The 2013 Center for Financial Studies

Modeling Competition

Dongyang Wang (661181660)

Yulin Li (661181660)

Wenhua Xiao (661128134)

Jingli Li (661138638)

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Contents1. Introductions:......................................................................................................................................2

2. Ascertain the Importance of the Anchors to Real Estate Portfolio (shopping center).........................2

3. Value lease obligations based on the information given and prevailing market conditions................3

4. Measure the Risk and Probability of Default.......................................................................................4

5. Hedging Strategies...............................................................................................................................6

6. Recommend Specific Derivatives.........................................................................................................7

7. Additional Risks Based on Hedging Strategies:....................................................................................9

8. Conclusions:.......................................................................................................................................10

Appendix...............................................................................................................................................11

1. Introductions:

This case’s target is to hedge the credit risk faced by Muck River Plaza because of there exist potential probability of default of two anchors in this plaza, Best Buy and Barnes & Noble. Beyond those anchors’ risk, other small tenants’ rent are mostly depend on the anchors behaviors and their rents are much higher than anchors’ rents, which is roughly 75% of the total rents. Our target is to quantify these risks and then devise proper strategy to hedging these risks efficiently. We use both accounting skills and computation skills to do the credit risk management and list some proper strategies Muck River Plaza may utilize to construct the portfolio and management the real estate.

2. Ascertain the Importance of the Anchors to Real Estate Portfolio (shopping center).

For the operation of the major anchors, we consider two scenarios for the Muck River Associates: good case and bad case. Good case represents that two major anchors, Best Buy and Barnes & Noble, run successfully and pay for the rent and surcharges and small anchors also rent the space as much as possible, while bad case represents that all anchors run into the worst case. Specifically, the two anchors, Best Buy and Barnes & Noble, will argue with Muck River Associate on interests for lease at most 5 dollars instead of 8 dollars. For the first year, the revenue, cost and profit are calculated and showed in the Table 1.

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Table 1 Profit of The Real Estate PortfolioScenarios Good Case Bad CaseTenants Major Anchors Small Anchors Major Anchors Small AnchorsRevenue $640,000 $1,760,000 $430,000 $647,000

Cost $950,000 $950,000Profit $1,450,000 $127,000

Compared with these two results, we find that the profit difference is $1,323,000, which means the profit of the bad case decreased by 91.24% of the profit under the good case. It demonstrates that the performance of both major anchors is important to the real estate portfolio.

3. Value lease obligations based on the information given and prevailing market conditions.

In this section, it is important to evaluate the lease obligations based on the scenarios: Best Buy and Barnes & Noble boom, cease or limit operations, and bankrupt. Correspondingly, the small anchors are combined with the situations of the major anchors.

We believe that for small anchors, they will stay boom or bankrupted when the major anchors go boom or bankrupted because for example major anchors, Best Buy and Barnes & Noble, attract people to go shopping and then the small anchors will benefit from it.

The Table 2 explains the expected profits for the Muck River Associate when two major anchors, Best Buy and Barnes & Noble go bankrupted while small anchors stay between boom and bankrupted and go bankrupted respectively.

Table 2 Expected Profit Under Bankruptcy For Two Major AnchorsScenarios Bankrupted

Expected Return $-814,771

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The Table 3 shows the expected profits for the Muck River Associate when two major anchors, Best Buy and Barnes & Noble cease or limit operations while small anchors boom, stay between boom and bankrupted and go bankrupted respectively

Table 3 Expected profit under ceased or limited operation in two major anchorsScenarios Stay between boom and bankrupted

Expected Return $176,484

The Table 4 shows the expected profits for the Muck River Associate when two major anchors, Best Buy and Barnes & Noble boom while small anchors boom, stay between boom and bankrupted and go bankrupted respectively. This is a little bit better scenario than that one in Table 3.

Table 4 Expected profit under boom in two major anchorsScenarios Stay between boom and bankrupted

Expected Return $328,928

4. Measure the Risk and Probability of Default

The risks faced by Muck River based on the lease payment uncertainties are the credit default risk and operational risk of the anchors. In this project, we use the KMV-Merton model to calculate the probability of default. The KMV-Merton default forecasting model produces a probability of default for the two firms at any given point in time. To calculate the probability, the model subtracts the face value of the firm’s debt from an estimate of the market value of the firm and then divides this difference by an estimate of the volatility of the firm (scaled to reflect the horizon of the forecast). The distance to default is then substituted into a cumulative density function to calculate the probability that the value of the firm will be less than the face value of debt at the forecasting horizon.

The KMV-Merton model employs the Merton bond pricing model. KMV model is based on the structural approach to calculate probability of default. KMV model is best when it is applied to publicly traded companies, where the value of equity is determined by the stock market. It translates firm’s stock price and balance sheet to an implied risk of default. It works best in highly efficient liquid market conditions. One assumption for Merton model is that the total value of the firm follows geometric Brownian motion(equation 1), whereV is the total value of the firm, μ is the expected continuously compounded return on V , σ V is the annualized firm value volatility, and dW is a standard Weiner process.

dV=μVdt+σ VVdW (1)

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What’s more, we price the value of equity using Black Scholes Merton formula. According to put-call parity, the value of the firm’s debt is equal to the value of a risk-free discount bond minus the value of a put option written on the firm, again with a strike price equal to the face value of debt and a time to maturity of T (see in equation (2)), where E is the market value of the firm’s equity, F is the face value of the firm’s debt, ris the instantaneous risk-free rate, N ( . )is the cumulative standard normal distribution function:

E=V N (d1 )−e−rT F N (d2 )(2)

d1=ln(VF )+(r+ σ V2

2 )Tσ V √T

d2=d1−σV∗√T

Because Merton’s model assumes the value of equity is a function of the value of the firm and time, it follows from Ito’s lemma:

σ E=(VE

) ∂ E∂V

σV

In the Black Scholes Merton model, we can get∂ E∂V

=N (d1 ), where d1is already defined above.

So, combining these two equations, we get the volatilities of the firm’s value and its equity are related by equation (3). E=current stock price∗sharesoutstanding. σ E is the volatility of equity. σ V is the volatility of firm value. V is the total value of the firm.

σ E=(Vc )N (d1 )σV (3)

From the equations above, we get the values of σ V and V , which are the volatility of firm value and the total value of the firm respectively. Then the distance to default can be calculated as equation (4), where μ is the expected return on firm’s value.

DD=ln(VF )+(μ−σ V2

2 )Tσ V √T

(4 )

So the corresponding implied probability of default is calculated in equation (5).

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PD=N (−ln(VF )+(μ−σV2

2 )TσV √T )=N (−DD )(5)

The most critical inputs to the model are clearly the market value of equity, the face value of debt, and the volatility of equity. As the market value of equity declines, the probability of default increases. This is both a strength and weakness of the model. For the model to work well, both the Merton model assumptions must be met and markets must be efficient and well informed.

In its promotional material, KMV points to the Enron case as an example of how their method is superior to that of traditional agency ratings. When Enron’s stock price began to fall, its distance to default immediately decreased. The ratings agencies took several days to downgrade Enron’s debt. Clearly, using equity values to infer default probabilities allows the KMV-Merton model to reflect information faster than traditional agency ratings. However, when Enron’s stock price was unsustainably high, KMV’s expected default frequency for Enron was actually significantly lower than the default probability assigned to Enron by standard ratings. If markets are not perfectly efficient, then conditioning on information not captured by π KMVprobably makes sense.

Table 5 Results from KMV Model (B: billion dollars)

Firm NameTotal Value of

Firm (V )Volatility of Total

Value (σ V)Distance of

Default (DD)Probability of Default (PD)

Best Buy 24.7289 B 0.3280 0.8960 0.1851

Barnes& Noble

2.7813 B 0.2266 -0.3097 0.6216

Table 6 Key Statistic for Best Buy and Barnes& Noble (B: billion dollars)

Firm NameDebt(F)

Value of Equity (E)

Volatility of Equity (σ E)

Expected Return on Firm Value (μ)

Risk Free Rate(r)

Time to Maturity(T )

Best Buy 12.26 B 14.66 B 0.5159 0.04490.0251 5Barnes&

Noble2.44 B 0.868 B 0.5635 -0.0319

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5. Hedging Strategies

Through our former analysis, we know that Best Buy has an impressive performance during last year and Barnes & Noble shut down a bookstore in July after more than 12 years. We try to hedge all these risks and the further risks derive from smaller sales volume. Small Tenants’ rent fees are generally higher then anchors and they also affects Muck River Plaza’s profit tremendously.

Based on the principle of simplicity and effectiveness, we want to find two kinds of derivatives to hedge two anchors’ risks separately. The advantages of using derivatives are: (1) Trading actively in the derivative market and easy to buy or sell depends on different strategies; (2) Derivatives have higher leverage ratio and the cost of strategy will be less;(3) Two derivatives are the simplest strategy to hedge credit default risk; (4) We can simply expand the numbers of derivatives contract to hedge the additionally risk derive from smaller tenants.

6. Recommend Specific Derivatives

After searching the market and collect the historical data, we found that CDS (Credit Default Swap) is the best way hedge the credit risk in this situation. Best Buy have an impressive performance on its equity in last year, it increased from $15 to $42 in this period. Its 5-year CDS is 227bp right now, which implies roughly an 18% chance of default in 5 years. 1 year ago, its 5-year CDS is about 890bp, which implies roughly a 50% chance of default in 5 years. Consequently, we just need to pay 2.27% of the notional amount annually to hedge the credit risk of Best Buy.

But the condition of Barnes & Noble is totally different. Barnes & Noble shut down a bookstore in this July after more than 12 years and its stock prices fluctuated upside down in this year. Essentially, Barnes & Noble doesn’t offer CDS to the market and then we can’t use the same strategy as Best Buy. After doing a lot of research of Barnes & Noble’s stock, we find out that its expected return on equity is -19%. If it keeps going down in the next year, Barnes & Noble may expand its shutdown plan in the future. Our Strategy is simple, buy the put option of Barnes & Noble’s stock in one year and using it to hedge the risk.

These Tables below are the calculation results based on the accounting method and we use these risk expose to calculate the number of contracts we need to buy. Furthermore, we also measure the risks derive form the small tenants and divide them equally to two company’s derivative by expanding the numbers of contracts. But it’s easily to find that the probability of default for Barnes & Noble is much higher than the probability of default for Best Buy, and that means we need to pay more to hedge the risk exposure of Barnes & Noble.

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Table 7 The Risk Exposure for Two AnchorsYear 1 Year 2 Year 3 Year 4 Year 5

Cash Flow (Normal) 320,000 333,200 346,400 361,600 376,800Cash Flow (Bankrupt) 64,500 68,460 72,420 199,480 204,040Risk Exposure 255,500 264,740 273,980 162,120 172,760

Table 8 The Risk Exposure for Other TenantsYear 1 Year 2 Year 3 Year 4 Year 5

Cash Flow (Normal) 1,760,000 1,786,400 1,812,800 1,843,200 1,873,600Cash Flow (Bankrupt) 352,000 357,280 362,560 368,640 374,720Risk Exposure 1,408,000 1,429,120 1,450,240 1,474,560 1,498,880

For Best Buy, we just need to pay 2.27% of the notional amount annually to hedge the credit risk. In this case our notional amount is $250,000 and then the price of Best Buy CDS is $5,675. Finally, multiplying a scalar of 3 to hedging additional risks derives from the small tenants and then we need to pay $22,700 totally.

For Barnes & Noble, we consider three put options expire in Jan 2015. Their strike price and option prices respectively are $15, $17, $20 and $3.42, $5.4, $7.6. The stock price of Barnes & Noble is $14.57 right now and then the risk premiums of these put options are different. Based on the principle of simplicity and effectiveness, we prefer the one with lower risk premium and then we choose the put option with strike price equal to $20. The expected of return on equity is -19% and our estimation of return on equity for Barnes & Noble is -40%. Then we use the risk exposure divided by estimated profit of our put option to calculate the numbers of derivatives we need to use to hedge the credit risk. Same procedure in CDS of Best Buy, the price of Barnes & Noble Put Option is $334,400. Finally, multiplying a scalar of 3 to hedging additional risks derives from the small tenants and then we need to pay $1,337,600 totally. The results are shown in Table 9.

Table 9 Dollar Values of Derivative ContractsAnchors' Risk Small Tenants’ Risk Total Risk

BBY CDS 5Y 5,675 17,025 22,700BKS Put(K=20) 1Y 334,400 1,003,200 1,337,600

Interpreting the results of hedging portfolios:

(1) CDS cost less to hedge the credit risk than put options;

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(2) The probability of default of Best Buy is smaller and that results the cost of hedging portfolio be lower in this year;

(3) The shutdown news of Barnes & Noble makes us pay more attentions to its credit risk and it also increases the cost of put options;

(4) Higher strike price of option price increases the cost of hedging portfolio but we can get most of them at the maturity because our option is deep in the money;

(5) The times to maturity of two derivatives are different and then we need to choose another put option after 1 year depends on Barnes & Noble’s performance at that time.

7. Additional Risks Based on Hedging Strategies:There are three risks in our Hedging Strategies: The first one is the residue risk exposure because the imperfect hedging; the second one is the interest rate risk based on principles we have paid on the derivatives; the third one is the operational risk derive form the Plaza’s business strategy.

In the first situation, we buy 44,000 contracts of put options of Barnes & Noble. But actually we just need to buy 43,840 contracts of put options. As the result show, we pay additional $1,215 in put options and that may produce additional risk. For the CDS of Best Buy, the notional amount of CDS should be 255,000 and our notional amount is 250,000. As the result show, we still need to pay another $125 to hedge the credit risk of Best Buy but we can’t do that because the imperfect hedging.

In the second situation, we pay $1,360,300 to construct hedging portfolio, which will lead to interest risk for this 1 year period. If the interest rate is 2.66%, the interest during 1year should be $36,670 at the interest rate exposure. But these risks are acceptable for Muck River Plaza because the payments are annually and then the interest rate risk is limited.

But these two additional risks are hard to be eliminated in this scenario. Our final target is to hedge the credit risk, and the additional risk exposure is roughly 3.01% of the total amount of principle of our hedging strategy.

The third one is the operational risk derive form Muck River Plaza’s business strategy. Best Buy and Barnes & Noble are kind of competitors to each other. Best buy are electronic store and Barnes & Nobles is bookstore. But actually they are all selling videos, music CD, etc. Essentially, Barnes & Noble’s selling their new product, NOOK, which is a kind of e-reader. This is a one source of major profit of Barnes & Noble and there are a lot of similar electronic devices, such as iPad, sold in Best Buy. For some extent, they are competitors and they depress the each other’s sales. So our suggestion to eliminate this risk is to substitute the Barnes & Noble to another different anchor, such as Wal-Mart, Macy’s or IKEA. We prefer to keep Best Buy because it has a pretty good performance in its stock market and it just needs a little bit money to hedge its default risk. After this substitution the cost of our hedging will be lesser and we can receive more payment from the anchors, because of they are diversified, and also more payment form other tenants because the volume of costumers increases.

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8. Conclusions:

Beside to consider the impact value of the credit risk management, the cost to manage the risk also must be balanced. The costs can be measured in actual monetary values. In this project, the implementation cost associated with our risk management strategy is $1,360,300. We use the credit default swap to hedge the default risk of Best Buy lease. The cost to manage this risk is 22,700. The cost is very low for the small probability of default based on the great performance of Best Buy business in these years. What’s more, we can see that the credit default swap used to hedge the risk is perfectly matched which makes us manage this risk easy. Furthermore, as no credit default swap is available for us, we use the put options to hedge the default risk of Barnes & Noble lease. The cost to manage this risk is 1,337,600. The cost is much higher than the cost to hedge the default risk of Best Buy lease. In order to balance the cost effectively, we buy the deep-in-the money put options to hedge the default risk. Deep-in-the money put option has the lower premium because the time value is small. After considering these factors for the risk management costs, we believe that the cost to manage the risk can be balanced against the impact value.

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Appendix

1. Codes of KMV Model

function [ F ] = cfsfun1(x,D,E,rf,sigma_E,T) d1=(log(x(1)/D)+(rf+x(2)^2/2)*T)/(x(2)*sqrt(T));d2=d1-x(2)*sqrt(T);F=[E-x(1)*normcdf(d1,0,1)+D*exp(-rf*T)*normcdf(d2,0,1);... sigma_E*E-x(1)*normcdf(d1,0,1)*x(2)]; endclear all;close all;cd 'C:\Users\LYL\Documents\financial computation';addpath 'C:\Users\LYL\Documents\financial computation'; %%%bestbuyD=12.26;E=14.66;rf=0.0251;sigma_E=0.515921506;T=5;miu=0.0449;x0=[1 1]; options=optimset('Display','iter'); % Option to display output[x,fval] = fsolve(@(x)cfsfun1(x,D,E,rf,sigma_E,T),x0,options); V=x(1,1);sigma_v=x(1,2); DD=(log(V/D)+(miu-(sigma_v^2)/2)*T)/(sigma_v*sqrt(T));p_default=normcdf(-DD,0,1);clear all; cd 'C:\Users\LYL\Documents\modeling';addpath 'C:\Users\LYL\Documents\modeling';%%%barnesD=2.44;E=0.868;rf=0.0251;sigma_E=0.563545029;T=5;miu=-0.0319;x0=[1 1]; options=optimset('Display','iter'); % Option to display output[x,fval] = fsolve(@(x)cfsfun1(x,D,E,rf,sigma_E,T),x0,options); V=x(1,1);sigma_v=x(1,2); DD=(log(V/D)+(miu-(sigma_v^2)/2)*T)/(sigma_v*sqrt(T));

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p_default=normcdf(-DD,0,1);

2. Best Buy 1 Year CDS Historical Data:

6/19/2

008

9/25/2

008

1/1/2

009

4/9/2

009

7/16/2

009

10/22/2

009

1/28/2

010

5/6/2

010

8/12/2

010

11/18/2

010

2/24/2

011

6/2/2

011

9/8/2

011

12/15/2

011

3/22/2

012

6/28/2

012

10/4/2

012

1/10/2

013

4/18/2

013

7/25/2

0130

100

200

300

400

500

600

Figure 1 BEST BUY 1 Year CDS From 2008 to 2013

BBY 1 Year CDS

3. Comparison between Best Buy’s and Barnes & Noble’s Stock Prices

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