calc $ stat questions

Name

Points Possible Points

Question 1 4

Question 2 4

Question 3 4

Question 4 4

Question 5 4

Question 6 4

Question 7 4

Question 8 4

Question 9 4

Question 10 4

Question 11 4

Question 12 4

Question 13 4

Question 14 4

Question 15 4

Question 16 4

Question 17 4

Question 18 4

Question 19 4

Question 20 4

Question 21 4

Question 22 4

Question 23 4

Question 24 4

Question 25 4

TOTAL 100

1) State the assumptions of traditional financial economics.

Consider the following game: to start the game the player pays $1 to the house. After this

payment, the house rolls 2 dice. If the average of the 2 dice is even, the house pays the

player $4 otherwise the house pays the player $0. Is this game fair?

Consider the wealth utility functions associated with 3 investors who each have $2:

Investor 1: 𝑢1(𝑤) = 3.6946 ∙ 𝑤

Investor 2: 𝑢2(𝑤) = 𝑒 𝑤

Investor 3: 𝑢3(𝑤) = 4.8750 ∙ 𝑤 0.6

Assuming the investors behave in accordance with the assumptions of traditional financial

economics, which, if any, of the investors would definitely play the game? Explain.

2) Complete the following table for a 2 year, semi-annually paying, fixed rate, coupon bond.

Calculate the yield to maturity for the bond above if its 8/16/2016 market price is 101.00.

What is larger the coupon or the yield to maturity? Why?

What is bigger a) the amount needed to purchase this bond on 8/16/2016, or b) the amount

the bond issuer pays back in aggregate over the life of the bond (with no credit risk)? Why?

3) Assume Elizabeth deposits $18,000 into an FDIC insured bank account tomorrow and then

annually thereafter. What will the balance in the account be immediately following the 10th

deposit if she earns a guaranteed rate of 2.00% compounded annually (on each anniversary

of her first deposit)?

Immediately after the 10th deposit, how much total interest will she have earned?

Fixed Coupon 0.75%

Date 8/16/2016 2/16/2017 8/16/2017 2/16/2018 8/16/2018

Balance $4,000,000

Accrual Time N/A

Interest Owed N/A

Payment N/A

Principal N/A

4) For each of the sequences below, state whether the sequence has a finite limit and, for any

of the below sequences that do have a finite limit, state the limiting value of the sequence.

𝑏𝑡+1 = 24 + 0.80 ∙ 𝑏𝑡 & 𝑏0 = 1

𝑏𝑡+1 = 1.25 ∙ 𝑏𝑡 − 12 & 𝑏0 = 40

𝑏𝑡+1 = 𝑏𝑡 2

+ 39

2 ∙ 𝑏𝑡 & 𝑏0 = 20

5) Find the level payment that exactly amortizes the given balance on 11/16/2016 assuming

that interest accrues at the stated fixed coupon according to the “30/360” day count

convention. Fill in the table rounding to the nearest cent at each step.

What is the total interest paid over the life of the loan above?

Fixed Coupon 3.00%

Date 8/16/2016 9/16/2016 10/16/2016 11/16/2016

Balance $900.00

Accrual Time N/A

Interest Owed N/A

Payment N/A

Principal N/A

6) Assume Ted wants to buy a $400,000 property and has decided to use a 30 year level-

payment (“30/360” day count) mortgage to finance part of the purchase. He goes to a bank

and the bank offers him 2 options: he can put down 20% and get a rate of 3.50% OR he can

put down 10% and get a rate of 4.50%. What is the difference in the monthly payment

between the 2 options?

What is the difference in the lifetime total interest paid between the 2 options?

7) Referring to Question 6) above, if Ted decides to put down 20% of the property value (and

ignoring any loan origination or other fees that may be associated with this activity):

How much will Ted pay to the bank over the 1st year of the loan (after 12 payments)?

What is the unpaid principal balance at the end of the 1st year (after 12 payments)?

How much total interest did he pay in the 1st year of the loan (after 12 payments)?

8) Assume the cash flow profile for a particular capital project is as given in the table below

and the firm’s risk adjusted, weighted average cost of capital is 10.00%.

What is the net present value of the capital project using the firm’s discount rate?

What is the internal rate of return of the capital project?

Should the firm undertake the capital project? Why or why not?

9) Define APR.

Calculate the APR associated with a fully amortizing loan of $1,000 and that has a payment

schedule of 12 equal monthly payments in the amount of $100/month.

10) Recall the central difference approximation of the derivative of 𝑓(𝑥) at 𝑥 = 𝑎 is

𝑓′(𝑎) ≈ 𝑓(𝑎 + ℎ) − 𝑓(𝑎 − ℎ)

2ℎ

Use this ratio to approximate the derivative of 𝑓(𝑥) = 𝑥4 where 𝑎 = 10 and ℎ = 1.

What is the exact value of the slope of the line tangent to 𝑓(𝑥) = 𝑥4 where 𝑎 = 10?

Year Cash Flow

0 -$4,000.00

1 $500.00

2 $500.00

3 $4,500.00

11) State the maximum output of the following function on the given interval:

𝑓(𝑥) = 16𝑥3 − 132𝑥2 + 120𝑥 − 29 [ 0 , 2 ]

12) State the condition under which profit is maximized.

Given the following price-demand relationship and total cost function for a particular

firm, determine the price the firm should charge to maximize profit.

𝑞(𝑝) = 1020 − 8 𝑝 & 𝑐(𝑞) = 2050 + 22.50 𝑞

13) What is the profit associated with the price level you found in Question 12)?

Is the so-called “Law of Demand” satisfied by the price-demand model in Question 12)?

Compute the optimized price level’s elasticity of demand using the models in Question 12).

14) Assume a consumer, who has $500 of disposable income, has the utility function

𝑢(𝑥, 𝑦) = 13 𝑦0.8 𝑥1.2

where 𝑥 represents the number of pints of beer ($6 per unit) and 𝑦 represents the number

of pairs of jeans ($100 per unit) consumed. Assuming exactly $500 is spent, what is the

budget constraint?

Which goods, if either, satisfy decreasing marginal utility?

Assuming the consumer is a rational, self-interested, utility maximizer, what basket of

goods will the consumer consume?

15) Assume you construct the mean-variance efficient portfolio with a portfolio value of

$260,000 where the target return is 6% and you use a combination of 3 assets: 2 risky

assets, 𝑋 and 𝑌, and another that earns the risk free rate. Assume the risk free rate is 1%,

the correlation in the returns of the risky assets is 0, the risk in asset 𝑋 is 30%, the risk in

asset 𝑌 is 40%, the expected return in asset 𝑋 is 6% and the expected return in asset 𝑌 is

11%. What is the dollar amount of the portfolio that you should put in the risk free asset for

this mean variance efficient portfolio?

Start by finding the values of 𝑥, 𝑦 and 𝑧 that

𝒎𝒊𝒏𝒊𝒎𝒊𝒛𝒆 𝑓(𝑥, 𝑦) = 9𝑥2 + 16𝑦 2 𝒔𝒖𝒄𝒉 𝒕𝒉𝒂𝒕 𝑥 + 2𝑦 = 1 𝐴𝑁𝐷 𝑥 + 𝑦 + 𝑧 = 1

16) Compute the Capital Asset Pricing Model (“CAPM”) beta of Alphabet relative to the S&P 500

using all the following data:

Date Alphabet S&P 500

Mid-March, 2016 $727 2022

Mid-April, 2016 $743 2062

Mid-May, 2016 $712 2064

Mid-June, 2016 $719 2095

Mid-July, 2016 $721 2152

17) What is the probability that the average of 2 (6-sided) dice is less than 3.66?

What is the probability that the average of 105 dice is less than 3.66?

18) Compute the probability that a mortgage currently in the Performing status will be in the

Delinquent status after 3 months given the following monthly transition matrix where the

1st row (and column) represents the Performing status, the 2nd row represents the

Delinquent Status, and the 3rd row represents the Default status.

𝑀 = ( 0.95 0.05 0.00 0.25 0.60 0.15 0.00 0.00 1.00

)

19) Assume gold is currently trading at $1350.00/oz. Assume that in each of the next 2 years,

and regardless of what actually transpires, the price of gold will either rise by 20% with a

probability of 40% or fall in price by 10%.

Build a 2 period binomial tree for the gold price process.

Calculate the fair premium to pay today for a (European) put option on 1 ounce of gold if the

put option’s strike price is $1,458 and the option expires in 2 years. Round to the nearest

cent and note the model implied risk free rate is 2% compounded annually.

Compute the fair premium for the otherwise contractually equivalent (European) call.

Use “Put Call parity” to compute the fair value of the corresponding forward contract.

20) Fit the following observations relating the borrower credit score, FICO, to the probability of

prepayment, SMM, assuming SMM is 𝑌 and FICO is 𝑋 in the linear regression model:

𝑴𝑶𝑫𝑬𝑳: 𝑌 = 𝛼 + 𝛽 ∙ 𝑋 + 𝜖

21) Assume there is an MBS collateral pool containing only 30 year, fixed rate, level payment,

monthly paying, fully amortizing mortgages and the following beginning period

characteristics:

 Beginning Period Unpaid Principal Balance in Pool = $5,000,000.00 ;

 Weighted Average Note Rate = 3.50% (in the “30/360” day count convention) ;

 Current Weighted Average Age = 60 Months ; and

 Weighted Average FICO = 720.

Further assume that the monthly default rate, MDR, is 2.00% and the Recovery Rate is

90.00% (with no recovery lag). Using the model you calibrated in Question 20) to estimate

the SMM based on the given Weighted Average FICO of the collateral pool, compute each

of the following dollar amounts:

 The gross defaults;

 The recoveries;

 The periodic interest;

 The periodic principal;

 The prepayments; and

 The ending period unpaid principal balance in the pool.

FICO SMM (%)

700 3.00

600 2.00

750 7.00

650 5.00

22) If a financial institution reports quarterly P&L information as in the table below, calculate

the 95% Value at Risk (VaR) assuming the P&L is a normally distributed random variable.

Use either the Maximum Likelihood or Unbiased estimators but identify which you used.

In your opinion, as between the Maximum Likelihood estimators approach and the

Unbiased estimators approaches, which would a prudential regulator prefer and why?

23) Assume a company’s variable production cost is a constant $22.50 per unit and its fixed

production cost is $2,050. Assume the company does some market research and obtains the

following price-demand data observations:

Assuming a linear price-demand model, what is the model implied, profit maximizing price?

Under the linear price demand model, compute the elasticity when profit is maximized and

state whether or not the “Law of Demand” is satisfied by the calibrated model.

Reporting Period P&L

Q4 $250m

Q3 $480m

Q2 -$200m

Q1 $100m

P ($) Q (Units)

50.00 650

60.00 500

70.00 450

80.00 400

24) Using the same data set as in Question 23), fit a constant elasticity price-demand model, and

compute the model implied, profit maximizing price the company should charge.

Under the constant elasticity model, compute the elasticity when profit is maximized and

state whether or not the “Law of Demand” is satisfied by the calibrated model.

25) An estimator of the historical Black Scholes (“BS”) volatility parameter is

𝜎ℎ𝑖𝑠𝑡 = √𝑉𝑎𝑟[ln(𝑆𝑡+∆𝑡 /𝑆𝑡 )]/∆𝑡

Based on all the gold price data below, and using the following table to the extent you

choose, estimate of the BS historical volatility parameter.

Date gold/oz 𝑺𝒕+∆𝒕/𝑺𝒕 𝐥𝐧 (𝑺𝒕+∆𝒕/𝑺𝒕) [𝐥𝐧(𝑺𝒕+∆𝒕/𝑺𝒕)] 𝟐

Mid-March, 2016 $1262 N/A N/A N/A

Mid-April, 2016 $1259

Mid-May, 2016 $1268

Mid-June, 2016 $1267

Mid-July, 2016 $1350