ParT 2: ValuaTION, rISk aNd reTurNa result of the warrants being exercised, then the price of the warrants equals the price of an identical
call option, $C, multiplied by the dilution factor, N 1 /(N 1 + N 2 ):Eq. 8.4 Warrant value = $C(N 1 / (N 1 + N 2 ))
exampleHere is a special example of warrant issues being
purchased on a very large scale by a single
purchaser motivated by a perceived governmental
need to help a large financial institution. As part of
the Troubled Asset Relief Program (TARP), in 2008
the US government paid Wells Fargo $25 billion in
exchange for preferred shares and warrants. The
government received 110.3 million warrants, and
Wells Fargo had 3,325 million outstanding shares
at the time. We will use the Black–Scholes formula
and the adjustment factor in Equation 8.4 to value
the Wells Fargo warrants. To value the warrants,
we must know the price of Wells Fargo shares, the
expiration date of the warrants, the strike price and
the risk-free rate. We also need an estimate of WellsFargo’s volatility. Here are the relevant figures:
share price = $31.22; strike price = $34.01; risk-free
rate = 2%; expiration = 10 years; standard deviation
= 94.7%.σ=
++
=
−+
=
=−=−=−
=−=
=− =
=
+
=
−ddd t
NN
Cln31.22
34.01
0.02
(0.947)
2
10
0.947 10
(0.086)(4.684)
2.995
1.535
1.535 2.995 1.460
(1.535)0.938, (1.459)0.072
31.22(0.938) 34.01(2.718) $27.2 8
Warrant3,325
3,325 110.3
(27.28) $26.40
1221(.02)(10)A convertible bond grants investors the right to receive payment in the shares of a company rather than
in cash. There are some convertible bond issues in Australia each year, although the volume is small:
Australia represents about 3% of the world total of convertible bonds.
Usually, the shares, which investors have the right to ‘purchase’ in exchange for their bonds, are
the shares of the company that issued the bonds. In some cases, however, a company that owns a large
amount of ordinary shares in a different company will use those shares as the underlying asset for a
convertible bond issue. In either case, a convertible bond is essentially an ordinary corporate bond with
an attached call option or warrant.
Recently, the software giant Microsoft announced a sale of four-year, zero-coupon bonds, which
would generate proceeds for the company of approximately $1.25 billion. Microsoft’s bonds offered
investors a yield to maturity (YTM) of just 1.85%, similar to the yields on four-year government bonds at
the time. How could a technology company borrow money at a rate comparable to that paid by the US
government? Investors were willing to buy Microsoft’s bonds despite their low yield, because the bonds
were convertible into Microsoft ordinary shares. Specifically, each Microsoft bond that had a market
value when issued of $1,000 could be converted into 29.94 shares of Microsoft ordinary shares.
Convertible bonds offer investors the security of a bond and the upside potential of ordinary shares. If
Microsoft’s shares increased in value at least six months after the issue date, its convertible bondholders
would redeem their bonds for Microsoft shares rather than cash. To see how far Microsoft’s shares would
have to rise before bondholders would want to convert, we divide the bond price by the conversion ratio.
The conversion ratio defines how many Microsoft shares bondholders will receive if they convert. In
this case, the conversion ratio is 29.94. Therefore, if bondholders choose to convert immediately, they
will effectively be paying a conversion price for Microsoft of:Conversionprice==
$1, 000
29.94$33.40
LO8.5convertible bond
A bond that gives investors
the right to convert their
bonds into shares
conversion ratio
The number of shares
bondholders receive if they
convert their bonds into
shares
conversion price
The market price of a
convertible bond, divided by
the number of shares that
bondholders receive if they
convert