Interest Rate Immunization: A Single Liability

When you're trying to plan fixed-income cash flows, interest rate risk is like a disease.

To understand why, recall that the pricing of a bond is like calculating the internal rate of return (IRR) of the bond's cash flows. Assume a simple, plain coupon bond. Maybe the IRR is 5.2%, and there's your yield to maturity. If there is no default, you will get those promised cash flows. But how does the reinvestment assumption of IRR apply to this bond?

No. Actually they must be in order for this to work.

No. There's only one market rate that's relevant in this calculation.

Exactly. The IRR is the only rate in the calculation, so if the bond will really provide the yield to maturity, then all cash flows have to be reinvested at the yield to maturity. If rates fall just after a bond is issued, what will happen to reinvestment income?

Yes! Those lower rates won't allow the same reinvestment income that was expected originally.

No, it will be too low. Those lower rates won't allow the same reinvestment income that was expected originally.

But what will this yield curve drop do to the bond's price?

Right.

No. When rates go down, bond prices go up.

Specifically, the bond price will jump up and then drop over time toward par as more and more coupons are paid. So how would you characterize the investor's position so far, due to this yield curve decline?

Absolutely. The investor gets a higher bond price but lower reinvestment income. Over time, these differences adjust and eventually balance each other out. And this is where the magical conclusion is. The bond price gain exactly offsets the reinvestment loss right at the Macaulay duration of the bond. The investor is __immunized__ against interest rate risk at this point. Immunization is minimizing the volatility of fixed income returns from the "disease" of interest rate risk.

No. Only in terms of reinvestment income.

No. Only in terms of price.

Suppose this is a 10-year bond, and the Macaulay duration is 8. The investor could choose this coupon bond and enjoy immunization if the investment horizon was how long?

You got it.

No, it would have to be eight years.

That Macaulay duration is where the offsetting effects would be seen, and so that's the investment horizon over which immunization occurs. The investor sells, and it's done. After that point, the bond price continues to decline toward par, and the reinvestment income continues to be too low. So then the investor would only lose.

Of course, there is a simpler way to immunize: just find a zero-coupon bond. There's no reinvestment risk with a zero, since there are no coupons to reinvest. Sure, the price may fluctuate, but nobody cares. The investor has an eight-year investment horizon, so if an eight-year zero-coupon bond can be found, just buy it, and wait. Yield curve changes won't make any difference, which eliminates **structural risk** or the risk that the yield curve twists or shifts in a way that mismatches the cash flow yield of the liability versus YTM of the zero-coupon bond. The face value will be received (barring default), and the expected yield will be obtained. Avoid the disease, or deal with the disease. There are multiple ways to manage.

To summarize: [[summary]]

Coupons must not be reinvested

Coupons must be reinvested at the risk-free rate

Coupons must be reinvested at the yield to maturity

It will be too low

It will be too high

Price goes up

Price goes down

The investor wins

The investor loses

There are competing effects

8 years

10 years

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