Topic 1: Financial Markets
1. You are among the OTC marketmakers in the stock of BioEngineering, Inc. and quote a bid of $102.25 and an ask of $102.50. 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.)
You have sold 10,000 shares at the ask price of $102.50. You bought 4,000 shares at a bid price $102.25. Thus, 6,000 shares are sold short (sold without already owning the security). Your revenue from the 4,000 “round trip” purchase and sale produces a profit equal to the bid minus the ask times the volume done. Hence, the profit on the round trip trades is $0.25 × 4, 000 = $1, 000.
The value of your inventory is equal to the value of your short position of 6,000 shares. Since there is both a bid and an ask price, this question can answered is various ways depending on what you assume:
The “conservative” valuation is to value your position at the ask price of $102.50. Then, you have a position of -$615,000. This conservative valuation is useful because, if you cover your short position by buying from another dealer at his ask price of $102.50, you would have to pay $615,000. (Also, in this example it is the price for which you sold the securities.)
The “aggressive” valuation is to value your position at the bid price of $102.25. Then, you have a position of -$613,500 (i.e. less negative than above). This value is implicitly based on an expectation that some investors will come to you and sell you 6000 shares at your current bid price.
Often, real-world market makers will value their inventory at the mid price, in this case $102.375. Then, you have a position of -$614,250.
(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 $110.25 and $110.50, respectively.
1
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?
You have bought 8,000 shares during Day 2 at $110.25 and sold 2,000 shares at $110.50. On the 2,000 you bought and sold during the day you earn 2, 000 × $0.25 = $500. You also added 6,000 shares to your inventory at a price of $110.25. Since you were short 6,000 shares at $102.50 from yesterday’s trading, your loss on these 6,000 shares is −$7.75 × 6, 000 = −$46, 500. Thus your total profit/loss over the two-day period is $1, 000 + $500 − $46, 500 = −$45, 000. Your inventory at the end of Day 2 is zero since you purchased 6,000 shares that offset the 6,000 share short position at the end of Day 1.
(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?
A marketmaker’s objective is to earn the bid-ask spread, and not (necessarily) to speculate on equilibrium price movements. The 6,000 share short position at the end of Day 1 left you vulnerable to a jump in quoted prices. Perhaps you should have increased the prices during Day 1 as you observed more buying than selling. Also, you could have reduced your short inventory position by buying from other dealers at $102.50 as the market closed on Day 1. There are two factors that may have prevented you from doing this: (1) Your expected profit of $.25 on the 6,000 shares if market prices remained unchanged at $102.25 (bid) and $102.50 (ask). (2) Unwillingness of other dealers to accommodate your purchase of 6,000 shares at the $102.50 ask price.
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? It sells for 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?
We have to solve T in 1000 ∗ 1.05−T = 325.57. Either use the financial calculator, 2 or notice that we can solve it as
1000 ∗ 1.05−T = 1.05−T = −T log {1.05} =
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. (b) Invest $550 for ten years in Chase at a guaranteed annual interest rate of 5.5%. The annual return on the bond is: r = (1000/550)1/10 − 1 = .0616
The interest on the Chase deposit is only 5.5%. Therefore, the bond is a better invest- ment. Note also that the Chase deposit after 10 years will have grown to:
F = 550(1.055)10 = 939.48, which is lower than the face value of the bond.
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 it does 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.
To determine whether it is better to get for free (a) or (b), we must calculate which security has the higher present value.
It is always helpful to show a time line of the cash flows (to simplify the picture we omit the 000s in 10,000).
1This property holds for both the 10-log and the natural log (often denoted ”ln”), so both can be used to solve this problem.
3
325.57 0.32557 log {0.32557} − log {0.32557} log {1.05} where the third equation follows from the second by using the property that log {xy } = y log {x}.1
T= T =
23

(a)
(b)
+10+10+10+10+10+10 0 0 ... ||||||||| |||||||||
012345678
0 0 0 0 0 0 0 0 0 0 0 +10 +10 ... |||||||||||||| ||||||||||||||
0 1 2 3 4 5 6 7 8 9 10 11 12 13 Cash flow (a) is a straightforward annuity whose present value is given by:
1 1 PV=C r−r(1+r)t
You can calculate the present value by entering the appropriate values for C, r, and t into the formula above or by using the annuity keystrokes programmed into your calculator. In both cases make certain to enter the numbers and do the calculations carefully. Given the values of C = $10,000 and t = 6, when the interest rate, r, is equal to 5%, the present value is $50,757. When an interest rate of 10% is used with the same cash flows, the present value is $43,552
Cash flow (b) is a perpetuity that begins 10 years from now. We can value it in two parts. First, we know that a perpetuity has a value, P, given by:
P=C r
Thus, with C = $10, 000 and r = .05 the perpetuity is worth $200,000. But it is worth $200,000 ten years from now, not today. We can get the present value of $200,000 received with a 10 year delay by treating that sum as though it were a zero coupon bond with a face value of $200,000 payable in 10 years. Hence,
PV = $200,000 (1 + r)10
When we substitute r = .05 in the formula above we find that the present value of the perpetuity beginning 10 years from now is $122,782. Hence, we prefer the perpetuity because its present value of $122,782 is larger than the $50,757 present value of the annuity in (a).
When r = .10 we have to first recalculate the value of the perpetuity: P = $10, 000 = $100, 000
.10 4

With $100,000 ’payable’ in 10 years at 10% interest we have P V == $100, 000 = $38, 554
1.1010
Thus at 10% we prefer the annuity in (a), with a present value of $43,552, to the perpetuity, which is worth only $38,554.
This result stems from the fact that even though the perpetuity has infinite cash flows compared with the annuity, those cash flows begin with a 10- year delay. At a 10% interest rate the delayed cash flows are penalized very heavily.
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 return on $100 invested in her fund if she allows you to reinvest in her fund the 1% you earn each day?
Allowing you to reinvest at 1% per day means that you are earning compound interest on your initial $100 investment. The formula for P growing to F for one year at a compound rate r per annum is:
F = P 1 + r n n where n is the number of compounding periods per year and hence r/n is the rate per compounding period. We are given r/n = 1% per day and are asked to calculate the annual yield. This is equivalent to asking for the effective annual rate.
EAR = (1 + .01)250 − 1 = 11.0321
Multiplying by 100 puts this into percentage terms: 1103.21% per annum.
Looked at another way, investing $100 in the hedge fund produces $100(1 + .01)250 = $1203.21 at the end of one year.
(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?
If the hedge fund manager insists on putting your daily 1% earnings into a zero-interest bearing checking account, then you will earn only the daily rate (1%) multiplied by the number of days, or,
1% × 250 = 250%
Notice that this is equivalent to the annual percentage rate (APR) calculation:
APR=periodicrate ×n=1%×250 5

The value at the end of the year includes interest earnings plus original investment, that is, 100+250=350.
(c) Can you summarize when it is proper to ”annualize” using APR (annual percentage rate) versus EAR (effective annual rate)?
Whether you use APR or EAR to annualize a periodic rate depends upon the process for reinvesting the proceeds of your investment. If you can reinvest at the periodic rate (as in (a)) then EAR is appropriate. If the reinvestment rate is zero (as in (b)) then APR is appropriate. Since the reinvestment rate is rarely zero, the APR usually understates the annual rate.