Ciia Final Exam II March11_se

EXAMINATION II:

Fixed Income Valuation and Analysis

Derivatives Valuation and Analysis

Portfolio Management

Solutions

Final Examination

March 2011

Question 1: Fixed Income Valuation and Analysis (43 points)

The following answers do not represent the only calculation methods or explanations that could be applied. Please mark everything as correct as long as the definitions, calculations and relationships make sense (please note that the final digit may vary according to the calculation method).

a) a1) The price P of a perpetual bond is equal to its fixed coupon (i.e. C) divided by its yield R:

(4 points)

a2) The Macaulay Duration D of a perpetual bond is defined as follows:

The Modified Duration Dmod is: (4 points)

Alternative solution:

Macaulay Duration:

Modified Duration: (4 points)

a3) The price P of the perpetual bond can be approximated as follows (y stands for yield):

As a result, the new price stands at 77.42 + 9.99 = 87.41 (4 points)

a4) Linear approximations of non-linear bond prices are only reliable for small parallel yield changes. A spread tightening of 100 bps is not small, i.e. the approximation is in this case not reliable and too conservative. The exact bond price after spread tightening amounts to 88.89 as opposed to the above approximated 87.41.

(5 points)

ACIIA® Solutions Examination Final II – March 2011

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b) b1) The economic loss (in EUR millions), in case of calling the bond, can be derived as follows:

- Call at 100% leads to pay out of original issuance volume of EUR 1,000 millions

- A non-call leaves, ceteris paribus, the bond at 77.42 or EUR 774.2 millions as a liability

=> Economic loss = EUR (1 – 0.7742) billion = EUR 225.8 millions (4 points)

[Note to the correctors: The answer “no loss” should also be accepted if motivated as follows. The question does not specify when mark-to-market profit/loss was recognized; so, based on mark-to-market, 225.8 million gain must have been recognized first and then calling the bond at 100% would cause 225.8 million loss and as a result, the total must be no loss, if everything happens during this fiscal year.]

b2) A spread over swap rates of 250bps would exactly match the 6% coupon thus leaving the bond at 100%, i.e. at this spread level the issuer is indifferent between calling and not calling.

(3 points)

b3) The interest expenses of a newly issued 7% bond amount to: 7% ∙ EUR 1 billion = EUR 70 million p.a. over the next 30 years.

The interest expenses of the (not called) perpetual bond after step-up amount to:(5% + 1%) ∙ EUR 1 billion = EUR 60 million p.a. over the next 30 years.

=> The present value of the incremental interest expenses of EUR 10 million p.a. amounts to:

(5 points)

b4) - Investors expecting a call might be reluctant to provide the bank with funding in the future

(reputational risk)

- Similar bond issues might decrease in value due to a non-call decision which could upset market participants who are invested in other bonds issued by the bank

- The bank might enjoy an excess regulatory capital and liquidity position and could therefore only be interested in minimizing the interest expenses at a lower balance sheet volume and shorter average tenor on the liability side

- The new coupon after step-up from 5% to 6% p.a. could be higher than the costs for shorter term liquidity which could still be sufficient from a liquidity risk perspective

- The non-call decision could be perceived by the market as a sign of weakness in terms of the bank not being able to raise the liquidity (and/or new regulatory capital) needed for redemption

(2 points for each bullet, max 4 points)


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c) c1) A yield spread is the difference between the yield to maturity on a particular bond and that on a Government bond of comparable maturity. Both brokers A and B quote the same price, therefore they quote the same yield spread.

The OAS measures the yield spread of a fixed income instrument that is not attributable to embedded options. It holds:OAS = spread – option value (in interest rate basis points).

Broker B quotes a lower volatility resulting in a lower call option price and, hence a higher OAS. (5 points)

c2) The convexity of callable bonds is in general lower than the convexity of bonds without a call feature. In case the callable bond price is close to its strike the convexity could even turn negative. (5 points)


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Question 2: Derivatives Valuation and Analysis (27 points)

a)

(4 points)

b) A currency swap is the exchange of one currency against another one at a determined price and to buy back the same amount of currency at a future date at a given price. The price difference expressed in pips or basis points show the interest rate differential between these two currencies from the spot to the forward date. As in this case both are equal, the spot and the forward/future price match, because both currencies bear the same interest rate. (5 points)

c)

contracts or 19,000,000 EUR (4 points)

A failure to adjust the hedge ratio after the interest rate shock would mean that the position is now no longer optimally hedged and the risk of the position will be higher than it would be in the case where the post shock hedge ratio is used. (3 points)

d)

(2.5 points)

(2.5 points)

Alternative solution [using the linear convention -money market use]:

(2.5 points)

(2.5 points)


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e) Here we have a major difference between the yield curve in EUR and the one in USD. The EUR yield curve is ascending where the USD one is descending. (2 points)

The main reasons why market participant anticipate these development are among other:- Business conditions (→ Money demand)- Money supply- Inflation expectations- Central Bank credit policies- Exogenous shock or turmoil (i.e. Sub prime crisis, Stock Market collapsing, Petrol Crisis)- Negative interest rate to avoid overvaluation of a currency like 1978 in Switzerland or

very high interest rate to avoid speculation against a currency like in 1995 within the EMS.

- Macroeconomic expectations.- ….

[Note to the correctors: Various other answers would be possible.](1 point per factor, max. 4 points)


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Question 3: Derivatives Valuation and Analysis (20 points)

a)

- Company A's default probability rose

- The expected recovery rate in the event of a default by Company A declined

- Company A's asset value declined

- The market CDS spread increased(2.5 points per factor, max. 5 points)

b) Assuming that there are no arbitrage opportunities in the market, the theoretical value of Company A's 5-year CDS spread is derived by subtracting the yield on 5-year government bond from the yield on Company A's 5-year corporate bond, or 3% - 1% = 2%. (4 points)

c) Since risky bond + CDS = risk free bond, Short Company A's 5-year corporate bond sell 5-year CDS protection for Company A invest in the 5-year government bond.

This results in -3% + 2.5% + 1% = 0.5%. (5 points)

d)

- Corporate bonds, government bonds and CDSs have differing degrees of liquidity (government bonds are the most liquid; corporate bonds the least; the liquidity of CDSs used to be in the middle, but has declined in recent years).

- Taxation is different for corporate bonds, government bonds and CDSs (in the United States, government bonds enjoy lower taxes).

- Corporate bonds cannot be shorted (or at least it is difficult to short them).

- Corporate bonds, government bonds and CDSs have different transaction costs (CDSs have collateral etc. as credit supplementation).

(2 points per reason, max. 6 points)

[Note to the correctors: The answer "because there are arbitrage opportunities" does not get any points.]


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Question 4: Portfolio Management (50 points)

a) Stratified sampling: - Partition the universe of benchmark stocks into cells based on the stock´s characteristics

(typically sector and size for national portfolios, and country and sector for international portfolios).

- Eliminating stocks below a threshold weight in each cell.- Distribute the weight of the eliminated stocks in each cell by overweighting the remaining

stocks (by way of a heuristic).(3 points)

Sampling by eliminating the stocks with smallest benchmark weight: - Rank the universe of benchmark stocks by benchmark weight.- Eliminate stocks below a weight threshold or simply eliminate a certain number of stocks

with the smallest weights.- Distribute the weight of the eliminated stocks in each cell by overweighting the remaining

stocks (by way of a heuristic).(3 points)

Optimised sampling:- Estimate the variance/covariance matrix of stock returns.- Solve the following optimization problem: minimize expected tracking error, subject to

weight constraint (portfolio weights have to sum up to 1) and a function, that limits the number of stocks.

- The constraint limiting the number of stocks is needed to prevent the optimizer from the trivial solution (full replication): alternative constraints can be formulated (e.g. based on liquidity requirements) that also prevent full replication.

(3 points)

b) Stratified sampling:

Advantages:- Ensures benchmark neutrality for most important risk factors, if characteristics for cell

formation are appropriately chosen, so tracking quality might be better compared to the second indexing approach.

- Rather simple approach, much less data requirements compared to optimised sampling.

Disadvantages:- More data requirements compared to second approach.- Risk control is limited to a small number of factors, which means that relevant risk factors

remain uncontrolled.- There is no ante tracking error estimate provided.

(3 points)


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Sampling by eliminating the stocks with smallest benchmark weight:

Advantages: - Very simple method with minimal data requirements.- Best benchmark overlap concerning single stock active weights for a given number of

stocks to be eliminated from the tracking portfolio, so tracking quality might be better than with other methods, if stock specific risks are a major driver of performance.

Disadvantages:- No account for systematic risk factors, the tracking portfolio might be heavily exposed to

important risk factors, e.g. sector biases.- Severe bias against small caps, which might be a systematic risk factor (creating a

systematic tracking error).(3 points)

Optimised sampling:

Advantages:Best tracking quality, as long as estimated variance/covariance matrix is not to far from the “true” matrix. The reasons are:- That we incorporate useful information, which other approaches do not use.- That quadratic programming solves the tracking error minimisation problem in

mathematical terms as it is, whereas the other approaches only indirectly (by posing restrictions) lead to low tracking errors.

Disadvantages:- Complex optimisation design, massive data requirements for estimating the covariance

matrix.- Less robust than other methods: if estimated covariance matrix contains errors, the

tracking solutions will react sensitively to these errors.(3 points)

c) The following statements should be discussed:

(1) On the basis of the “fundamental law of active management (FLAM)”, the information coefficient (=quality of forecasts, not identical to but could be proxied by the hit ratio) is as important for the expected information ratio as is the number of independent forecasts. With the market timing model, only 4 independent forecasts are produced in a given year, therefore the information coefficient must be very high to generate an attractive information ratio. In most cases, it makes sense to combine different alpha sources (as long as there is at least some forecasting quality), as this increases the number of independent forecasts. Therefore it should be questioned, whether the described investment approach is really compatible with the FLAM.

(2) One disadvantage of variance/covariance forecasts is, that estimation errors can seriously affect the quality of the optimization results – which is the opposite of being robust. Although the greater impact tends to be in the expected returns rather than risk dimension.


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(3) The portolio weights have to add up to 1, not 0.

(5) By increasing v, the solution is tilted in favour of avoiding risks. So the statement is wrong.

(6) The “earnings surprise” anomaly states, that typically stocks with positive surprises tend to outperform directly after publication.

(7) This statement should be at least moderated, to account for the uncertainty of the incremental returns. While it is possible, that a sophisticated options overwriting approach may provide expected outperformance on average, the outperformance clearly will be conditional on the market movements. In rapidly rising markets, part of the upside will not be captured with this approach, leading to likely underperformances versus the index in such a scenario.

(8) The success of this return enhancing method is dependent on the market direction. So, this approach is not free from return forecasts (at least implicitly) and should not be considered as very robust.

(2 points each, max.10 points)

d) Information ratio of M2:Annualized active return = 12 ∙ 0.25% = 3.0% p.a. (1 point)Annualized tracking error = 120.5 ∙ 0.9% = 3.1% p.a. (1 point)Information ratio (annualized) = 3.0% / 3.1% = 0.96 (1 point)

Information ratio of M1 = 3.3%/3% = 1.1 (2 points)

On first view, when only comparing the information ratio numbers, the track record of M2 does not seem to be attractive, as the information ratio is lower than what M1 can offer. That said, the track record of M2 is much longer. The track record of M1 with a length of 36 months may be too short to derive conclusive statements, whereas 108 months of track record for M2 are suitable to confirm the quality of this investment approach. (3 points)

e) Assets are divided into two parts:- passive „core“ portfolio - (very) active „satellites“ portfolio

Typically, the core part has large weight in the overall portfolio and is indexed to the strategic asset allocation of the investor (which may be derived from an asset liability perspective of this investor).

The satellites part is typically constructed of a number of very active portfolios that cover disjoint subsets of the overall benchmark of the investor.

(3 points)


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Advantages core/satellite approach:- Potential for reduction of overall management fees (as large core part pays passive fees).- Potential for finding specialised asset managers that provide better risk return profiles.- Suffers less from active positions that cancel each other out.

(3 points)

Disadvantages core/satellite approach:- Problem of asset allocation between sub-portfolios is left with the investor, whereas in the

„generalist“ solution the external managers provide their services in this important field, too. (3 points)

f) Conclude, that in the alternative portfolio the weight of Al3 has to be 50% (in order to cover the bond exposure of 50%).

Equate the active return of alternative portfolio to 1.5%:(2 points)

Take into account, that x1 + x2 + 0.5 = 1 (1 point)

Solve for x1 and x2:

(about 1%)

The weight of Al1, Al2, and Al3 should be 1%, 49%, and 50%, respectively. (2 points)


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Question 5: Derivatives and Derivatives in Portfolio Management (40 points)

[Note to the corrector:The question does not specify whether the returns in this question are measured on a simple or continuously compounded basis. The proposed solution only shows the result using simple returns; however, both interpretations should be accepted. As a consequence, please be careful during your correction and accept both as correct.]

The following notations are used in the solutions:S0: Nikkei average at Time 0 (present)ST: Nikkei average at Time T (3 months later)

: Futures price at Time 0 (present)CK: call option price with strike price at Time 0PK: put option price with strike price at Time 0

a) We have

(or )Therefore futures price = 10,100 yen.

(4 points)

b) A futures and (European) options with the same underlying asset and maturity will, under no-arbitrage conditions, satisfy the put-call parity: . However, this does not hold true under the given conditions.From ,we know that there is an arbitrage opportunity. (4 points)

One example of an arbitrage trade is to sell 1 trading unit of the (overpriced) call option and purchase 1 trading unit of a put option while at the same time taking a 1 trading unit long position in the futures and lending 99,000 yen [= (10,100 - 10,000) / 1.01 ∙ 1,000 yen] for 3 months at an annualized interest-rate of 4%. In this transaction, the payoffs will offset each other to zero at maturity, and at the current point in time an arbitrage profit of 21,000 yen will be earned.

TODAY POSITION Index Points JPY

1. Sell call with strike 10,000 +665 +665,000

2. Buy put con with strike 10,000 – 545 – 545,000

3. Buy Nikkei future @ 10,100

4. Lend 1’000∙(10,100-10,000)/1.01 JPY – 99 – 99,000

Total: + 21 + 21,000


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IN 3

MONTHS

Index Points

ST ≤10,000 ST > 10,000

1. Sell call with strike 10,000 0 – (ST –10,000)

2. Buy put con with strike 10,000 (10,000 – ST) 0

3. Buy Nikkei future @ 10,100 ST–10,100 ST–10,100

4. Repayment of

1,000∙(10,100-10,000)/1.01 JPY 100 100

Total: 0 0(6 points for an example)

c) As one example, there is a long straddle formed by purchasing 1 trading unit each of put and call options with strike prices of 10,000. The cost of buying both options to create the position is There will be a loss if the Nikkei average does not change much from the current price, but either a large decline or a large rise will cause the price of the in-the-money option to rise, earning you a profit.

At , the payoff diagram below illustrates the payoff at maturity (per 1/1,000 trading unit) of the total position derived by borrowing funds to purchase options at the current point in time and repaying them at maturity. This position will be profitable if the price of the underlying asset (the Nikkei average) experiences a large move either upwards or downwards.

(5 points for the proposal)

(3 points for the graph)


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10,000

Value of the Nikkei average at maturity

Payoff

A

0

d) The fund's value is always 100,000 times the Nikkei average (because it has the same composition as the Nikkei average). However, 1 trading unit of the option is 1,000 times. Therefore, to set the required floor, it is necessary to purchase 100 put options (= 100,000/1,000).

The cost of purchasing 100 trading units of put options at the strike price is If this is borrowed at the risk-free rate and repaid in 3 months, expenses at

the end of 3 months will be Therefore, the floor set by this trade is

(5 points for the calculation)

(2 points for the graph)

e) Selling call options and investing the proceeds at the risk-free rate will increase the floor at the end of 3 months. Obviously, this is a gain in the floor at the expense of the returns to be earned if the Nikkei average increases, and compared to d), the value of the position is smaller when the Nikkei average is high. (5 points)

f)

The futures delta is: because there are 3 months to maturity.

The put option delta is .

To synthesize 1 unit of put options, a position of units in the

futures is required. You must purchase 100 trading units of the put option to create a floor, so multiplying by 100 gives a short futures position of 19.8 trading units.

(4 points)


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Position value

Value of the Nikkei average

K

Regarding position adjustments, to build the floor it is necessary to increase sales of futures when the Nikkei average declines and to buy them back when it increases. (More accurately, if we assume the put option gamma to be positive, a decline in the put option will cause the delta to decline (the absolute value increases, which increases the degree of negativity), making it necessary to increase sales of the futures to maintain the delta.) (2 points)


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Ciia Final Exam II March11_se

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