1. Advanced Bond Concepts: Introduction
  2. Advanced Bond Concepts: Bond Type Specifics
  3. Advanced Bond Concepts: Bond Pricing
  4. Advanced Bond Concepts: Yield and Bond Pricing
  5. Advanced Bond Concepts: Term Structure of Interest Rates
  6. Advanced Bond Concepts: Duration
  7. Advanced Bond Concepts: Convexity
  8. Advanced Bond Concepts: Formula Cheat Sheet
  9. Advanced Bond Concepts: Conclusion

You can use the bond yield formula to determine the return you’ll realize by holding a bond to maturity. The required yield, conversely, is the return a bond must offer to make it worthwhile to investors, and it’s usually the same yield offered by other plain vanilla bonds in the market with similar credit quality and maturity. Once you’ve decided on the required yield, you can figure out the yield.

Calculating Current Yield

The current yield is an investment’s annual income divided by its current price. It represents the return you would expect if you held a bond for a year. You can calculate current yield using the following formula:

[Note: An asterisk (*) means "multiply."]

So, for example, if you bought a bond with a par value of $100 for $95.92 and a coupon rate of 5%, the calculation would look like this:

This formula gives accurate results as long as you pay par value for the bond. Otherwise, you’ll have to modify the formula by adding the result of the current yield to the gain or loss given by the price: [(par value bond price) ÷ years to maturity]. The modified current yield formula takes into account the discount or premium at which you buy the bond, and is calculated as:

As an example, we’ll recalculate the yield of the bond in our first example, which matures in 30 months and has a coupon payment of $5:

The adjusted current yield of 6.84% is higher than the current yield of 5.21%. That’s because the bond’s discount price ($95.92 instead of $100) gives you more of a gain on the investment. Keep in mind: If you buy a bond between coupon payments, you should use the dirty price in place of the market price in the equation.

Current Yield and Zero-Coupon Bonds

Zero-coupon bonds have only one coupon payment, so we have to use a different calculation.


n = years left until maturity

As an example, assume you’re considering a zero-coupon bond that matures in two years and has a future value of $1,000. You can buy it for $925. We can calculate its current yield by plugging these values into the following formula:

Calculating Yield to Maturity

The current yield calculation show us the return the annual coupon payment gives us, but it doesn’t take into account the time value of money. For this reason, when investors and analysts discuss yield, they’re usually referring to the yield to maturity (YTM), which is the total anticipated return if a bond is held until maturity – including any capital gains or losses due to price fluctuation.

Because calculating YTM is complicated and time consuming, investors (both private and professional) typically rely on financial calculators to do the math. A good one to try is Investopedia’s Yield to Maturity Calculator. To calculate YTM, enter the par value, market value, annual rate (%) and maturity in years, and choose the frequency of payments. An example of Investopedia’s calculator is shown below. 

Still, some investors prefer to tackle the math on their own. Before revealing the equation, here are three things to keep in mind:

  1. As a bond’s price increases, its YTM falls. For example, if you buy a bond with a par value of $100 and a 10% coupon, its yield would be the coupon rate divided by the par value (10 ÷ 100 = .10) or 10%. If the bond price dropped to $90, the yield would become (10 ÷ 90 = .11) or 11%. You’d still earn the same amount of interest because the coupon is based on the par value.
  2. With premium bonds, the coupon rate is greater than market interest rates. With discount bonds, the coupon rate is less than market interest rates.
  3. YTM is the yield you’ll receive if you reinvest all coupons received at a constant interest rate.

 

Here’s the formula:

Here’s an example, assuming a bond with a par value of $100, a current yield of 5.21% and a price of $95.92. The bond matures in 30 months, and its coupon is 5%, paid semi-annually. Remember, the cash flow is the amount you receive for each coupon payment. Since the coupon rate is 5% and it’s paid twice a year, the cash flow would be $2.50.

Solving for 'i'

Now we get to the hard part because we need to solve for “i” – the interest rate. Instead of picking random numbers, we remember that when a bond is priced at par, the interest rate is equal to the coupon rate. When a bond is priced above par, the coupon rate is greater than the interest rate. In our example, the bond is priced at a discount, so the annual interest rate must be greater than the 5% coupon rate. With this in mind, we can plug some annual interest rates (that are higher than 5%) into the formula and solve it. Here are the results:

 

Annual Interest Rate

Semi-Annual Interest Rate

Bond Price

10.0%

5.0%

$89

9.0%

4.5%

$91

8.0%

4.0%

$93

7.0%

3.5%

$95

6.0%

3.0%

$98

 

We know our bond is priced at $95.92, so we know the interest rate is somewhere between 6.0% and 7.0%. Next, we’ll use our formula to try interest rates in 0.1% increments:

 

Annual Interest Rate

Semi-Annual Interest Rate

Bond Price

7.0%

3.5%

$95.48

6.9%

3.45%

$95.70

6.8%

3.4%

$95.92

6.7%

3.35%

$96.15

6.6%

3.30%

$96.37

Since we know our bond price is $95.92, we can now see the interest rate must be 6.8%.

Calculating Yield for Callable and Puttable Bonds

Callable and puttable bonds have additional yield calculations: a callable bond has to account for the issuer’s ability to call the bond on the call date; likewise, a puttable bond has to account for the buyer’s ability to sell the bond at the put date.

The yield for callable bonds is called yield-to-call (YTC), and it’s the interest rate you’d receive if you held the bond until the call date. The yield for puttable bonds is called yield-to-put (YTP), and it’s the rate you’d receive if you held the bond until its put date. The calculation requires two small modifications to the YTM formula (highlighted in bold):

Note that maturity value has been replaced by call value, and now “n” represents the time until the call date. To find YTP, you’d change call value to put value, and “n” would represent the time until the put date.

 

 

 


Advanced Bond Concepts: Term Structure of Interest Rates
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