Valuation of a Convertible Bond and Comparison of Risk-Return Characteristics
Using the rate tree in Table 3, from an estimated rate volatility is 12%, the value of Bond A is _closest_ to:
Incorrect.
One way of getting this answer is to ignore the option on the bonds.
With Year 1 calculated, the tree looks like this:
| | Year 0 | Year 1 | Year 2 | Year 3 |
|----------|------------|--------|--------|--------|
| | | | | 103.50 |
| | | | 99.64 | |
| | | 100.57 | | 103.50 |
| | Bond Value | | 100.44 | |
| | | 101.00 | | 103.50 |
| | | | 101.00 | |
| | | | | 103.50 |
Next, calculate the bond value by using the same formulas for Year 0.
$$\displaystyle \mbox{Year 0}{_{Up}} = \frac{\mbox{Year 1 Coupon + Year 1}_{Up}\mbox{ Value}}{\left ( 1 + r_{Year 0}\right )} $$
$$\displaystyle = \frac{\left (\mbox{3.50 + 100.575} \right )}{\left ( 1 + 0.021000\right )} = 101.93$$
$$\displaystyle \mbox{Year 0}{_{Down}} = \frac{\mbox{Year 1 Coupon + Year 1}_{Down}\mbox{ Value}}{\left ( 1 + r_{Year 0}\right )} $$
$$\displaystyle = \frac{\left (\mbox{3.50 + 101.00} \right )}{\left ( 1 + 0.021000\right )} = 102.35$$
Finally, you have for Year 0
$$\displaystyle (0.5)101.93+ (0.5)102.35 = 102.14$$.
Incorrect.
You might have arrived at this answer by not using a valuation tree to calculate the bond value under volatility.
Right!
The rate tree in Table 3 gives the rates after volatility is applied, but the first step is to calculate the cash flows in each period—that is, the coupon payments and the return of par value plus the final coupon payment in the third year:
| | Year 1 | Year 2 | Year 3 |
|-----------|--------|--------|--------|
| | 3.50 | 3.50 | 103.50 |
Next, build the valuation tree, starting with the final cash flows in Year 3:
| | Year 0 | Year 1 | Year 2 | Year 3 |
|----------|------------ |-------------|-------------------|---------|
| | | | | 103.50 |
| | | | Year 2 Up, Up | |
| | | Year 1 Up | | 103.50 |
| | Bond Value | | Year 2 Up, Down | |
| | | Year 1 Down | | 103.50 |
| | | | Year 2 Down, Down | |
| | | | | 103.50 |
Now, find the Year 2 values for each branch of the valuation tree using the formula
$$\displaystyle \mbox{Year 2}{_{Up, Down}} = \frac{\mbox{Year 3 Cash Flow}}{\left ( 1 + r\right )}$$
and including the call option, which puts an upper price limit of 101.00 at each node. Do this for each branch.
$$\displaystyle \mbox{Year 2}{_{Up, Up}} =min\left ( 101.00,\frac{\mbox{103.50}}{\left ( 1 + 0.038786\right )} \right )$$
$$\displaystyle \mbox{Year 2}{_{Up, Down}} =min\left ( 101.00,\frac{\mbox{103.50}}{\left ( 1 + 0.030510\right )} \right )$$
$$\displaystyle \mbox{Year 2}{_{Down, Down}} =min \left ( 101.00,\frac{\mbox{103.50}}{\left ( 1 + 0.024000\right )} \right )$$
Now the valuation tree looks like this:
| | Year 0 | Year 1 | Year 2 | Year 3 |
|----------|------------|--------------|--------|--------|
| | | | | 103.50 |
| | | | 99.64 | |
| | | Year 1 Up | | 103.50 |
| | Bond Value | | 100.44 | |
| | | Year 1 Down | | 103.50 |
| | | | 101.00 | |
| | | | | 103.50 |
Calculating the nodes for Year 1 adds a twist since the Year 2 cash flows vary. Now, discount each branch's value in Year 3 and take the average (effectively done for Year 3, but since all of the possible cash flows in Year 3 were the same, you can get away with rolling both averaged branches into one). For the Year 1Up node, calculate two possible cash values using cash flows discounted back to Year 1, using discount rates from the interest rate tree (Table 3).
$$\displaystyle \mbox{Year 1}{_{Up, Up}} = \frac{\mbox{Year 2 Coupon + Year 2}_{Up,Up}\mbox{ Value}}{\left ( 1 + r_{Year 1, up}\right )} $$
$$\displaystyle = \frac{\left (\mbox{3.50 + 99.64} \right )}{\left ( 1 + 0.029493\right )} = 100.1852$$
$$\displaystyle \mbox{Year 1}{_{Up, Down}} = \frac{\mbox{Year 2 Coupon + Year 2}_{Up,Down}\mbox{ Value}}{\left ( 1 + r_{Year 1, up}\right )} $$
$$\displaystyle = \frac{\left (\mbox{3.50 + 100.44} \right )}{\left ( 1 + 0.029493\right )} = 100.9623$$
The valuation in Year 1 is the weighted-average of the two values.
$$\displaystyle (0.5)(100.19)+(0.5)(100.96)= 100.575$$
This is because the tree assumes that in the Year 1Up node, there is a 50% chance that Year 2 rates will be higher and a 50% chance that they'll be lower. Also apply the price limit that the call option imposes, if necessary. The Year 1Up node value is then
$$\displaystyle min\left (101.00, 100.575 \right ) = 100.575$$.
Year 1Down is calculated the same way using the Year 1Down rate and the Year 2Up, Down (which is the same as the Year 2Down, Up) and Year 2Down, Down values.
$$\displaystyle \mbox{Year 1}_{Down} = min\left (101.00, [(0.5)1.58 + (0.5)102.13] \right ) $$
$$\displaystyle = min(101.00,101.85) = 101.00$$
USD 101.17.
USD 102.14.
USD 102.58.
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