Fixed Income Investments - Spot Rates and Bond Valuation

On some occasions, such as with non-U.S. government bonds which pay annual interest compared to semi-annual interest in the U.S., an adjustment needs to be made in order to compare their yields.
The computation is as follows:

Formula 14.11

Bond-equivalent yield of an annual-pay bond = 2[(1 + yield on annual-pay bond) to the .5 power - 1]

Example:
Assume that the YTM on an annual-pay bond is 8%.

Answer:
Bond-equivalent yield = 2 [(1 + .08) to the .5 power - 1]
= 2 [.03923]
= .078461 = 7.95%


Look Out!

The bond equivalent yield will always be less than the annual-yield.

Example:
Now if you want to convert the bond equivalent yield of a U.S. bond into an annual-pay bond the calculations are as follows:

Formula 14.12

Yield on annual-pay basis = [(1 + yield on bond-equivalent basis/2)2-1

Example:
The yield of a U.S. bond quoted on a bond-equivalent basis of 8%:

Answer:
Yield on annual-pay basis = [(1 + 8/2 to the 2nd power) -1]
= [(1.04) to the 2nd power - 1]
= .0816 = 8.16%


Look Out!

The yield on an annual-pay basis is always greater than the yield on a bond-equivalent basis. This is because of compounding.

Example: Computing the Value of a Bond Using Spot Rates
Suppose you have a bond that matures in 1.5 years that has a coupon rate of 8% and the spot curve is 5% for six months, 5.25% for 1 year and 5.50% for 1.5 years.

Answer:
Bond price = 40/ (1.05) + 40 / (1.0525) to the second power + 1040 / (1.055) to the third power.
Bond Price = 38.09 + 36.12 + 931.06
Bond Price = 1005.27

This can be applied to any maturity; all you need to do is to continue theformula out to that maturity to discover the price of the bond.

Example: Compute the Theoretical Treasury Spot Rate Curve Using Bootstrapping
Again let's look at an example to get through this LOS. We have a six month annualized yield of 4% and similarly of the 1 year Treasury Security the rate is 4.40%. Given these two rates we can compute the 1.5 year theoretical spot rate of a zero coupon bond. For our example let's use a coupon of 6% with them selling at par.

Answer:
First let's get the cash flows:
0.5 year = .06 x $100 x .5 = 3.00
1.0 year = .06 x $100 x .5= 3.00
1.5 year = .06 x $100 x .5 = 3.00 +100(par value) = 103

On to the next step:

3.00/ 1.02 + 3 / (1.02) to the second power + 103 / (1 +x3) to the third power = 100
2.94+ 2.88 + 103 / (1 + x3 ) to the third power = 100
103/ (1 +x3) to the third power = 94.18
(1 + x3) to the third power = 103 /94.18

Limitations of the Nominal Spread
As we discussed earlier, a nominal spread is the spread between a non-treasury bond's yield and the yield to maturity on the comparable Treasury security in terms of maturity. For example, if an IBM is trading at a YTM or 6.25% and the comparable Treasury is at 5%, then the nominal spread is 125 basis points. This spread measure takes into consideration the extra credit risk, option risk and any liquidity risk that may be associated with the non-treasury security.

Even though this is a quick and dirty way to describe the yield difference, it has two drawbacks. They are:

1. For bond bonds, the yield does not take into consideration the term structure of spot rates.
2. In the case of callable/puttable bonds, expected interest-rate volatility may change the cash flows of the non-treasury security.

Differentiating Between Spreads
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