Financial Risk Management

Convertible BondsAn example of how to determine value of company by capital

structure

By Philip CorsanoGnostam Consulting

www.gnostamconsulting.comTel 206 384 0069

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Firm Capital StructureModigliani & Miller, [“MM”] proposition 1: Value of firm

is independent of capital structure, [i.e. you don’t create value by capital structure alone, in absence of tax effects];

Modigliani & Miller: It does not matter what risk preferences are for investors.

Assume Investors have the ability to borrow and lend for their own account (and at the same rate as firms) so that they can “undo” any changes in firm’s capital structure

Prepared by Philip Corsano, Gnostam Consulting Training, Tel: 206 384 0069, pcorsano@gmail.com

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M&M Proposition 2Bonds are almost risk-free at low debt levels

rD is independent of leveragerE increases linearly with debt-equity ratios and the

increase in expected return reflects increased risk

As firms borrow more, the risk of default risesrD starts to increaserE increases more slowly (because the holders of

risky debt bear some of the firm’s business risk)

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Leverage and Returns

securities all of uemarket val

income operating expectedr assets on return Expected a

EDA r

ED

Er

ED

Dr

5

r

DE

rD

rE

M&M Proposition II

rA

Risk free debt Risky debt

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Capital StructurePV of Tax Shield =

(assume perpetuity)

D x rD x Tc

rD

= D x Tc

Example:

Tax benefit = 1000 x (.10) x (.40) = $40

PV of 40 perpetuity = 40 / .10 = $400

PV Tax Shield = D x Tc = 1000 x .4 = $400

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Capital StructureFirm Value =

Value of All Equity Firm + PV Tax Shield

Example

All Equity Value = 600 / .10 = 6,000

PV Tax Shield = 400

Firm Value with 1/2 Debt = $6,400

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U.S. Tax CodeAllows corporations to deduct interest payments

on debt as an expense

Dividend payments to stockholders are not deductible

Differential treatment results in a net benefit to financial leverage (debt)

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U.S. Tax CodePersonal taxes bias the other way (toward equity)

Income from bonds generally comes as interest and is taxed at the personal income tax rate

Income from equity comes partly from dividends and partly from capital gains

Capital gains are often taxed at a lower rate and the tax is deferred until the stock is sold and the gain realized.

If the owner of the stock dies – no capital gain tax is paid

On balance, common stock returns are taxed at lower rates than debt returns

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U.S. Tax Rates 2013Top bracket (over $450,000 for a married

couple)Personal rates: 39.6%Capital gains: 20% [+3.8% investment income

surtax for high income earners] (holding period of <12 mos, otherwise taxed at marginal income tax rate)If stock is held for less than 1 year capital gain is

taxed at the personal rateIf stock is held for over 1 year capital gains tax is

between 20 - 23.6%, but can be less if earn less that $15,000 in taxable income.

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Financial Distress

Debt/Total Assets

Mar

ket V

alue

of

The

Fir

m

Value ofunlevered

firm

PV of interesttax shields

Costs offinancial distress

Value of levered firm

Optimal amount of debt

Maximum value of firm

Financial ChoicesTrade-off Theory - Theory that capital structure is

based on a trade-off between tax savings and distress costs of debt.

Pecking Order Theory - Theory stating that firms prefer to issue debt rather than equity if internal finance is insufficient.

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M&M with taxes and bankruptcy

WACC now is more hump-shaped (similar to the traditional view – though for different reasons).

The minimum WACC occurs where the stock price is maximized.

Thus, the same capital structure that maximizes stock price also minimizes the WACC.

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Introduction to Covertible Bonds

A convertible bond = standard corporate bond with an option (to buy the underlying equity of the company).

Conversion feature allows holder of the bond to convert or exchange the bond into a predetermined number of shares of common stock (known as the conversion ratio).

A convertible bond [“CB”] is sensitive to the interest rate (corporate yield curve), [duration and convexity], the credit spread over the treasury rate [credit risk] as well as the volatility of the underlying equity.

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Capital Structure and Financial Distress

Costs of Financial Distress - Costs arising from bankruptcy or distorted business decisions before bankruptcy.

Market Value = Value if all Equity Financed

+ PV Tax Shield

- PV Costs of Financial Distress

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Advantage of convertible is that can issue shares at conversion price which is above “current” share price. This is also referred to “premium” above current price;

Reduces dilution, [because of premium];

Access to investor segment normally precluded from equity risk, attractive for many bond investors;

Lower straight coupon for issuer, [because of conversion option]

Less impact on P & L statement than equity because of tax deductibility of interest on bond.

Companies: Why Issue a CB?

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Investors: Why they Need to Buy CB?

A CB offers lower risk, [much less volatile than equity issue];

It has a built in protection in a risky market;

A CB has a higher running yield than a share dividend;

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Callable CB: A callable CB allows an issuer to buy back the bond some time prior to the maturity at a pre-determined price. A “soft call” means that the issuer can only call the bond if the price of the underlying stock is above the strike price by at least a certain percentage;

Puttable CB: A puttable CB means that the investor can sell the CB back to the issuer within a certain timeframe before the maturity of the CB at a certain price; a put option raises the value of the CB;

Resettable CB: If the strike price is resettable, CB investors can gain additional exposure to the equity component; if the price of the underlying stock falls, the parity value of the CB falls as well and therefore by resetting the strike price, or raising the conversion ratio, the CB’s parity value increases. (Example: CBs issued by Japanese corporations in the mid-90s; these can be analyzed by path dependent options)

Types of Convertible Bonds

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Convertible Bond Pricing Model

CallIVCB

sri

i

Par

i

CIV

t

n

tt

)1()1(1

tTdd

tT

tTqrKSd

KdNeSedNCall tTrtTq

12

2

1

2)()(

1

2ln

*

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Conversion Ratio & Conversion Price

Conversion ratio: # shares of common stock that the bondholder will receive from exercising the “call” option of CB; conversion privilege may extend for all or only some portion of the bond’s life, and the stated conversion ratio may change over time (it is always adjusted proportionately for stock splits and stock dividends).

EXAMPLE: JBB Corp issued a convertible bond with a conversion ratio of 25.32 shares. The par value of the bonds is $1000. This means that for each $1000 of the par value of this issue the bondholder exchanges for JBB common stock, he will receive 25.32 shares;

Stated Conversion Price = (Par Value of the CB)/(Conv Ratio)

= $1000/25.32

= $39.49 conversion price per share, above stock price at issue date.

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Strike Price

Further suppose that the JBB convert has a maturity of 5 years, coupon of 6% per annum (payable annually) and that the current risk free rate is 2.5%; the CB has no dividend yield and the credit spread is zero;

This will give the Investment Value (IV) of the CB as $1,162.60 (discounting for 5 years at the risk free rate of interest);

The Strike Price, K of the CB is therefore equal to $45.92 and is found out by:

K = (CB’s Investment Value)/(Conversion Ratio)

= $1,162.60/25.32

= $45.92

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JBB Convert Pricing(See Spreadsheet for details)

Dividend yield 0%

Stock price $40.00

Stock volatility 20%

Face value $1,000.00

Coupon 6.00%

Risk free 2.50%

Spread 0.00%

R + S 2.50%

1 $60.00 0 $60.00 $58.54

2 $60.00 0 $60.00 $57.11

3 $60.00 0 $60.00 $55.72

4 $60.00 0 $60.00 $54.36

5 $60.00 $1,000.00 $1,060.00 $936.89

Bond (Investment) Value $1,162.60

Strike Price $45.92

Call Value $6.87

Convert Value $1,169.47

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Complications: Call Provisions

Almost all CB issues are callable by the issuer;

Typically there is a “non-call” period from the time of issuance. During this time if stock goes above the conversion price by a sufficient premium, should convert, otherwise hold convertible as allows for accrual and collection of fixed coupons;

Some issues have a provisional call feature that allows the issuer to call the issue during the non-call period if the stock reaches a certain price;

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Convertible Valuation as Stock-plus Method

Can value a CB as a combination of an issuer’s stock, with a relatively high yield, plus a European put option;

Instead of viewing a CB as a fixed income instrument with an embedded call option, because of its convertible feature one can think of it as a stock with a yield greater than its dividend, and discount this “higher” dividend at appropriate “rate”;

The Investment Value can be looked upon as a floor, [“put], or ability to sell a put on company assets = to credit worthiness of assets coverage, [only is net asset value of company covers value of outstanding corporate debt]; the stock value is simply the conversion value (stock price multiplied by the conversion ratio) and the put value represents the fixed income value of the convertible.

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Binomial Tree for Convert Pricing

t = 0 t = 1 t =2 t = 3 t = 4

108.7282.50%

84.6779 2752.9862.50%

65.948 2212.1043.16% 65.948

1750.105 2.50%51.361 51.361 1669.7983.91% 4.02%

1388.061 40.00 1311.17640.00 4.88% 40.000

4.54% 1076.826 6.00%1118.940 31.152 31.152 1000.000

5.37% 6.00%919.140 24.261614 927.743

6.00% 24.262860.708 6.00%

18.895 1000.0006.00%

927.74314.7166.00%

1000.000

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Black-Scholes Framework for Convert Valuation

Coversion ratio 28.993

Dividend yield 0%

Stock price $50.75

Stock volatility 20%

Face value $1,000.00

Coupon 10.25%

Risk free 5.00%

Spread 0.00%

R + S 5.00%

1 $102.50 0 $102.50 $97.62

2 $102.50 0 $102.50 $92.97

3 $102.50 0 $102.50 $88.54

4 $102.50 0 $102.50 $84.33

5 $102.50 $1,000.00 $1,102.50 $863.84

Bond (Investment) Value $1,227.30

Strike Price $42.33

Call Value $19.39

Convert Value $1,246.68

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Binomial Pricing Model - continued

3,585.35

$224.08 $3,585.35 1.00

2,656.09

166.01

1,967.68

122.98 1,967.68

1,457.70 1,457.70 $122.98 $1,967.68 1.00

91.11 91.11

1,079.89 1,079.89 1,079.89

67.49 67.49 $67.49 $1,079.89 1.00

800 800.00 800.00

$50.00 50.00 50.00

592.65 592.65 592.65

37.04 37.04 $37.04 $1,060.00 0.00

439.05 439.05

27.44 27.44

325.26

325.26 $20.33 $1,060.00 0.00

20.33 240.96

15.06

178.50

$11.16 $1,060.00 0.00

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Convertible Greeks

1)( dNe tTq

tTS

edN Ttq

)(1

)(1* tTqedNtTSv

)(12

)()(

1

2tTqtTr

tTq

edqSNdNrKetT

edNS

28

Greeks - Continued

2)()( dNetTK tTr

q

CB

RR

CBuupsilon

FX

CB

OAS

CBo

)(

)(

)(

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Zero Coupon Convertibles

The most bond like convertible is the zero coupon CB. The zero CB doesn’t pay any cash interest but it carries a series of (synthetic) accreting put options;

In effect the buyer has paid for a series of put options with the coupon streams that he has forgone;

The valuation of a zero CB must include a series of puts as well as series of calls that both the buyer and the issuer can claim as their right (the basic long stock plus long put model helps here);

The zero retains more bond like features at issue because the put option provides a bond floor that is close to the current value and this bond floor (put) accretes each year , helping to reduce the downside equity risk;

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Sony Zero-coupon CB Trading History (18 June 2003 – 18 June 2004)

see spreadsheet analysis

Fujitsu Convertible Bond6 J an 1995 - 8 Dec 1995

800,000

900,000

1,000,000

1,100,000

1,200,000

1,300,000

1,400,000

600,000 700,000 800,000 900,000 1,000,000 1,100,000 1,200,000 1,300,000

(Parity)

(Convert Value)

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Sony CB (see spreadsheet analysis)

527,797*49.0

ParityCB

cmxy

324.0

49.0

ScholesBlack

Trading

3980

16.177,6

85.2821

001685.0

spot

strike

vega

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CB Asset Swap

Trader / Investor Broker Bond Buyer

Convertible Bond

CB Call Option

CB Investment Value

Swap Trader

LIBOR + Spread

CouponsCoupons

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Example of a Vanilla Swap. Break even rate =

5.5%

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Convertibles CDS

CDS Buyer CDS Seller

Zero Payment

Contingent Settlement

No credit event

Credit event

Spread Fee

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Delta and Volatility

Convertibles with very little or no call protection remaining can be subject to a perverse effect of increased volatility;

As vol increases it has the effect of reversing the time value of an option and as volatility decreases it has the effect of increasing the time value of the option;

yearstradingDay

yprobabilitNORMSINV

ParityLogTriggerLogTime *

*

2

probNORMSINVdtParityTrigger CallCall *1*

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Example

If the CB has no call protection remaining and will only be called to force conversion, then the trader can estimate how much above the call price the parity level (trigger) should move before it may be called with a given probability and expected volatility.

For example, if the trader has determined that the parity level must be 120% of the call price for the company to safely call the issue, then he can estimate – using the previous formula – the amount of time premium that should be built into the CB’s embedded option;

For example: how much time will it take with an 80% probability for the trigger level to be reached for the CB with a parity of 102 and a trigger level of 120 and a 3-month annualized vol of 40%;

Time value is equal to 23% of the number of trading days in a year or roughly 59 trading days

59255*%23

3366.0

1625.0

84162.0*40.0

)102()120(22

Time

LogLogTime

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Example - continued

A trader can work with this formula in another way: say, a callable convertible with a 30-day call notice period has a parity level of 102 and a 3 month vol of 60%. The trader wants an 80% probability (of the trigger happening); then what would be the trigger level?

66.119))84162.0*25530*60.0(1(*102 Trigger

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Delta Neutral Arbitrage using Leverage(see spreadsheet)

XYZ Co.

(Non-Investment Grade BBB Convert) Value at Risk

Hedge

Settlement 25-Feb-04 long convert 1,000 $1,050,000 Correlation 0.85

Maturity 25-Feb-05 short stock 16,000 ($624,000) $623,700 Long Convert $61,559

Short Stock ($261,752)

Stock price $39 Annual Cash Flows

stock beta 0.85 95% VaR $211,923

convert par price $1,000 (annualized)

convert price $1,050 105% coupon $60,000

convert coupon 6% short interest rebate $31,200 95% 3 Day VaR $23,123

conversion premium 19.65% stock dividend $0

delta 0.594 Total Cash Flow $91,200 Credit Loss

implied vol 30%

short credit interest 5.00% Capital Required for Hedge Expected Credit Loss $1,764

Maximum Loss $1,050,000

margin for leverage 15% Levered un-hedged LMV $63,945 95% Maximum Credit Loss $672,000

borrowing rate 7.00% plus lesser amount of: $93,555 Unexpected Credit Loss $670,236

margin*delta*LMV

Credit Data (LMV - Parity)*delta

Rating BBB

1 year prob of default 0.28% Total Capital Required $157,500

Recovery Rate 40%

95% vol of Recovery Rate 10% Carrying cost of the position $62,475

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Example - Amazon

Amazon CB combined with the company’s straight debt was an interesting trade in March, 2000; Amazon 4.75% CB due 2009 was trading at 40% of par with a yield of over 19% (but with a very little value assigned to the embedded call option);

At the same time the 10% straight bond due 2008 was trading at 58% of the par with a YTM of 15% (the bond did not actually pay a coupon of 10%, since it was zero coupon with a clause to start paying cash interest payment on March 1, 2003);

Traders were long 145 CB at 40.00 and short 100 straight high yield at 58 thus creating an equal dollar offsetting investment netting to zero;

By mid-July 2000 the Amazon CB traded at 54 (gain on the long CB) and the straight high yield traded at 66 (loss on the short position) thereby realizing a net gain on $12,300 on an investment of zero.

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Learning Outcome MM is not actually relevant to most corporate situations, in

that Taxes play an important role in planning;

While the equilibrium capital structure may be defined by the point at which increasing leverage increases risk of default such that “firm” value decreases, for most practical purposes, a stable capital structure usually implied a ma debt load = (1-Tax Rate);

Convertible bonds are actually quite useful for maximizing value if used in conjunction with an effective asset development plan. Provide cheaper funding, though eventually will convert to expensive equity;

In the end it is always about identifying and properly risk managing positive cash flow projects to be brought on stream.

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Thank youFor more information about Capital structure

consulting, please contact Philip Corsano on:

206 384 0069, pcorsano@gmail.com

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