Hello!
The most important concept you need to know in order to answer these
questions is that the value of a bond is equal to the present value of
the cash flows provided by the bond (yearly coupon payments + capital
payment at maturity date), using the yield to maturity (YTM) as the
discount rate.
Let's address the first question. I hadn't read correctly the question
when I asked about the coupon rate. The problem states that the firm
"issued bonds at face value at a yeild to maturity of 7 percent". If
the bonds were issued at face value, then the coupon rate is identical
to the YTM. So the coupon rate for these bonds was 7%.
When the YTM rises, the market value of the bonds fall. Basically,
when the YTM rises, we're using a higher discount rate for the present
value calculation mentioned in the first paragraph. Therefore, this
present value (the value of the bond) must fall. We can find exactly
what will be the price of the bond after the YTM rises to 15%. Let's
assume that the face value of the bond is $100. The coupon payment is
thus $7. Therefore, the value of the bond, at 8 years to maturity, if
the YTM becomes 15% will be (assuming the last coupon payment was paid
yesterday):
PV = 7/1.15 + 7/1.15^2 + 7/1.15^3 + ... + 107/1.15^8 = 64.10
Thus the value of the bond FALLS from face value (=$100) to $64.10
when the YTM rises from 7% to 15%.
For the second question, we need to find the discount rate that makes
the present value of the cash flows from this bond equal to its market
value. Assuming that the stock is currently trading at $64.10, as we
found in the previous question, the equation we must solve now is:
64.10 = 7/(1+r) + 7/(1+r)^2 + 7/(1+r)^3 + ... + 87/(1+r)^8
Notice that the last payment is $87: $7 from the coupon and $80
because the bond will pay 80% of the face value.
This equation is very difficult to solve analytically, so I'll use an
YTM calculator. You can find one online at:
http://www.investopedia.com/calculator/AOYTM.aspx
Enter the following values:
Par Value = 80 (this is the face value; however, we'll be paid only
80% of it, so we must enter $80)
Market Value = 64.10
Annual Rate = 7/80 = 8.75% (this is the coupon rate as a percentage of
the "par value" - since the coupon payment remains at 7, while we
entered 80 as the par value, then we must tell the calculator that the
annual rate is 8.75%, as 8.75% of $80 is equal to $7)
Maturity in Years = 8
Payments = Annualy
We get that the YTM on a bond with these features is 12.87%.
Google search terms
yield to maturity calculator
://www.google.com.ar/search?hl=es&q=yield+to+maturity+calculator&meta=
I hope this helps! If you have any doubt regarding my answer, please
don't hesitate to request clarification before rating it. Otherwise, I
await your rating and final comments.
Best wishes!
elmarto |