variance utility by settings its derivative with respect to w to zero.

αP = (1 − w) f + wα σP2 = w2σ 2

∂U

∂w≡ 0 = α − f −γwσ 2 w* =

α − fγσ 2 eq 4

U =(1−w) f +wα −12 γw

2σ 2

Where w* represents an optimized portfolio holding.

9

Importance of Tobin Equation

w* =α − fγσ 2

•Holdings are proportional to expected excess return.

•Bigger holdings if optimistic or if interest falls

•Inversely proportional to volatility and risk aversion

•Own less to sleep better

•Reduce holdings when volatility increases

•Additional risk generally causes selling.

10

Three Asset DiversificationAdd the general market (S&P 500) with αm and σm

2.

U =(1−wm −w1) f +wmαm +w1α1

−12γ(wm

2σ m2 +w1

2σ12 + 2wmw1σ1m)

∂U

∂wm

≡ 0⇒ γ(wmσ m2 + w1σ 1m) = α m − f

γ(wmσ 1m + w1σ 12 ) = α 1 − f

These are two equations with two unknowns. Solve with algebra.

eq 5

eq 6∂U

∂w1

≡ 0 ⇒

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Three Asset Results

wm =σ1

2 αm− f( )−σ1m α1 − f( )γ σ m

2σ12 −σ1m

2( )

Solution to simultaneous equations 5 and 6

eq 7

eq 8

Notice that this is like determinant theory which points to a matrix notation.

w1 =σm

2 α1 − f( ) −σ1m αm − f( )γ σ m

2σ12 −σ1m

2( )

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Markowitz Matrix Notation

σ =

σ12 σ 12 σ 1n

σ 12 σ 22 σ 2n

σ 1n σ 2n σ n2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

w =

w1

w2

wn

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

α − f =

α 1 − f

α 2 − f

α n − f

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

γσw = α − f or w * =1γ

σ−1

α − f( ) eq 9

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Markowitz Efficient Portfolio

Equation 9 is the Markowitz efficient portfolio. Notice that risk aversion γ does not effect your allocation between stocks. Your γ is determined by how much you desire risk free. The variances and covariances can be measured from historical data, and are thought to be fairly constant into the immediate future. Thus your portfolio allocation depends mostly on α, your own expectations of future growth. Actually Markowitz did a lot more work to constrain his portfolios to no short selling and no borrowing.

w * =1γ

σ−1

α − f( ) Eq. 9

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Markowitz Admits Large Holdings

Pure Markowitz Stock and Market Holding

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

alpha expected of risky stock

fraction of wealth held

risky stock

market

€

w1 wm( ).03 .01

.01 .04

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ =

13

α1−.03 .07−.03( ) w1 represents volatilities and covariances like SAIC

If your expectations of SAIC exceed the market, then strong holdings are recommended by the Markowitz theory.

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CAPM Capital Asset Pricing Model

What α’s does the market assume? In eq 7, the market must see α1 to be large enough so that people just want to start buying it. All current stock prices must be set so that the numerator of eq 7 is positive but close to zero.

0 =σ m

2 α1 − f( )−σ1m αm − f( )γ σ m

2σ12 −σ1m

2( )σm

2 α 1 − f( ) = σ 1m α m − f( )

define beta

β1 ≡σ 1m

σ m2 α1 = f + β1 α m − f( )

CAPM - when you can’t guess α1, assume

eq 12

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Observables

“I think of expectations as observable, at least in principle.”

Fischer Black, 1995

Instead

“Honest agents examining the same firm will see a distribution of expectations perhaps centered on CAPM, but with a width of estimates consistent with σ.”

Bill Scott, 1998

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Arbitrage Pricing Theory APTUsually this is derived from the assumption that an efficient market allows no risk free profits from self-funded hedges; i.e. there are no arbitrage opportunities.

α1 − f

β1

=α 2 − f

β2

=α n − f

βn

Instead, this equation also derives directly from CAPM, in that the current buyers and sellers set the current price. This implies that there is one price regardless of risk aversion or optimism.

eq 13

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Modigliani and Miller Theorem

•Corporate finance does not affect value to shareholders.

•Corporations can raise capital several ways

•Cutting dividends

•Selling more stock

•Borrowing by selling bonds

•Paying employee bonuses with options rather than cash

•These each have different risk-return effects

•Such risk-return changes leave share value unchanged - APT

•All change both the risk and the expected return such that the APT ratio does not change -- thus the stock price is unaffected by finance.

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Rational Investing•Invest some risk free for your own personal comfort, your γ

•Predict technology and future fads.

•Search for firms whose α will exceed CAPM

•Your α estimates may well be better than CAPM!

•Allocate an optimized portfolio using eq 9, Markowitz.

•Rebalance the portfolio frequently.

•Sell the high growth stocks in order to buy more laggers?

•Reallocate only when you’ve revised your α estimates.

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Portfolio Diversification

• If you can’t predict α and use CAPM.– a very small investment in everything

• However ifαi > α m and βi < 1 can cause wi ⇒ 10%−40%

If you think you know what you’re doing, portfolio diversification theory admits to very aggressive holdings.

Search for high growth, low volatility investments.

21

Utility and Probability Theory

•Simple Portfolio Analysis•Utility and Probability•Bankruptcy•Lifetime Savings and Consumption

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Pratt Constant Relative UtilityU(WT ) −C =

WT1−γ

1−γdU

dWT

=1

WTγ so that

utility vs wealth and risk aversion

-2

-1

0

1

2

0 0.5 1 1.5 2 2.5 3

future wealth

utility0

1

2

4

risk aversion gamma

Utility, more wealth is always better, but less is a lot worse. Most employees are γ = 2 to 4. Plot of U vs WT with C set so that the utility of present wealth is always 0.

Time Separable and State Independent

eq 14

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Log Normal Probabilities

Normal distribution ofrates of return vs optimism and volatilityσ

Same log normal distributionsof stock prices at 5 years

Normal Probabilities of rate of return

0

1

2

3

4

5

-1 -0.5 0 0.5 1

annualized continuous compounded rate of return

probability density

(0.2,0)

(0.2,0.1)

(0.5,-0.06)

(0.5,0.12)

(σ,μ)

5 Year Log Normal Stock Price

0

1

2

3

4

5

0 2 4 6 8 10

5 year stock price

probability density

(0.2,0)

(0.2,0.1)

(0.5,-0.06)

(0.5,0.12)

(σ,μ)

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Log Normal Probability Math

Ε(WT ) ≡ W0eα pT

= W0

T

2πσ p2 drp∫ exp −

rp − μ p( )2T

2σ p2

⎛

⎝ ⎜

⎞

⎠ ⎟e

rp T

Expectation of wealth grows probabilistically with expected return.

Normal probability of possible returns integrates to give

=W0 exp (μ p + σ p2 / 2)T[ ]

so that α p = μ p +σ p

2

2Expected return is the most likely return plus half the volatility!

eq 15

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Log Normal Utility Math

Ε WT1−γ

1− γ

⎛

⎝ ⎜ ⎞

⎠=

W01−γ

1− γT

2πσ p2

drp∫ exp −rp − μ p( )

2T

2σ p2

+ (1 − γ )rpT ⎛

⎝ ⎜

⎞

⎠ ⎟

=W0

1− γ

1− γexp (1− γ )(α p −

12

γσ p2 )T

⎛ ⎝

⎞ ⎠

The Pratt utility of wealth can also be integrated under the log normal probability distribution.

Maximizing the Pratt utility under lognormal expectations is the same as maximizing the mean-variance utility.

eq 16

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Pratt and simple are the same

• Pratt and simple mean-variance utilities maximize the same function, thus– Pratt and mean variance γ are the same

• avoiding volatility and losing wealth hurts more than gaining wealth.

• Both are independent of the time.

– Rebalance portfolios over time to maintain w* until you recalculate your α’s and β’s.

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Derivative Pricing

• Simple Portfolio Analysis

• Utility and Probability

• Bankruptcy

• Lifetime Savings and Consumption

28

Considering BankruptcyLet b be the annual probability of bankruptcy of the wb asset. Let T be either a fixed period or the bankruptcy time.

€

Ε U WT( )( ) =1−bT+bT 1−wb( )

1−γ

1−γT

2πσ p2 drpe

−rp−ˆ r p( )

2T

2σ p2

e1−γ( )rpT∫

€

WT =erpT

No Bankruptcy

€

1−bT

€

WT = 1−wb( )erpT

Bankruptcy

€

bTprobability

outcome

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Markowitz with Bankruptcy

€

r w * =

t σ ( )

−1

γr α −f −

δibb

1−wb( )γ

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

r w * ≈

1γ

t σ +

t I r b ( )

−1 r α − f −

r b ( )

In limit of low wb*, b acts as risk

Which is non linear in wb and saturates

€

∂∂wb

αp −γσp2 2( ) ≈

δibbT

1−wb( )γ

Performing the integral and setting derivative to zero

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Bankruptcy Limits Holdings to 50%

€

wb wm( ).03 .01

.01 .04

⎛

⎝ ⎜ ⎞

⎠ ⎟ =

13

αb −.03 .07−.03( )

Bankrupty effect on fraction of risky held vs alpha

0

0.2

0.4

0.6

0.8

1

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

alpha

w

0

0.003

0.01

0.02

0.03

0.06

b

w1 represents volatilities and covariances like SAIC

If your expectations of SAIC exceed the market, then strong holdings admit.

With any bankruptcy risk, holdings saturate at 50%

31

Lifetime Savings and Consumption

• Simple Portfolio Analysis

• Utility and Probability

• Bankruptcy

• Lifetime Savings and Consumption

32

Retirement Planning• How much can be spent per year during

retirement? How much needs to be saved while working? Merton 1971

• Annuity equation - wealth P– Can spend R each year– Assumes certainty r– Spends last dollar at D

• mortgage equation

R =rP

1−e−rD eq 22

$10,000

$100,000

$1,000,000

$10,000,000

0 5 10 15 20 25 30

C

P

r = 7.5%

33

Inflation Adjusted Retirement

Variable annuity equation to keep income adjusted for inflation

R(t) =(r −i)P(t)

1−e−(r−i )( D−t)

Live on the amount that interest exceeds inflation, the rest then keeps up with inflation. Thus each year’s spending can grow with inflation.

$10,000

$100,000

$1,000,000

$10,000,000

0 5 10 15 20 25 30

C inf

P nf

r = 7.5%, i = 3%

34

Retirement under Uncertainty

Assume that your wealth is efficiently invested by the Markowitz equation which is frequently rebalanced so that expected return is αP and expected volatility is σP

2. Utility of inflation adjusted spent dollars is maximized by the variable annuity equation (Merton)

R(t) =aP(t)

1−e−a(D−t)

Notice, once the portfolio is maximized for utility, certainty returns are replaced by half the expected excess returns. Volatility isn’t in the equation.

eq 23a =(α P + f )

2−i

35

Uncertain Retirement

R(t) =aP(t)

1−e−a(D−t)

Lognormal random walk of portfolio returns. Variable annuity equation generates steady retirement income likely to adjust to inflation and unlikely to deplete early. Smoothing in early years.

a =(α P + f )

2−i

$10,000

$100,000

$1,000,000

$10,000,000

0 5 10 15 20 25 30

P

Ca = 3.75%, rr = 8.02%

36

Continuous Standard of Living

Save C fraction of your salary while working. Salary assumed to grow with inflation. Retire in T years and spend R for D-T more years. Wealth P to finance retirement is in units of today’s annual salary.

C =e−aT −e−aD −aP

1−e−aD

After retirement spend R R =aP

1−e−a(D−t)

Inflation adjusted retirement equals inflation adjusted salary less savings.

eq 24a =

(α P + f )2

−i