Chapter 2 (1)
2.3 What risks does a bank take if it funds long-term loans with short-term deposits?
The risk is that interest rates will rise so that, when deposits are rolled over, the bank has to pay a higher rate of interest. The
rate received on loans will not change. The result will be a reduction in the bank’s net interest income
2.4 Suppose that an out-of-control trader working for DLC bank loses $7 million trading foreign exchange. What do you think will
happen?
DLC’s loss is more than its equity capital and it would probably be liquidated. The subordinated long-term debt holders would
incur losses on their $5 million investment. The depositors should get their money back
2.5 What is meant by net interest income?
The net interest income of a bank is interest received minus interest paid.
2.6 Which items on the income statement of DLC Bank in Section 2.2 are most likely
to be affected by (a) credit risk, (b) market risk, and (c) operational risk?
Credit risk primarily affects loan losses. Non-interest income includes trading gains and losses. Market risk therefore affects
non-interest income. It also affects net interest income if assets and liabilities are not matched. Operational risk primarily
affects non-interest expense.
2.16 Explain the moral hazard problems with deposit insurance. How can they be overcome?
Deposit insurance makes depositors less concerned about the financial health of a bank. As a result, banks may be able to take
more risk without being in danger of losing deposits. This is an example of moral hazard. (The existence of the insurance
changes the behaviour of the parties involved with the result that the expected payout on the insurance contract is higher.)
Regulatory requirements that banks keep sufficient capital for the risks they are taking reduce their incentive to take risks. One
approach (used in the U.S.) to avoiding the moral hazard problem is to make the premiums that banks have to pay for deposit
insurance dependent on an assessment of the risks they are taking.
Chapter 5 (1)
5.1 What is the difference between a long forward position and a short forward position?
When a trader enters into a long forward contract, she is agreeing to buy the underlying asset for a certain price at a certain
time in the future. When a trader enters into a short forward contract, she is agreeing to sell the underlying asset for a certain
price at a certain time in the future
5.2 Explain the difference between hedging, speculation, and arbitrage
A trader is hedging when she has an exposure to the price of an asset and takes a position in a derivative to offset the exposure.
In a speculation, a trader has no exposure to offset. She is betting on the future movements in the price of the asset. Arbitrage
involves take a position in two or more different markets to lock in a profit.
5.3 What is the difference between entering into a long forward contract when the forward price is $50 and taking a long
position in a call option with a strike price of $50?
In the first case, the trader is obligated to buy the asset for $50, the trader does not have a choice. In the second case, the
trader has an option to buy the asset for $50 (the trader does not have to exercise the option)
5.4 Explain carefully the difference between selling a call option and buying a put option
Selling a call option involves giving someone else the right to buy an asset from you for a certain price. Buying a put option
gives you the right to sell the asset to someone else
5.7 Suppose you write a put contract with a strike price of $40 and an expiration date in three months. The current stock price is
$41, and the contract is on 100 shares. What have you committed yourself to? How much could you gain or lose?
You have sold a put option. You have agreed to buy 100 shares of $40 per share if the party on the other side of the contract
chooses to exercise the right to sell for this price. The option will be exercised only when the price of stock is below $40.
Suppose, for example, that the option is exercised when the price is $30. You have to buy at $40 shares that are worth $30, you
lose $10 per share, or $1,000 in total. If the option is exercised when the price is $20, you lose $20 per share, and so on. The
worst that can happen is that the price of the stock declines to almost zero. You would lose $4,000.
1
,5.8 What is the difference between the over-the-counter market and the exchange-traded market? Which of the two markets do
the following trade in: (a) a forward contract, (b) a futures contract, (c) an option, (d) a swap, (e) an exotic option?
The OTC market is a telephone- and computer-linked network of financial institutions, fund managers, and corporate treasurers
where two participants can enter into any mutually acceptable contract. An exchange-traded market is a market organized by
an exchange where traders either meet physically or communicate electronically and the contracts that can be traded have
been defined by the exchange.
Chapter 12 (2)
2
,3
,4
, 12.1 What is the difference between expected shortfall
and VaR? what is the theoretical advantage of expected shortfall over VaR?
VaR is the loss that is not expected to be exceeded with a certain confidence level. Expected Shortfall is the expected loss
conditional that the loss is worse than the VaR level. Expected Shortfall has the advantage that it always satisfies the
subadditivity condition, that diversification is good.
12.2 What conditions must be satisfied by the weights
assigned to percentiles in a risk measure for the
subadditivity condition to be satisfied?
For the subadditivity condition to be satisfied, the weight
assigned to the qth quantile must be a non-decreasing
function of q
12.3 A fund manager announces that the fund’s one-month
95% VaR is 6% of the size of the portfolio being
managed. You have an investment of $100,000 in the
fund. How do you interpret the portfolio manager’s
announcement?
There is a 5% chance that you will lose $6,000 or more during a one-month period
12.4 A fund manager announces that the fund’s one-month 95% ES is 6% of the size of the portfolio being managed. You have the
same investment as in 12.3. How do you interpret this announcement?
Your expected loss during a ‘bad’ month is $6,000. Bad months are defined as the months where returns are less than the five-
percentile points on the distribution of monthly returns
12.5 Suppose that each of two investments has a 0.9% chance of a loss of $10 million and a 99.1% chance of a loss of $1 million.
The investments are independent of each other.
(a) What is the VaR for one of the investments when the confidence level is 99%?
(b) What is the expected shortfall for one of the investments when the confidence level is 99%?
(c) What is the VaR for a portfolio consisting of two investments when the confidence level is 99%?
(d) What is the ES for a portfolio consisting of two investments when the confidence level is 99%?
(e) Show that in this example VaR does not satisfy the subadditivity condition, whereas expected shortfall does
(a) 1 million
(b) The expected shortfall is 0.9 * 10 + 0.1 * 1 = $9.1 million
(c) There is a probability of 0.0092 = 0.000081 of a loss of $20 million, a probability of 2 * 0.009 * 0.991 = 0.017838 of a loss of
$11 million, and a probability of 0.9912 = 0.982081 of a loss of $ 2 million. The VaR when the confidence level is 99% is
therefore $11 million
(d) The expected shortfall is (0.000081 * 20 + 0.009919 * 11) / 0.01 = $11.07 million
(e) Because 1 + 1 < 11, the subadditivity condition is not satisfied for VaR
Because 9.1 + 9.1 > 11.07, it is satisfied for expected shortfall
12.6 Suppose that the change in the value of a portfolio over a one-day time period is normal with a mean of zero and a standard
deviation of $2 million; what is (a) the one-day 97.5% VaR, (b) the five-day 97.5% VaR, and (c) the five-day 99% VaR?
(a) 2 * 1.96 = $3.92 million
(b) SQRT (5) * 2 * 1.96 = $8.77 million
(c) SQRT (5) * 2 * 2.33 = $10.40 million
12.8 Explain carefully the differences between marginal VaR, incremental VaR, and component VaR for a portfolio consisting of a
number of assets
Marginal VaR is the rate of change of VaR with the amount invested in the th asset.
Incremental VaR is the incremental effect of the ith asset on VaR (I.e. the difference between VaR with and without the asset)
Component VaR is the part of VaR that can be attributed to the ith asset (the sum of component VaR equals the total VaR)
12.10 Explain what is meant by bunching
5
, Bunching is the tendency for exceptions to be bunched rather than occurring randomly throughout the time period considered
12.13 Suppose that each of two investments has a 4% chance of a loss of $10 million, a 2% chance of a loss of $1 million, and a 94%
chance of a profit of $1 million. They are independent of each other.
(a) What is the VaR for one of the investments when the confidence level is 95%?
(b) What is the expected shortfall when the confidence level is 95%?
(c) What is the VaR for a portfolio consisting of the two investments when the
confidence level is 95%?
(d) What is the expected shortfall for a portfolio consisting of the two investments
when the confidence level is 95%?
(e) Show that, in this example, VaR does not satisfy the subadditivity condition,
whereas expected shortfall does.
(a) A loss of $1 million extends from the 94 percentile point of the loss distribution to the 96 percentile
point. The 95% VaR is therefore $1 million.
(b) The expected shortfall for one of the investments is the expected loss conditional that the loss is in the 5
percent tail. Given that we are in the tail there is a 20% chance than the loss is $1 million and an 80%
chance that the loss is $10 million. The expected loss is therefore $8.2 million. This is the expected
shortfall.
(c) For a portfolio consisting of the two investments there is a 0.04 × 0.04 = 0.0016 chance that the loss is
$20 million; there is a 2 × 0.04 × 0.02 = 0.0016 chance that the loss is $11 million; there is a 2 × 0.04 ×
0.94 = 0.0752 chance that the loss is $9 million; there is a 0.02 × 0.02 = 0.0004 chance that the loss is $2
million; there is a 2 × 0.2 × 0.94 = 0.0376 chance that the loss is zero; there is a 0.94 × 0.94 = 0.8836
chance that the profit is $2 million. It follows that the 95% VaR is $9 million.
(d) The expected shortfall for the portfolio consisting of the two investments is the expected loss conditional
that the loss is in the 5% tail. Given that we are in the tail, there is a 0.0016/0.05 = 0.032 chance of a loss
of $20 million, a 0.0016/0.05 = 0.032 chance of a loss of $11 million; and a 0.936 chance of a loss of $9
million. The expected loss is therefore $9.416.
(e) VaR does not satisfy the subadditivity condition because 9 > 1 + 1. However, expected shortfall does
because 9.416 < 8.2 + 8.2.
12.16 The change in the value of a portfolio in three months is normally distributed with a mean of $500,000 and a standard
deviation of $3 million. Calculate the VaR and ES for a confidence level of 99.5% and a time horizon of three months.
The loss has a mean of −500 and a standard deviation of 3000. Also, N−1(0.995) =2.576. The 99.5% VaR in $’000s is
−500+3000×2.576) =7,227. We are 99.5% certain that the loss will not be greater than $7.227 million.
The ES is
2
e−2 .576 /2
500+3000 =9 , 176
- √ 2 π ×0 . 005
The expected loss conditional that it is in the 0.5% tail of the distribution is $9.176 million.
Chapter 13 (2)
6
, 13.1 What assumption is being made when VaR is calculated using the historical simulation approach and 500 days of data?
The assumption is that the statistical process driving changes in market variables over the next day is the same as that over the
last 500 days
13.3 Suppose we estimate the one-day 95% VaR from 1,000 observations (in millions of dollars) as 5. By fitting a standard
distribution to the observations, the probability density function of the loss distribution at the 95% point is estimated to be
0.01. What is the standard error of the VaR estimate?
The standard error of the estimate is
1 0.05∗0.95
13.4
0.01 √1,000
=0.69
The one-day 99% VaR for the four-index example is calculated in Section 13.1 as $253,385. Look at the underlying
spreadsheets on the author’s website and calculate (a) the 95% one-day VaR, (b) the 95% one-day ES, (c) the 97% one-day
VaR, and (d) the 97% one-day ES.
(a) The 95% one-day VaR is the 25th worst loss. This is $156,511.
(b) The 95% one-day ES is the average of the 24 highest losses. It is $209,310.
(c) The 97% one-day VaR is the 15th worst loss. This is $172,224.
(d) The 97% one-day ES is the average of the 14 highest losses. It is $240,874.
7
