Tip: Working through these problems with a formula sheet handy is a great way to learn and get comfortable with the formula sheet. This will help you use the formula sheet quickly and confidently during the exam.
You can download my paper formula sheets at https://robmunger.com/2000share. If this is your first time using them, the “Basic” (not “Text”) sheet has all of the formulas and may be slightly more readable. There is also an online version on this website.
✏️ What is the price of the following bond?
F = $5000
c = 12%
i = 13%
T = 4
✔ Click here to view answer
Key equations from the formula sheet:
The coupon payment is
Here’s a shorter way to calculate it:
✏️ What is the price of the following bond?
F = $5000
c = 0
i = 13%
T = 10
✏️ What is the price of the following bond?
F = $5000
c = 12%
i = 13%
T = ∞
✏️ What is the YTM of the following bond?
F = $5000
c = 12%
T = ∞
PB = $8000
✔ Click here to view answer
Key equations from the formula sheet:
To answer a YTM problem, you write down the bond pricing formula and solve it for i.
Algebra (Switcheroo):
The following would be a stretch question. I wouldn’t expect a question this hard on the exam.
✔ Click here to view answer
Key equations from the formula sheet:
For any YTM problem, you write down the bond pricing formula and solve for i.
Algebra/Switcheroo:
Recopy:
Take the 1/10 power of both sides:
You can enter this into most spreadsheets as 1.6667^(1/10) or 1.6667^.100. For many calculators, you would replace the caret, ^, with a dedicated button, such as a or button. Whether you use a calculator or spreadsheet, be careful with exponents that aren’t round numbers, though. Don’t forget the parentheses around (1/10)!
Subtract 1 from both sides:
We can check our math by substituting 5.241% back into the bond pricing equation and confirming that we get a price close to our original bond price of $3000. In this case, I’ve done the calculations with high precision to ensure that the answer comes out very close to $3000.
✏️ Suppose that for a given firm, β=1.3. The expected return on the market is 13%, and the risk free rate is 5%. What is the Expected return for this stock?
✏️ Suppose that for a given firm, β=1.3. The Market Risk Premium 10%, and the risk free rate is 5%. What is the Expected return for this stock?
✔ Click here to view answer
Key equations from the formula sheet:
“Risk Premium” means you subtract off the risk free rate.
Market risk premium = Expected risk premium of market
We have been given the market risk premium, so
If , then , so .
Now, we can apply the formula:
EMH Stock Price
Section titled “EMH Stock Price”✏️ You are considering purchasing a share of XYZ Corp. There is a 30% chance that in one year, the stock price will be $110, a 40% that it will be $120, and a 30% that it will be $130. You expect to receive a $1 dividend in one year. Given the riskiness of XYZ corp, similar corporations are earning a return of 14%. What is the fair price of XYZ stock right now?
✔ Click here to view answer
Options
Section titled “Options”✏️ Draw a short call with a strike price of $45 and a premium of $2
✔ Click here to view answer
✏️ What is your P/L on the above short call if S=$38 at expiration? If it equals $52 at expiration? Also, what are your total profits or losses per contract at these two prices.?
✔ Click here to view answer
Key equation from the formula sheet:
P/L from Selling a Call = Pr - Max (S-K, 0)
When S=$38:
P/L from Selling a Call = Pr - Max (S-K, 0) = $2 - $0 = $2 per share
When S=$52:
P/L from Selling a Call = Pr - Max (S-K, 0) = $2 - Max($52-$45,0) = $2 - $7 = -$5 per share
Your total profits or losses from a full contract are $2×100=$200 and -$5×100=-$500, respectively.
✏️ Suppose S=$43 on the above option and premium is $2. What is the IV and TV?
✔ Click here to view answer
Key equations from the formula sheet:
Call IV = Max (S-K, 0)
Premium = Intrinsic Value + Time Value
Call IV = Max (S-K, 0) = Max($43-$45, $0) = $0
Premium = Intrinsic Value + Time Value
$2 = $0 + Time Value, so the time value is $2.
✏️ Suppose that S=$42 on the above K=$45 option and the time value of the option is $3. What is the premium?
✔ Click here to view answer
Call IV = Max (S-K, 0) = Max ($42-$45, 0) = Max (-$3, $0) = $0
Premium = Intrinsic Value + Time Value = $0 +$3 = $3
Option Strategies
Section titled “Option Strategies”✏️ Consider a bull spread with the following options.
A K=$35 call has a premium of $4
A K=$40 call has a premium of $1
Construct a bull spread from these options. (Just say which options you would buy/sell.)
✔ Click here to view answer
Buy the $35 Call and Sell the $40 Call. See the formula sheet for a cheat sheet on exactly what to buy and sell.
✏️ Suppose you Buy the $35 Call and Sell the $40 Call. This allows you to construct a bull spread. The $35 Call has a premium of $4 and the $40 call has a premium of $1. Suppose S=$33.
What is your P/L of the Call you purchased? Show your work.
✔ Click here to view answer
You purchased a $35 Call for $4. Because S=$33, you will not exercise this call. It will expire worthless. However, you will have paid $4 for the call, so your P/L is -$4.
✏️ What is your P/L of the Call you sold? Show your work.
✔ Click here to view answer
You sold a $40 call and received a premium of $1. Because S=$33, your counterparty will not exercise the call. You get to keep the premium of $1, so your P/L is $1. (If your counterparty exercised the call, they would be purchasing a stock worth only $33, but they would pay $40. They don’t want to do this!)
✏️ What is your P/L of the combined bull spread?
✔ Click here to view answer
For the call we purchased, P/L is -$4.
For the call we sold, P/L is $1
Combining them, -$4 + $1 = -$3
✏️ Suppose you Buy the $35 Call @$4 and Sell the $40 Call @$1 to construct a bull spread.
Suppose S=$45. what is your P/L on each of the calls and on the bull spread?
✔ Click here to view answer
Before, both calls were out of the money. Now, both calls will be in the money because the current stock price is higher than both of their strike prices.
P/L of Long K=$35 Call @$4:
P/L = Max(S-K,0) - Prem = Max(45-35,0) - Prem = Max(10,0) - 4 = 6
P/L of Short K=$40 Call @$1:
P/L = Prem - Max(S-K,0) = Prem - Max(45-40,0) = 1 - Max(5,0) = -4
Total P/L for the option strategy = $6 - $4 = $2
Extra Practice
Section titled “Extra Practice”(I call this extra practice because it goes beyond the level of difficulty I would anticipate on the exam. If you can do it, you are in great shape!)
✏️ Graph the bull spread in the previous examples, above.
✔ Click here to view answer
Futures
Section titled “Futures”✏️ Suppose that you are bullish on wheat and are sure that the price of wheat will rise in the coming weeks. Would you purchase or sell a wheat futures contract?
✔ Click here to view answer
Buying a futures contract is a bet that the futures price will rise, so you would buy the futures contract.
✏️ Suppose that you purchase the contract at 632 and that by the time the contract expires, the price has fallen to 629. The contract size is 3000 bushels. What is your total profit or loss?
✔ Click here to view answer
ΔContractPrice = (NewContractPrice - OldContractPrice)
ΔContractPrice = 620-632 = -3
Buy the contract:
P/L = ΔContractPrice * ContractSize
P/L = -9,000
✏️ Continuing the above problem, suppose that you deposited $10,000 in your margin account to open the trade. How much will remain in your margin account afterward? Suppose the initial margin is $10,000 and the maintenance margin is $5,000 Will you get a margin call? If so, how much must you deposit?
✔ Click here to view answer
Initially, there was $10,000 in your margin account. You lost $9,000, so there is only $1,000 in your margin account. Because $1,000 < $5,000, you will get a margin call and will need to deposit additional money. You need to refill the coffee cup until the balance is equal to the initial margin of $10,000. To refill the cup from $1,000 to $10,000 will take a $9,000 deposit.
Conceptual Question
Section titled “Conceptual Question”*I don’t think that the following questions would be illustrative of a typical question on the final. However, I am including them because it will help you review concepts that could be tested in other conceptual questions. *
If you want to see what typical conceptual questions will be on the exam, look at the problem sets and at the midterm exam. They will be much better examples of what you want to prepare for. These examples are instead better for making sure that you have all of the concepts that might be tested in a conceptual question.
✏️ Discuss multiple types of risk, and then analyze government bonds in terms of those types of risk. Discuss the risk-free from the CAPM rate and connect it to the risk profile of government bonds.
✔ During that class, we covered two types of risk: credit risk and interest rate risk.
Credit risk is the risk that a bond issuer will not make the payments that they have committed to make. Credit risk is assessed by the credit rating agencies.
Government bonds are often considered free from credit risk for two reasons:
- They can raise tax rates or divert tax revenues in order to service the debt
- If they can also increase the money supply (create money) to service the debt.
Interest rate risk is the risk that the market value of a bond may fluctuate when interest rates change. This is because interest rates are often used as the discount rate when calculating the price of a bond. Therefore, when interest rates change, bond prices change: Sovereign debt is still subject to interest rate risk, just as any other bond is.
However, short-term sovereign debt, such as United States T-bills, has such a short maturity that its interest rate risk is effectively zero. Therefore, the yield on US T-Bills is often used as the risk-free interest rate in the CAPM.
✏️ What type of investor would prefer municipal bonds? Someone in a higher tax bracket or in a lower tax bracket? Use formulas from class to explain your answer.
✔ Munis decrease your tax bill, therefore they will be more attractive to people with higher tax rates.
Our key formula from the Muni slides is:
You can see from the formula that your marginal tax rate is subtracted from your return. Therefore, someone with a higher marginal tax rate will have more of their interest subtracted via government taxes than will someone with a lower marginal tax rate.
An example may help to see this.
Suppose you have a Taxable bond with a return of 10% and you are in a 20% tax bracket. The 20% tax will subtract 20% of your return, lowering your return from 10% to 8%:
If your marginal tax rate were higher, then the same taxable bond would be equivalent to a muni with a higher return. For example, if your MTR=30%, the 30% tax will subtract 30% of your return, lowering your return from 10% to 7%:
In contrast, if your marginal tax rate were lower, then the same taxable bond would be equivalent to a muni with a higher return. For example, if you MTR=10%, the 10% tax will subtract 10% of your return, lowering your return from 10% to 9%:
Clearly higher tax rates hurt taxable bonds much more. Therefore, people with higher tax rates have a stronger incentive to purchase municipal bonds rather than taxable corporate bonds.
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