money and banking

Money and Banking I

Fall 2013

(BEE3043 & BEE2028)

Yoske Igarashi Office: Streatham Court 0.51 Office hours: Tue 14:00-15:00

Wed 11:00-12:00 Thu 11:00-12:00

1

Greenbaum & Thakor, Contemporary Financial Intermediation (2nd Edition) I looked at 15 textbooks and G&T is by far the most suitable for this module

(microeconomics of money and banking, not macroeconomics). Still, it is far from perfect.

Because of this reason, class attendance is crucial. Reading through the textbook will not replace missed classes!!

About the textbook

Topics covered by the textbook

Topics covered by the module.

2

Announcement If you find any errors, typos, grammar or spelling mistakes, etc., please let me know. The lectures are recorded for your convenience. (We do not take responsibility for any

failure in recording due to system errors, so please try to attend the class as much as possible.)

There will be an assignment (a collection of small questions) which counts 20% of the final mark. The questions will be different between BEE2028 students and BEE3043 students. The details will be posted later.

The final exam will be a closed-book/note exam. Only calculators will be permitted.

3

Math foundations

Summation notation () http://www.youtube.com/watch?v=8i9-9zHbW6g

Matrix addition and transpose http://www.youtube.com/watch?v=U74zP3BsAbM

Matrix multiplication http://www.youtube.com/watch?v=kuixY2bCc_0

Expectation and variance of a random variable http://www.youtube.com/watch?v=OvTEhNL96v0

Covariance and correlation http://www.youtube.com/watch?v=35NWFr53cgA

Graphing a linear function http://www.youtube.com/watch?v=52RiUx5B-ms

Graphing a linear inequality http://www.youtube.com/watch?v=5h6YzRRxzO4

Differentiation (Derivatives) http://www.youtube.com/watch?v=54KiyZy145Y

Solving a system of linear equations by Excel http://www.youtube.com/watch?v=-EnecLe_0B4 4

Topics (plan)

Preliminaries 1: Risk diversification Revision: Expectation, Variance, Covariance, etc. Portfolio choice Insurance

Major risks faced by banks Credit risk Interest risk Collateral risk Liquidity risk

Banking and asymmetric Information Belief update Hidden type

Adverse selection Adverse selection in banking and use of collateral General lesson: How to separate types?

Hidden action Moral hazard in banking and use of collateral Moral hazard in banking and use of capital

Off-balance-sheet banking Options No-arbitrage pricing Loan commitment

Loan sales and Securitization

The Deposit Contract and Bank Regulation The origin of banking (Anecdote) Money creation Bank as a liquidity provider Fixed-income claimant vs. Residual claimant Reserve requirement and capital requirement (Basel II)

5

The plan for 1st week (23/06/13)

Risk management (Portfolio managers) Risk management (Insurance companies)

Revisions/preliminaries Return on stocks/shares and portfolios Short-selling Expectation, variance, covariance, standard deviation, and correlation Some matrix algebra

Expectation and variance of w1X1 + .. + wnXn

Short break Portfolio optimization Excel demonstration for portfolio optimization Insurance and law of large number

6

(Rate of) Return For a stock (share),

Return on portfolio Suppose you have 100. You invest 50% in Stock A, 25% in Stock B, and 25% in Stock C. What is the return on your portfolio (RP)?

Suppose the returns on A, B and C turn out 5%, 20%, and -10%, respectively.

1111 +

= +++t

ttt P

DPR

date-t price

date-(t+1) price

dividend per share

net

CBAP RRRR 25.025.05.0 ++=

Stock A Stock B Stock C Total

Invested amount 50 25 25 100

Realized return 5% 20% -10% 5%

Final value 52.5 30 22.5 105

05.0)10.0(25.0)20.0(25.0)05.0(5.0 =++=PR

Revision: Return

7

Short-selling

A principle of capital gain

Expectation of price increase Buy it, and sell it after the price increases. Expectation of price decrease Borrow it and sell it. After the price decreases, buy and return

it. (= short-selling) Investors can sell a thing even if they dont have it. (In practice,

there are regulations and limitations for short-selling.)

Revision: Short-selling

8

Expectation, Variance, Covariance, Standard deviation, and correlation

Prob. X Y

State 1: Hot

0.5 9K 8K

State 2: Mild

0.3 5K 6K

State 3: Cold

0.2 3K 4K

Weighted Average _ 6.6

K 6.6 K

Variance and covariance

E[X] = 0.5 x 9 + 0.3 x 5 + 0.2 x 3 = 6.6 K

E[Y] = 0.5 x 8 + 0.3 x 6 + 0.2 x 4 = 6.6 K

X Y

11

means


Prob. X Y X X (X X)2

State 1: Hot

0.5 9K 8K 2.4 K 5.76 (K)2

State 2: Mild

0.3 5K 6K -1.6 K 2.56 (K)2

State 3: Cold

0.2 3K 4K -3.6 K 12.96 (K)2


K 6.6 K

0 6.24 (K)2


Var[X] = 0.5 x 5.76 + 0.3 x 2.56 + 0.2 x 12.96 = 6.24 (K)2

X Y

14

means

deviation



Y - Y (Y Y)2

State 1: Hot

0.5 9K 8K 2.4 K 5.76 (K)2 1.4 K 1.96 (K)2

State 2: Mild

0.3 5K 6K -1.6 K 2.56 (K)2 -0.6 K 0.36 (K)2

State 3: Cold

0.2 3K 4K -3.6 K 12.96 (K)2 -2.6 K 6.76 (K)2


K 6.6 K

0 6.24 (K)2 0 2.44 (K)2


Var[Y] = 0.5 x 1.96 + 0.3 x 0.36+ 0.2 x 6.76 = 2.44 (K)2

X Y

15

means

variance

deviation



Y - Y (Y Y)2

State 1: Hot

0.5 9K 8K 2.4 K 5.76 (K)2 1.4 K 1.96 (K)2

State 2: Mild

0.3 5K 6K -1.6 K 2.56 (K)2 -0.6 K 0.36 (K)2

State 3: Cold

0.2 3K 4K -3.6 K 12.96 (K)2 -2.6 K 6.76 (K)2


K 6.6 K

0 6.24 (K)2 0 2.44 (K)2


X Y X

50.224.6K=

= Y

56.144.2K=

=

16

Standard deviation

means

variance

deviation



Y - Y (Y Y)2

(X - X)(Y - Y)

State 1: Hot

0.5 9K 8K 2.4 K 5.76 (K)2 1.4 K 1.96 (K)2 3.36 (K)2

State 2: Mild

0.3 5K 6K -1.6 K 2.56 (K)2 -0.6 K 0.36 (K)2 0.96 (K)2

State 3: Cold

0.2 3K 4K -3.6 K 12.96 (K)2 -2.6 K 6.76 (K)2 9.36 (K)2


K 6.6 K

0 6.24 (K)2 0 2.44 (K)2 3.84 (K)2


Cov[X,Y] = 0.5 x 3.36 + 0.3 x 0.96 + 0.2 x 9.36 = 3.84 (K)2

X Y X

50.224.6K=

= Y

56.144.2K=

=

X,Y = 984.056.150.2)(84.3),( 2

=

=KK

KYXCovYX

Correlation Coefficient

18

Standard deviation

means

variance

covariance

deviation

Correlation Coefficient X,Y

0 1 -1

No correlation

Perfect correlation

Perfect negative correlation


More intuitive indicator of co-movement or links than covariance.

Most studies report standard deviations and correlation coefficients, but not variances or covariances. If you need variances and covariances, you need to recover them from st. deviations and correlation coefficients.

19 Std. deviations Correlation coef.

variances covariance

equivalent


Y - Y (Y Y)2

(X - X)(Y - Y)

State 1: Hot

0.5 8K 2K

State 2: Mild

0.3 7K 6K

State 3: Cold

0.2 4K 9K

Weighted Average _

Exercise2. What is the expectation and standard deviation of the number you get from rolling a die?


X Y X = Y = X,Y =

Exercise1. Complete the following table.

Tutorial questions

20

Means and variances of many random variables

Suppose there are 5 stocks. (Stock A, B, C, D and E.) The expected (monthly) returns are expressed as a vector.

How about the variability or risk? Is it a vector of variances?

No. You need a 5 x 5 matrix to express variability. (Variance-covariance matrix)

Aviva British Gas Capita Diageo Easyjet

1.10% 0.90% 1.05% 1.15% 1.20%


290 %2 131 %2 58 %2 24 %2 113 %2

Aviva BG Capita Diageo Easyjet

Aviva 290 %2 96 %2 34 %2 31 %2 53 %2 BG 96 %2 131 %2 46 %2 27 %2 45 %2 Capita 34 %2 46 %2 58 %2 15 %2 24 %2 Diageo 31 %2 27 %2 15 %2 24 %2 22 %2 Easyjet 53 %2 45 %2 24 %2 22 %2 113 %2 21

Means and variances of many random variables

Equivalently, you can have the vector of standard deviations and the correlation matrix.

Std. deviations

Correlation matrix


17% 11% 8% 5% 11%

Aviva BG Capita Diageo Easyjet

Aviva 1.000 0.495 0.263 0.368 0.294 BG 0.495 1.000 0.529 0.482 0.368 Capita 0.263 0.529 1.000 0.392 0.301 Diageo 0.368 0.482 0.392 1.000 0.417 Easyjet 0.294 0.368 0.301 0.417 1.000

22

A linear combination of X1, X2, . , Xn EX) 3X1+0.5X2-10X3+8X4 EX) E[3X1+0.5X2-10X3+8X4] = 3 E[X1] + 0.5 E[X2] -10 E[X3] + 8 E[X4] If we denote the expectation vector by , and the coefficients by then the expectation of a linear combination is simply

Expectation of linear combination Variance and covariance

][][][][ 22112211 nnnn XEwXEwXEwXwXwXwE +++=+++

23

),,,( 21 n ,21

nw

ww

w

.w

.2211 nnwwww +++=


Why is this formula useful? Suppose you already know the expected returns of stocks A, B, C, D and E. Suppose you construct your own portfolio composed of these stocks. EX) 3,000 to w = (40%, 20%, 20%, 15%, 5%). Then, you can compute the expected return on your portfolio, using the formula!! Portfolio return = w = 0.40(1.10%)+0.20(0.90%)+0.20(1.05%)+0.15(1.15%)+0.05(1.20%) = 1.0625% How about the variance of the portfolio return? Can we calculate it if we know variances of individual stock returns?

24

][][][][ 22112211 nnnn XEwXEwXEwXwXwXwE +++=+++


1.10% 0.90% 1.05% 1.15% 1.20%

Expectation of linear combination

=

][),(2][][ YVarYXCovXVarYXVar ++=+

][),(2][][ 22 YVarbYXabCovXVarabYaXVar ++=+

222 2)( yxyxyx ++=+

22222 2)( ybabxyxabyax ++=+

][][ 2 XVaraaXVar =

Variance formulas Variance and covariance

MNEMONICS

EX) If Var[X]=15, Var[Y]=15 and Cov(X,Y)= 10, then Var[X+Y]=10.

MNEMONICS

EX) Assume the same X and Y as above and let a = b = 0.5. Then, Var[0.5X+0.5Y] = (0.5)2x15+2x0.5x0.5x(-10)+ (0.5)2x15 = 2.5.

25

Variance formulas Exercise3. (Ref. G&T p16-18) X and Y are stock returns of companies A and B so they are random. It is known that means of X and Y are 10% and 20% respectively, and standard deviations are 20% and 40%, respectively. Moreover, the correlation coefficient between X and Y is 0.4. If one mixes X and Y by fifty-fifty, what is the standard deviation of his portfolio?

Exercise4. Guess the formula for . (Express it in terms of variances and covariances of )

][ 332211 XwXwXwVar ++.,, 321 XXX


26

.),(2][][,...,1

211

Variance formula in the matrix form If the variance-covariance matrix of is given by an n by n

(symmetric) matrix V, then the variance of a linear combination of Xs is given by:

where .

nXXX ,, 21


Vwww

wVwwXwXwVar

n

nnn

=

=++

1

111 )(][

1xn nxn nx1

1x1

nw

ww

w

2

1

29

Matrix addition and transpose http://www.youtube.com/watch?v=U74zP3BsAbM Matrix multiplication http://www.youtube.com/watch?v=kuixY2bCc_0

Variance formula in the matrix form EX) Suppose that the variance-covariance

matrix of is given by

Then

321 ,, XXX


84.381.203.07.425.03.572.01.207.423.57

)3.05.02.0(

3.0245.0152.0273.0155.0582.0463.0275.0462.0131

)3.05.02.0(

3.05.02.0

2415271558462746131

)3.05.02.0(

]3.05.02.0[ 321

=++=

=

++++++

=

=

++ XXXVar

30

.2415271558462746131

=V

Exercise 5. There are three random variables X1, X2, and X3.

Their means are given by = (3.5, 4.5, 7.5).

Their standard deviations are given by = (2, 4, 8)

The correlation matrix is given by .

(1) Find the variance-covariance matrix.

(2) What is the expectation and variance of 0.2X1 + 0.5X2 + 0.3X3? What is the standard deviation?

139.048.039.0153.048.053.01

31

Portfolio choice problem Suppose there are n assets available.

The return ( ) of each asset is random.

Suppose that by experience one knows the expectations and variance-covariance matrix of the returns of the n assets.

One can choose her own portfolio:

subject to

111

+

++

t

ttt

PDPR

),,,( 21 n

nnn

n

V

1

111

),,,( 21 nwwww =

=n

iiw

11


33

1. Set the minimal acceptable expected return on the portfolio and minimise its variance:

2. Set the maximal acceptable variance on the portfolio return and maximise its expected return:

3. Maximise the combination of the portfolio return and variance:

Portfolio choice problem

Vwww

min%10w

=

=n

iiw

11

ww

max

s.t.

s.t. 2%)30(Vww

=

=n

iiw

11

Vwwww

21max s.t.

=

=n

iiw

11

A number you can choose.

A number you can choose.

34

Portfolio optimisation by MS Excel

Matrix algebra in MS Excel Naming a range of cells

Select a range of cells and directly type in the name bar. To unname the range, go to Formula Name manager

Matrix operation functions = TRANSPOSE(range name) = MMULT(range name 1, range name 2) = MINVERSE(range name)

To use matrix operation functions, first type a function in a single cell and press enter. Then select the whole range for the output, click on the function bar, and press Ctr+Shft+Entr.

Solver with Excel http://www.youtube.com/watch?v=uQU1KfsE5us

35

Adding Solver to Excel

First, you have to add Solver to Excel. From the Excel main menu, go to

Excel Options Add-Ins Solver Add-Ins and click Go.

In the list that appears, check Solver Add-In and click OK. Now under the Data tab, you can use Solver. The demonstration is provided in the class.

36

Risk diversification by insurance companies

If are independently and identically distributed (IID) with a common mean and std. , then the average has the same mean and, its std. is .

Proof) In our variance formula, all the covariance terms are zero due

to independence. Therefore,

EX) The average of 100 dice. (For one die, E[X1]=3.5 and =1.71)

nXX ,,1

n

.2

1

nnXXVar n =

++

37

Proposition

nXXX n++ 1

SMALL!!

Insurance What is risk of an asset or a project?

1. variance or standard deviation of its return

2. how its return is correlated with your own income

3. the probability of very bad events

People are willing to buy an insurance even if the premium is more expensive than the expected payoff of the insurance plan.

For insurance to work well, outcomes of people should be IID from one another. Works well for car accident, illness, fire, etc. Why is it that for earthquake insurances, the premium is higher and insurance

payment is lower than for normal fire insurances?

Why is it that house insurances typically exclude damages caused by earthquakes, hurricanes, floods, wars, terrorisms, etc?

38

Two types of risk

Idiosyncratic risk/shock Risk that people face which is independent of one another. Examples are non-epidemic disease, regular fire, car accident,

unemployment for personal reasons, etc. Such risks can be diversified out so insurance works very well.

Aggregate risk/shock

The common risk that affect many people altogether. Examples are natural disasters, wars, pandemic disease,

economic recessions, financial crises, etc. Such risks are hard to diversify out and insurance does not work

well.

Firms and hence their stock returns suffer both idiosyncratic and aggregate risks.

39

Money and Banking IFall 2013 (BEE3043 & BEE2028)About the textbookAnnouncementMath foundationsTopics (plan)The plan for 1st week (23/06/13)(Rate of) ReturnShort-sellingExpectation, Variance, Covariance, Standard deviation, and correlationExpectation, Variance, Covariance, Standard deviation, and correlationExpectation, Variance, Covariance, Standard deviation, and correlationExpectation, Variance, Covariance, Standard deviation, and correlationExpectation, Variance, Covariance, Standard deviation, and correlationExpectation, Variance, Covariance, Standard deviation, and correlationExpectation, Variance, Covariance, Standard deviation, and correlationExpectation, Variance, Covariance, Standard deviation, and correlationExpectation, Variance, Covariance, Standard deviation, and correlationExpectation, Variance, Covariance, Standard deviation, and correlationCorrelation Coefficient X,YSlide Number 20Means and variances of many random variablesMeans and variances of many random variablesExpectation of linear combinationExpectation of linear combinationVariance formulasVariance formulasVariance formulasVariance formulasVariance formula in the matrix formVariance formula in the matrix formSlide Number 31Slide Number 32Portfolio choice problemPortfolio choice problemPortfolio optimisation by MS ExcelAdding Solver to ExcelRisk diversification by insurance companiesInsuranceTwo types of risk

money and banking

Documents

excel http

correlation http

transpose http

matrix multiplication

linear inequality http

linear function http

random variable http

banks credit risk