## Thursday, November 29, 2018

### Bond Prices, Interest Rate Changes and Coupon Size

Bond Prices, Interest Rate Changes, and Size of Bond Coupon

Question:   Consider four bonds each with semi-annual payments, a 10-year maturity and a face value of \$1,000 redeemed at maturity.   The bonds differ in semi-annual coupon – \$100, \$50, \$25, and \$0.

What are bond values at issue date if the yield to maturity is 3 percent?

What are bond values at issue date if the yield to maturity is 6 percent?

How did the change in interest rates impact the change in bond prices across the four bonds?

What are the durations of the four bonds?

How do bond durations appear to vary with changes in bond prices stemming from an interest rate change in this example?

Calculations:

Here are the bond prices at a 3 percent interest rate.

 Bond Prices at 3.0 percent interest rate Maturity Semi-Annual Payments 20 20 20 20 Coupon Payment 100 50 25 0 Par Value 1000 1000 1000 1000 YTM 0.03 0.03 0.03 0.03 PV \$2,459.33 \$1,600.90 \$1,171.69 \$742.47

Here are bond prices at 6 percent interest rate.

 Bond Prices at 6 percent interest rate Maturity Semi-Annual Payments 20 20 20 20 Coupon Payment 100 50 25 0 Par Value 1000 1000 1000 1000 YTM 0.06 0.06 0.06 0.06 PV \$2,041.42 \$1,297.55 \$925.61 \$553.68

Here are bond prices changes for when the interest rate goes from 3 percent to 6 percent.

 Change in bond prices when interest rates go from 3 to 6 percent PV @ 3 percent \$2,459.33 \$1,600.90 \$1,171.69 \$742.47 PV @ 6 percent \$2,041.42 \$1,297.55 \$925.61 \$553.68 Diff (\$417.91) (\$303.35) (\$246.07) (\$188.79) Percent Diff. -17.0% -18.9% -21.0% -25.4%

Here are bond duration estimates (weighted average of times cash flow is received) for the four bonds.

 Bond Duration When Interest Rates are at 3 percent Settlement Date 1-Jan-18 1-Jan-18 1-Jan-18 1-Jan-18 Maturity Date 1-Jan-28 1-Jan-28 1-Jan-28 1-Jan-28 Coupon rate 10.00% 5.00% 2.50% 0.00% Yield 3.00% 3.00% 3.00% 3.00% Frequency 2 2 2 2 Duration 7.32 8.17 8.88 10.00

Note that the higher duration bonds realized a large percent decline in bond price when interest rates rose from 3 percent to 6 percent.

Notes:

The bond price is the negative of the PV of the cash flows.  The PV function itself provides a negative number reflecting that the investor is paying for the bond and at the time of purchase money is leaving the investor’s hands.

A bond duration is the weighted average of times when the cash flows of the bonds are received.  The duration of a zero-coupon bond is equal to its maturity as there is only one cash flow date, the maturity of the loan.

Bonds with higher coupon rates have lower durations because cash flows are received earlier.   An increase in the coupon rate leads to lower duration.

Bond durations are a measure of the sensitivity of the price of bonds to interest rate changes.   When interest rate changes are very small the duration can provide a close estimate to the impact of a change in interest rates on bond prices. This approximation is less exact for large interest rate changes.

Authors Note:   Please consider reading my finance blog.   Start with this essay on problems with the U.S. retirement system.

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