Asset Pricing
What happens to the (i) price of the bond and (ii) the sensitivity of the bond when the yield increases?
The price declines, and so does the sensitivity to the yield.
What is the optimal portfolio of risky assets?
The portfolio that maximizes the Sharpe ratio, the additional return (over the risk-free rate) per additional risk (sigma).
I use the delta normal approach to calculate the VaR of my portfolio. I find that my 95% VaR has increased in 2024. What caused this?
The volatility (sigma) of my portfolio must have increased, because the delta normal VaR is measured as the critical z times sigma.
Explain what is meant by Distance to Default of a firm or bank. What does it allow us to calculate?
Distance to default is the difference between the value of assets and liabilities (or the distress point). Importantly, it is expressed as a multiple of the standard deviation of assets.
That is, it tells us how by many standard deviations the firm is from default. Therefore, it allows us to calculate the probability of default.
Comparing currency forward and futures contracts, which of the two (or both) usually eliminates all exchange rate risk of a position you have in foreign currency?
Forward contracts. Futures usually contain basis risk, because the timing and the asset (or currency) may not coincide perfectly with your position.
Suppose the same firm issues two 3-year bonds, one a premium bond and another a discount bond. If I intend to hold to maturity, which of the two do I prefer to purchase?
Neither, they will both have the same yield, or return.
I hold an asset A. I decide to add a second asset B to form a portfolio. Will this help to reduce risk (sigma of the portfolio return)?
Yes, as long as B is not perfectly correlated with A.
I calculated the 95% VaR for my portfolio: $30,000. Over the next 2 years, on how many weeks should I expect to lose more than $30,000? (Assume 1 year = 50 weeks)
I should expect to lose > $30,000 on (5% x 100 weeks) = 5 weeks.
True or False: Absence of arbitrage implies that two portfolios with the same value have the same expected payoff.
False: Absence of arbitrage implies that two portfolios with the same payoff have the same value.
A long position in a swap contract is equivalent to holding a portfolio consisting of i) long position in a fixed coupon bond and a short position in a floating rate note, ii) long position in a floating rate note, and a short position in a fixed coupon bond, iii) none of the above.
The correct answer is ii) long position in a floating rate note, and a short position in a fixed coupon bond.
An increase in the firm's Beta has what effect on its stock price?
It reduces the stock price; recall that Beta is in the denominator of the DDM formula.
What is a perfect hedge?
An asset that has perfect negative correlation (rho = -1) with an asset that you hold. By adding it to your portfolio you can eliminate risk (sigma)
Is VaR a worst-case scenario for my portfolio?
No, it is a floor or minimum loss that I can experience with a given probability.
The Merton approach to credit risk uses option pricing theory to determine the credit risk of a firm. Explain how a firm's balance sheet can be interpreted as an option.
A firm's equity can be viewed as a call option on the firm's assets.
If the assets (A) are greater than the firm's liabilities (L), then equity holders get the value of the difference between the two (A - L). If A < L, then equity holders get nothing.
In a swap contract, the PV of the fixed rate payments is ----- the PV of the floating rate payments at inception.
i) larger than
ii) less than
iii) equal to
The correct answer is iii) equal to
A firm reduces its dividend payout ratio. Will this reduce the stock price?
Not necessarily. A reduction in the dividend payout ratio is an increase in the plowback ratio, which will increase the stock price if ROE > k.
According to portfolio theory, should all investors share the same optimal portfolio P* of risky assets?
Yes. However, different investors will choose different combinations of P* with the risk-free asset according to their degree of risk aversion. This is called the Separation Principle.
Explain how the Credit Metrics approach can be used to calculate the 95% VaR of the credit risk of a loan to a rated firm.
Hint: three steps
1. Use the transition matrix to determine the minimum downgrade that the firm could suffer with 5% probability.
2. Using the forward curve for bonds of different ratings, find the YTM for the for the bond of the rating found in the first step.
3. Discount the value of the loan using this rate. The difference between this value and the current value of the loan will be the VaR.
You are given the following yield curve r1=0.05, r2=0.06, r3=0.07. Compute the price of a 2 year bond paying a 10% coupon with a face value 100 at t=0.
P=10/1.05+110/(1.06^2)=107.423
Using options only, how can we create a portfolio that replicates a forward contract?
Combining a long call and a short put with the same strike price is equivalent to a long forward.
A 2-year bond with a face value of $100 and with a YTM of 5% is valued today at $92.00. Does it pay coupons?
Yes. The price of the two-year zero coupon bond with that yield is $90.70. Since the price of $92.00 is greater than $90.70, it must pay coupons.
Consider two portfolios, M and B:
M is the minimum variance portfolio, expected return = 3%, sigma = 5%
B: expected return = 6%, sigma = 9%
Which portfolio do I prefer?
(Hint: draw a diagram)
I would prefer B. Any portfolio with a higher return and sigma than the minimum variance portfolio would have a higher Sharpe ratio.
Suppose I have 1,500 observations of daily returns on my portfolio. How can I use the historical simulation method to calculate the 99% daily VaR? Once I have calculated this, how do I obtain the 99% weekly VaR?
I rank the returns and find the 15th lowest return. To find the weekly VaR, I multiply the weekly VaR by the square root of 5.
You are given the following yield curve r1=0.05, r2=0.06, r3=0.07.Compute the forward rate f2 (between 2 and 3)
f2=(1+0.07)^3/(1+0.06)^2-1=0.09
The delivery price of a forward contract with one-year maturity is 70$. The underlying asset is a stock, with initial price equal to 65$. The interest rate is 10%. Compute the no-arbitrage forward price? Design a strategy to exploit the arbitrage opportunity.
The no-arbitrage forward price is equal to 65* (1+0.1) =71.5.
Strategy: long position on the forward, short sell the stock, invest 65
Payoff: 65 (1.1)-70=1.5