Topic 1: Financial Markets

1. You are among the OTC market makers in the stock of Bio-Engineering, Inc. and quote a bid of 102 1/4 and an ask of 102 1/2. Suppose that you have a zero inventory.

(a) On Day 1 you receive market buy orders for 10,000 shares and market sell orders for 4,000 shares. How much do you earn on the 4,000 shares that you bought and sold? What is the value of your inventory at the end of the day? (Hints: It is possible to have negative inventory. Further, there is more than one correct way to value an inventory, but please state what assumption your valuation is based on.)

I buy 4000 shares at the price of 102 ¼, for which the total cost is $409,000. I sell the inventory of 4000 shares at a price of 102 ½ and the profit from the sale is $410,000. The difference between the buy and the sale is $1000.
If the assumption is to execute the full buy order of 10,000 shares, I am left with 6000 shares that I can execute as a buy order at the end of the day. If I decide to do that, I can value the inventory at the bid or ask price. If I value at the bid price, then I have -$613,500 (6,000 x 102 ¼ = $613,500) of inventory value at the end of the day.
If I use the ask price, then the value of 6000 shares is (6000 x $102 ½) = - $615,000.

(b) Before trading begins on Day 2 the company announces trial testing of a cure for acne in mice. The quoted bid and ask jump to 110 1/4-1/2. During Day 2 you receive market sell orders for 8,000 shares and buy orders for 2,000 shares. What is your total profit or loss over the two-day period? What is the value of your inventory at the end of Day 2?

I sell 2,000 shares at a price of $110 1/2 for $221,000 . I buy 8,000 shares at a price of 110 1/4 for $882,000. I sold 6,000 of the shares from yesterday for a price of $615,000 (using the ask price). Inventory at the end of the 2nd day is 0. Losses were $221,000 + $615,000 - $882,000 = -$46,000.

(c) What is a market maker's objective? Is there anything you could have done during Day 1, consistent with a market maker's objective that would have improved your performance over the two-day period?

The market-maker’s objective is to make money on the bid-ask spread, by buying stock at a lower price than the price at which they sell it, or selling the stock at a higher price than they buy it back. To improve performance over the two day period, I could have avoided completing a full buy order on the 1st day to avoid the short balance.

Topic 2: Performance Measures
2. Suppose a 5-year zero-coupon Treasury bond with face value $1000 has a 5% yield (annually compounded).
(a) What price does this bond sell for? FV = $1000 t = 5 years r = 5% PV= FV / (1 + r)^t = $1000 / (1.05)^5 = $783.53 (b) Suppose another zero-coupon Treasury bond also has a 5% yield, but sells for $325:57. What is the maturity of this bond? FV = PV (1 + r)^t $1000 = $325.57 (1.05)^t Solve for (t) -> t = 23 year maturity

3. Which of the following investments do you prefer?
(a) Purchase a zero-coupon bond, which pays $1000 in ten years, for a price of $550. $1000 = $ 550 (1 + r )^10 R = 6.166% is the implied interest rate
(b) Invest $550 for ten years in Chase at a guaranteed annual interest rate of 5.5%. FV = $ 550 * (1.055) ^ 10 = $939.47%
In both options, the value we pay today is $550. We calculated the implied interest rate which was 6.166% on part a). The first investment is a better deal because it provides the higher return for 10 years.

4. Suppose you get for free one of following two securities: (a) an annuity that pays $10,000 at the end of each of the next 6 years; or (b) a perpetuity that pays $10,000 forever, but payments do not begin until 10 years from now (the first cash payment from this security is 11 years from today).
Which security would you choose if the annual interest rate is 5%?
Does your answer change if the interest rate is 10%? Explain why or why not. a) Annuity : (pays a fixed cash flow C, for T periods)
PV = C/R * (1 – 1/(1+R)^t)
PV = C * PV Factor = $10,000 * 5.07 = $50,700
PV Factor = [1 – 1/(1+R)^t ] / R = [1 – 1/(1.05)^6 ] / 0.05 = 5.07

b) Perpetuity: PV = (C/R) – pays a fixed cash flow every period forever. The value of perpetuity which begins in 11 years:
PV = C/R ( 1/ (1+R)^t) = $10000/ 0.05 * (1/1.05^10) = $122,784.78
Perpetuity is a better deal

SOLVING for 10% interest rate

Annuity => PV = PV = C/R * (1 – 1/(1+R)^t) = $ 43,552.61
Perpetuity => PV = C/R ( 1/ (1+R)^t) = 10000/ 0.1 * (1/1.1^10) = 38,544.32
Annuity is a better deal when interest rate is 10%.

5. Suppose a hedge fund manager earns 1% per trading day. There are 250 trading days per year. Answer the following questions:
(a) What will be your annual rate of return on $100 invested in her fund if she allows you to reinvest in her fund the 1% you earn each day? 1% compounds 250 times a year.
EAR = (1.01)^250 – 1 = 11.03%

(b) What will be your annual return assuming she puts all of your daily earnings into a zero-interest- bearing checking account and pays you everything earned at the end of the year?

APR = n*[(EAR+1)^(1/n) - 1] =250 * [(0.11 +1)^1/250 – 1] = 10%

(c) Can you summarize when it is proper to "annualize" using APR (annual percentage rate) versus EAR (effective annual rate)?
EAR is used when comparing investments with different compounding periods per year, whereas the APR does not include the intra-year compounding. APR ignores interest on interest which we don’t withdraw for that period.

6. Here are some alternative investments you are considering for one year. (i) Bank A promises to pay 8% on your deposit compounded annually. (ii) Bank B promises to pay 8% on your deposit compounded daily. Compare the effective annual rate (EAR) on these investments.
The option which compounds results more frequently will provide the highest EAR. We can calculate the EAR for both options to prove the point: 8% compounded annually results in an EAR of 8%. 8% compounded daily results in an EAR of (1 + (8/365)%)^365 - 1 = 8.328%.

7. (a) Suppose that you have purchased a 3-year zero-coupon bond with face value of $1000 and a price of $850. If you hold the bond to maturity, what is your annual rate of return?
FV (t) = C0(1 + R)^T + C1(1+ R)^1 + C2(1+ R)^2 + C3(1+ R)^3
FV(t) = C0 (1 + R )^ T
(C1 = C2 = C3 = 0)

T= 3 years
No coupon payments
FV = 1000
PV = 850
R = (FV/PV )^ 1/3 – 1 = ($ 1000/$ 850)^1/3 – 1 = 5.56%

(b) Now suppose you have purchased a 3-year bond with face value of $1000, a 7% annual coupon, and a price of $975. Assuming that you hold the bond to maturity, is the IRR greater or less than the return on the bond in part (a)?
Annual payment = coupon rate x Face Value = 0.07% x $1000 = $70

0
1
2
3
70
1,000
70
70
-975

FV (t) = C0(1 + R)^T + C1(1+ R)^1 + C2(1+ R)^2 + C3(1+ R)^3
-$975 + ($70/1 + irr) + [$70/(1 + irr)^2] + [$70 + $1000/(1 + irr)^3] = 0
IRR = 7.97%
IRR is greater than the rate of return on part given that we are getting a return on each yearly coupon as well.

8. Excel Question. Download the monthly S&P 500 prices from January 1950 until today http://finance.yahoo.com/ (click investing, click S&P 500, click historical prices, click monthly, click get prices, click download to spreadsheet).

(a) What is your best estimate for next month's return?
Best estimate for next month’s return is calculated via the arithmetic average. From excel spreadsheet, we calculated the average of all Monthly Returns from 1950 till today which turned out to be 0.68%.

(note: excel notebook can be provided upon request)

(b) What would have been your annualized HPR if you invested as of the start of the index?
Ann HPR = (V(t)/V0 )^ 1/t – 1
Price at V0 = $ 17.05
Price at Vt = $ 1518.2
T = 2013 – 1950 = 63 years
Ann HPR = (V2013/V1950 )^ 1/t – 1 = ($1518.2/$17.05)^1/63 - 1 = 0.0738
(c) In what month occurred the lowest monthly return? What happened? – (-16.94%)
10/01/2008 corresponds to 2008 economic crisis.