Download Innovations in Financial Engineering Solution to Homework One 3

FINA690G — Innovations in Financial Engineering
Solution to Homework One
3. We assume the yield to to be λ when the call provision is an advantage to the issuer (F = Principal ;
C =Coupon= 10%F ). Present value of the sum of the principal and the future coupon payment in the
next 15 years (bond price at the end of the 5th year) should be at least equal to the call price:
X
C
F
+
and set equal to Callprice = (1 + 5%)F so that
15
(1 + λ)
(1 + λ)k
15
P5 =
k=1
F
+
(1 + λ)k
15
X
k=1
10%F
= (1 + 5%)F
(1 + λ)k
[We can solve the equation by Excel or Matlab.]
λ = 9.366%.
The yield at the end of the 5th year must be less than 9.366% to induce the call.
6. In this case σ = 0.25, δt = 1, r = 0.05 and the tree parameters are u = 1.2840, d = 0.7788, a =
1.0513, p = 0.5394, and 1 − p = 0.4606. The upper number at each node is the stock price, the second
number is the equity component of the value, the third number is the debt component of the value,
and the fourth number is the total value. The convertible is valued at each node immediately after the
coupon payment.
(a) The instrument cannot be converted at the final nodes. The value at each of these nodes is
therefore 100. At the two-year point the instrument is worth (100 + 5)e−0.07 = 97.90 if it is not
converted. The instrument should be converted at node D since conversion will improve the value
to 41.22 × 3.5 = 144.27.
At node C the value of the instrument, if not converted, is (97.90 + 5)e−0.7 = 95.94. Because
19.47 × 3.6 is less than this, the instrument should not be converted at node C. At node B,
assuming no conversion, the equity value is 144.27 × 0.5394 × e−0.05 = 74.02. The debt value is
(97.90 × 0.4606 + 5)e−0.07 = 46.71.
The total value is 120.73. If converted the instrument is worth 32.10×3.6 = 115.56. The instrument
should not therefore be converted.
At node A, the equity component of the value is 74.05 × 0.5394 × e−0.05 = 37.98. The debt
component is
(46.71 × 0.5394 + 95.94 × 0.4606 + 5)e−0.07 = 69.36.
The value of the instrument is therefore 107.34.
(b) Without the conversion option the instrument is worth (95.94 + 5)e−0.07 = 94.12. The value of the
conversion option is, therefore, 107.34 − 94.12 = 13.22.
(c) The bond would be called at node A. It would then be converted and the value at node A would
be 115.56. The value of the equity components at node A would then be 115.56 × 0.5394 × e−0.05 =
59.29. The value of the debt component would be (95.94 × 0.4606 + 5)e−0.07 = 45.86. The total
value of the instrument would be reduced to 105.15. The value of the conversion option (net of the
call option) would be reduced to 105.15 − 94.12 = 11.03.
(d) The tree would be constructed to reflect the dividends. This would involve subtracting the present
value of the dividends from the initial stock price, constructing the tree in the usual way, and then
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adding in the present value of future dividends at each node. When calculating a value-if-converted
number at a node, the present value of the dividend six months later would be taken into account.
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