18 Non-Callable Bonds
18.1 Overview and Terminologies
A corporation issues bonds as a way of borrowing funds. When the corporation issues a bond, it agrees to pay a redemption value \(C\) at the end of \(n\) units of time, and it also agrees to pay a coupon payment equal to the coupon rate \(r\) times the Face value \(F\) (also known as the par value) at the end of each unit of time until the bond matures at time \(n\).
Most bonds pay coupons semiannually and sometimes annually. A bond that does not pay coupon is called a zero-coupon bond.
Some terminologies:
- \(F\): Par value (Face amount) of the bond
- \(r\): Coupon rate per payment period
- \(Fr\): Amount of each coupon payment
- \(C\): Redemption value of the bond. \(F = C\) unless stated otherwise
- \(i\): Yield to maturity/Yield rate
- \(n\): Number of coupon payments.
- \(P\): The bond’s price
18.2 Price of a Non-Callable Bond
The price of a bond is the present value of all coupon payments and the redemption value, discounted at the bond’s yield \(i\). \[P = Fr \times a_{\overline{n|}i} + Cv^n\] Remark: Bond price moves inversely to the yield rate, i.e. the higher the bond price, the lower the yield and vice versa.
Example: A 5-year bond makes semiannual coupon payments at an annual rate \(10\%\) per year. The coupon payments are based on a par value of \(\$1,000\), and the redemption amount is \(\$1,100\). The yield, which is compounded semiannually, is 7% per year. What is price of the bond?
Solution: Since the bond makes semiannual coupons, we have: * Coupon rate per payment period: \(r = 5\%\) * Yield rate per payment period: \(i =3.5\%\) * Number of payment period: \(n = 5\times 2 = 10\)
Thus, the bond price is: \[P = Fr \times a_{\overline{n|}i} + Cv^n = 50 \times a_{\overline{10|}3.5\%} + 1,100 \times (1.035)^{-10} = 1,149.64\] We can either use the TVM keys on the BA II, or use the Bond Worksheet:
The calculator’s worksheet always assumes that the par amount is $100, so we use the bond worksheet ot find the price per $100 of par, and then we multiply by 10:
[2nd] [BOND] 12.3100 [ENTER] ↓(set the start date)10 [ENTER] ↓(enter the coupon rate)12.3105 [ENTER] ↓(set maturity date)110 [ENTER] ↓- (Use
[2nd] [SET]until display shows 360)↓ - (Use
[2nd] [SET]until display shows2/Y)↓ 7 [ENTER] ↓[CPT] [×] 10 [=]Result is1,195.64
18.4 Bond Portfolio
A portfolio of bonds consists of multiple bonds, and we can find the portfolio yield, which is the internal rate of return of the portfolio.
Example: Three bonds are purchased to create a bond portfolio. The time until maturity, the coupon rate, and the price of each bond are shown below:
| Maturity | Coupon | Price |
|---|---|---|
| 5 | 6% | 1,000.00 |
| 10 | 9% | 1,140.47 |
| 15 | 0% | 315.24 |
The 5-year and the 10-year bonds make annual payments, and all three bonds have par values of \(\$1,000\). What is the portfolio yield of the portfolio?
Solution:
The cost of the portfolio is: \(1,000.00+1,140.47 +315.24 = 2,455.71\).
The BA II can be used to find the internal rate of return of the portfolio with the Cash flow Worksheet. Note that the cash flows are consecutive:
| Cash Flow | Value | Frequency |
|---|---|---|
| 0 | -2,455.71 | (1) |
| 1 | 150 | 4 |
| 2 | 1,150 | 1 |
| 3 | 90 | 4 |
| 4 | 1,090 | 1 |
| 5 | 0 | 4 |
| 6 | 1,000 | 1 |
Then press [IRR] [CPT] to get the yield rate of \(7.03713\%\).