9.19 More on Price vs. Reinvestment Risk Proof
Remember:
Reinvestment and price risks are inversely related. That is, if rates go up, the investor will benefit from a higher reinvestment rate, but be hurt in terms of price risk – and vice versa if rates go down.
Step One: If a simple Par bond is purchased and considered to have been instantaneously sold at the duration point, there will be perfect price- and reinvestment-rate offset. This is a necessary condition since we have not assumed any change in yields.
Let’s first use the earlier example of a 10% (semi) coupon. What if the YTM and reinvestment rate were equal to the coupon of 10%? The duration, if calculated as done earlier, is four years – specifically 4.05 years, as found above.
| Period (p) | Coupon (.05 semi) | FVF @ .05 | FVCF | PVF @ .05 | PVCF |
|---|---|---|---|---|---|
| 1 | $50 | 1.4071 | 70.36 | ||
| 2 | 50 | 1.3401 | 67.00 | ||
| 3 | 50 | 1.2763 | 63.81 | ||
| 4 | 50 | 1.2155 | 60.78 | ||
| 5 | 50 | 1.1576 | 57.88 | ||
| 6 | 50 | 1.1025 | 55.13 | ||
| 7 | 50 | 1.0500 | 52.50 | ||
| 8 | 50 | 1.0000 | 50.00 | ||
| 477.46 | |||||
| 9 | 50 | .9524 | 47.62 | ||
| 10 | 1050 | .9070 | 952.35 | ||
| 1,000.00 |
RCY = (MIRR) = [(477.46 + 1,000) ÷ 1,000]1/8 – 1 = .05 (semi)
If the investor sells the bond at the time of its duration, their RCY would be the same as the initial YTM.
What would the RCY be if you sold the bond after three years? Well, the terminal value of the coupon would then be $477.46 – ($70.36 + $67.00) = $340.10; the bond would be sold at par. Hence,
RCY = [(340.10 + 1,000) ÷ 1,000]1/6 – 1 = .05 (semi)
We would still earn our “original” RCY!
Step Two: Suppose we indeed assume an instantaneous change in yield. Now, we will have a non-Par bond. We will observe that if the bond is sold at the Duration point, there will be a perfect offset. RCY = YTM. This too is a necessary condition of our Theorem. It is however insufficient as we would also need to show – in the negative – that if this bond is sold at other than the Duration point there will be no precise offset and that the RCY will not equal the YTM.
Let’s use the same example of a 10% (semi) coupon and a “new” YTM of 8% (semi, of course); assume an “instantaneous” change in yield. We are implicitly assuming that YTM equals the Reinvestment rate, so REIN changes as well; this is because the YTM assumes reinvestment at the YTM, so if YTM changes, so too, it is assumed, will REIN change. The duration here is (again, as originally) 4 years (eight periods). Let’s say that the investor sells the bond after 4 years – at the time of the bond’s duration. Naturally, we assume ceteris paribus. Recall that we are dealing with the abstract notion of instantaneous changes in yield.
| Period (p) | Coupon (.05 semi) | FVF @ .04 | FVCF | PVF @ .04 | PVCF |
|---|---|---|---|---|---|
| 1 | $50 | 1.3159 | 65.80 | ||
| 2 | 50 | 1.2653 | 63.27 | ||
| 3 | 50 | 1.2167 | 60.84 | ||
| 4 | 50 | 1.1699 | 58.50 | ||
| 5 | 50 | 1.1249 | 56.25 | ||
| 6 | 50 | 1.0816 | 54.08 | ||
| 7 | 50 | 1.0400 | 52.00 | ||
| 8 | 50 | 1.0000 | 50.00 | ||
| 460.74 | |||||
| 9 | 50 | 48.10 | |||
| 10 | 1050 | 971.25 | |||
| 1,019.35 |
RCY = [(460.74 + 1,019.35) ÷ 1,000] 1/8 – 1 = .05 (semi)
The investor would have “in hand” $460.74 and would receive proceeds from the bond’s sale of $1,019.35. In other words, the price and reinvestment risks would exactly offset one another and the RCY would still be 5% (semi), i.e., the same as the original YTM. Nothing has changed in terms of his realized yield in spite of the “instantaneous” change in YTM.
Indeed, certain financial institutions “Immunize” their portfolios from interest rate / price risks by engaging with this strategy. The concept of Immunization will be discussed again briefly later.
(Incidentally, we could have done this calculation the short way, using annuity multipliers. Can you do that?)
Step Three: We will assume both yields have changed and that the bond is sold at any point in time other than the Duration point. The RCY will not equal the YTM. This condition is both necessary and sufficient. Case closed. Let’s see.
What if we sold the bond after six periods? What would be the realized compound yield or actual return if the yield and reinvestment rates were 8%?
| Period (p) | Coupon (.05 semi) | FVF @ .04 | FVCF | PVF @ .04 | PVCF |
|---|---|---|---|---|---|
| 1 | $50 | 1.2167 | 60.84 | ||
| 2 | 50 | 1.1699 | 58.50 | ||
| 3 | 50 | 1.1249 | 56.25 | ||
| 4 | 50 | 1.0816 | 54.08 | ||
| 5 | 50 | 1.0400 | 52.00 | ||
| 6 | 50 | 1.0000 | 50.00 | ||
| 331.67 | |||||
| 7 | 50 | 0.962 | 48.10 | ||
| 8 | 50 | 0.925 | 46.25 | ||
| 9 | 50 | 0.889 | 44.45 | ||
| 10 | 1050 | 0.855 | 897.75 | ||
| 1,036.55 |
RCY = [(331.67 + 1,036.55) ÷ 1,000]1/6 – 1 = 0.0536 ≠ 0.05
Here, since rates (instantaneously) went down (as compared with the initial YTM at time of the bond’s purchase) and, importantly, the holding period was less than the bond’s duration, the benefit from the resulting higher bond (sale-) price made up for the reduced reinvestment rate and shorter reinvestment length of time – as of the six-period horizon. (Compare the bold numbers in this scenario with the prior.)
Price went up from $1,019 to $1,036 (Good News) and the future value of the reinvested coupon went down from $460 to $331 (Bad News). The net effect of these two changes is that the RCY increased above the YTM. Price and Reinvestment risks did NOT offset one another.
Once again, Duration is the point at which price and reinvestment risks exactly offset one another. If market yields are assumed to have changed “instantaneously,” and the bond is expected to be sold, ceteris paribus, at any point other than the duration point, the RCY ≠ YTM, as originally expected.
It is suggested that the reader return now to “More on Price vs. Reinvestment Risk – Prelude” in order to review what was just presented from a bird’s eye perspective. The next page provides a summary of our findings.
