In: Finance
2. Robert Campbell and Carol Morris are senior vice-presidents of the Mutual of Chicago Insurance Company. They are co-directors of the company’s pension fund management division, with Campbell having responsibility for fixed income securities (primarily bonds) and Morris being responsible for equity investments. A major new client, the California League of Cities, has requested that Mutual of Chicago present an investment seminar to the mayors of the represented cities. Campbell and Morris, who will make the actual presentation, have asked you to help them by answering the following questions. a. What is the value of a 1-year, $1,000 par value bond with a 8% annual coupon if its required rate of return is 12%? What is the value of a similar 10-year bond? b. What would be the value of the bond described in part (a) if, just after it had been issued, the expected inflation rate rose by three percentage points, causing investors to require a 15% return? Is the security now a discount bond or a premium bond? c. What would happen to the bond’s value if inflation fell, and rd declined to 8%? Would it now be a premium bond or a discount bond? d. What is the yield to maturity on a 10-year, 8% annual coupon, $1,000 par value bond that sells for $880.00? That sells for $1,130.50? What does the fact that a bond sells at a discount or at a premium tell you about the relation¬ship between rd and the bond’s coupon rate?What would happen to the value of the 10-year bond over time if the required rate of return remained at (i) 12% or (ii) remained at 8%?
Value of a bond = present value of its coupons and redemption value
a. Value of one year bond = 80/1.12 + 1000/1.12 = $964.29
Value for a 10 year bond= 80/1.12 + 80/1.122 + ... + 80/1.1210 + 1000/1.1210
= $773.99
b) If the bond has a required rate = 15%
Value of one year bond = 80/1.15+ 1000/1.15 = $939.13
Value for a 10 year bond= 80/1.15 + 80/1.152 + ... + 80/1.1510 + 1000/1.1510
= $648.69
The security is now at a lesser price than its par value i.e. a discount bond
c) If the bond has a required rate = 8%
Value of one year bond = 80/1.08+ 1000/1.08 = $1000
Value for a 10 year bond= 80/1.08 + 80/1.082 + ... + 80/1.0810 + 1000/1.0810
= $1000
The security is now at a price same as its par value i.e. it is neither a premium nor a discount bond
d) The YTM of a bond can be calculated approximately as
YTM = { Annual Coupon amount + ( Redemption value - Market value) / no of years to maturity } / (0.4 * Redemption value + 0.6 * Market value)
So, YTM = { 80 + ( 1000- 880 ) / 10 } / (0.4*1000+ 0.6 *880 )
= 92/928 = 0.09914 = 9.914%
By hit and trial, we find YTM = 9.95%
In case , bond is selling at 1130.5
YTM = { Annual Coupon amount + ( Redemption value - Market value) / no of years to maturity } / (0.4 * Redemption value + 0.6 * Market value)
So, YTM = { 80 + ( 1000- 1130.5 ) / 10 } / (0.4*1000+ 0.6 *1130.5 )
= 66.95/1078.3 = 0.062088 = 6.21%
By hit and trial, we find YTM = 6.21%
The fact that a bond sells at a discount means that the required rate is higher than the coupon rate and
The fact that a bond sells at a premium means that the required rate is lower than the coupon rate
The 10 year bond with a coupon rate of 8% and required rate of 12% will gradally increase in price and will finally be equal to par value on maturity.
The 10 year bond with a coupon rate of 8% and required rate of 8% will remain equal to its par value throughout its maturity.