2013 CFS Modeling Competition

1. The 2013 Center for Financial Studies
Modeling Competition
Dongyang Wang (661181660)
Yulin Li (661181660)
Wenhua Xiao (661128134)
Jingli Li (661138638)
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2. Contents
1. 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
potentialprobability 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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3. Scenarios
Tenants
Revenue
Cost
Profit
Table 1 Profit of The Real Estate Portfolio
Good Case
Bad Case
Major Anchors
Small Anchors
Major Anchors
Small Anchors
$640,000
$1,760,000
$430,000
$647,000
$950,000
$950,000
$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 ForTwo Major Anchors
Scenarios
Bankrupted
Expected Return
$-814,771
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4. 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 anchors
Scenarios
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 anchors
Scenarios
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), where is the total value of
the firm, is the expected continuously compounded return on ,
is the annualized firm value
volatility, and
is a standard Weiner process.
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5. 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 (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, is the instantaneous risk-free rate,
is
the cumulative standard normal distribution function:
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:
In the Black Scholes Merton model, we can get
, where
is 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).
. is the volatility of
equity. is the volatility of firm value. is the total value of the firm.
From the equations above, we get the values of and , 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 is the expected return on firm’s value.
So the corresponding implied probability of default is calculated in equation (5).
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6. 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
probably makes sense.
Table 5 Results from KMV Model (B: billion dollars)
Firm Name
Best Buy
Barnes&
Noble
Total Value of
Firm( )
24.7289 B
Volatility of Total
Value( )
0.3280
Distance of
Default ( )
0.8960
Probability of
Default( )
0.1851
2.7813 B
0.2266
-0.3097
0.6216
Table 6 Key Statistic for Best Buy and Barnes& Noble (B: billion dollars)
Firm Name
Debt
( )
Value of
Equity( )
Volatility of
Equity( )
Expected Return
on Firm Value ( )
Best Buy
12.26 B
14.66 B
0.5159
0.0449
Barnes&
Noble
2.44 B
0.868 B
0.5635
Risk Free
Rate( )
Time to
Maturity( )
-0.0319
0.0251
5
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.
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7. 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 5year 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.
Table 7 The Risk Exposure for Two Anchors
Year 1
Year 2
320,000
333,200
Cash Flow (Normal)
64,500
68,460
Cash Flow (Bankrupt)
255,500
264,740
Risk Exposure
Year 3
346,400
72,420
273,980
Year 4
361,600
199,480
162,120
Year 5
376,800
204,040
172,760
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8. Table 8 The Risk Exposure for Other Tenants
Year 1
Year 2
1,760,000
1,786,400
Cash Flow (Normal)
352,000
357,280
Cash Flow(Bankrupt)
1,408,000
1,429,120
Risk Exposure
Year 3
1,812,800
362,560
1,450,240
Year 4
1,843,200
368,640
1,474,560
Year 5
1,873,600
374,720
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 premiumsof 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 Contracts
Anchors' Risk
Small Tenants’ Risk
5,675
17,025
BBY CDS 5Y
334,400
1,003,200
BKS Put(K=20) 1Y
Total Risk
22,700
1,337,600
Interpreting the results of hedging portfolios:
(1) CDS cost less to hedge the credit risk than put options;
(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.
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9. 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& Nobleand 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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10. 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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11. 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)];
end
clearall;
closeall;
cd'C:UsersLYLDocumentsfinancial computation';
addpath'C:UsersLYLDocumentsfinancial computation';
%%
%bestbuy
D=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);
clearall;
cd'C:UsersLYLDocumentsmodeling';
addpath'C:UsersLYLDocumentsmodeling';
%%
%barnes
D=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);
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12. DD=(log(V/D)+(miu-(sigma_v^2)/2)*T)/(sigma_v*sqrt(T));
p_default=normcdf(-DD,0,1);
2. Best Buy 1 Year CDS Historical Data:
Figure 1 BEST BUY 1 Year CDS From 2008 to
2013
600
500
400
300
200
BBY 1 Year CDS
100
0
3.Comparison between Best Buy’s and Barnes & Noble’s Stock Prices
12