4.4 Maturity Pull: Continued
What about a non-zero-coupon bond? It too would experience maturity pull.
In the case of a par bond (Case 1 earlier and again below), after one year (i.e., two semi-annual periods) it would have the present values of two coupons less (those for periods 9 and 10) and hence the PV of all the remaining coupons would be that much less. However, the large face value payment, now increasing in present value because comes in sooner than before, would, in this case, exactly make up for the loss of the coupons. This is summarized below.
Remember that we are concerned with present values. (Assuming a uniform discount rate over the bond’s life, the bond’s price must always be Par.) Below is the table of the bond discussed.
| Case 1 | |||
|---|---|---|---|
| Period | Coupon | PVF @ .05 | PVCF |
| 1 | $50.00 | .9524 | $47.62 |
| 2 | 50.00 | .9070 | 45.35 |
| 3 | 50.00 | .8638 | 43.19 |
| 4 | 50.00 | .8227 | 41.14 |
| 5 | 50.00 | .7835 | 39.18 |
| 6 | 50.00 | .7462 | 37.31 |
| 7 | 50.00 | .7107 | 35.54 |
| 8 | 50.00 | .6768 | 33.84 |
| 9 | 50.00 | .6446 | 32.23 |
| 10 | 50.00 | .6139 | 30.70 |
| 1,000.00 | .6139 | 613.90 | |
| 1,500.00 | 1,000.00 | ||
Note that the PV of the par bond will remain flat, i.e., at Par Value, over the entire life of the bond.
| Period | Cash Flow | Present Value Factor | Present Value | Difference |
|---|---|---|---|---|
| 9 | 50 | .6446 | ($32.30) | ($62.90) |
| 10 | 50 | .6139 | ($30.70) | |
| 10 | 1000 | .6139 | ($613.90) | ($62.90) |
| 8 | 1000 | .6768 | $676.80 | |
| 000 | 000 |
Not all bonds are priced at Par. Let’s look at the discounted case (#2) presented earlier and shown here.
| Case 2 | |||
|---|---|---|---|
| Period | Coupon | PVF @ .06 | PVCF |
| 1 | $50.00 | .9434 | $47.17 |
| 2 | 50.00 | .8900 | 44.50 |
| 3 | 50.00 | .8396 | 41.98 |
| 4 | 50.00 | .7921 | 39.61 |
| 5 | 50.00 | .7473 | 37.37 |
| 6 | 50.00 | .7050 | 35.25 |
| 7 | 50.00 | .6651 | 33.26 |
| 8 | 50.00 | .6274 | 31.37 |
| 9 | 50.00 | .5919 | 29.60 |
| 10 | 50.00 | .5584 | 27.92 |
| 1,000.00 | .5584 | 558.40 | |
| 1,500.00 | 926.43 | ||
In the discount bond case, the “loss” or “disappearance” of the coupons (in PV terms) is more than made up for by the increase in the PV of the par value. See the summary calculations below. Note how the 10th period’s face value row was condensed into one row. The now nearer maturity “pulls” the price upward.
| Period | Cash Flow | Present Value Factor | Present Value | Difference |
|---|---|---|---|---|
| 9 | 50 | .5919 | ($29.60) | ($57.52) |
| 10 | 50 | .5584 | ($27.92) | |
| 8/10 | 1000 | .6274 – .5584 | $69.00 | $69.00 |
| + $11.48 | + $11.48 |
The price of this bond next year, assuming the same discount rate, will be $926.43 + $11.48 = $937.91.
Notes
We know that:
- In one year, two coupons are paid. The PVs that “disappear” are those for periods 9 and 10 because the entire table moves up and with only 8 periods to go, the PVs for periods 9 and 10 must be eliminated from the price calculation. What were before periods 3 and 4, are now 1 and 2, and so forth.
- The price of a discount bond must rise over time.
- The “disappearance” of the present values of the two coupons over a year’s time, results in a reduction to the bond’s present value.
- Therefore, the increase in the present value of the Face Value ($69) therefore must more than offset the decrease – in time – of the present value of the “lost” coupons ($57.52).
- And it does! $69 – $57.52 = +$11.48
- The opposite would be the case for a premium bond.
- Here once again, we have assumed no change in market yields in order to focus on maturity pull alone.
The exact opposite would pertain for a premium bond; the price would decrease over time until it hits Par at maturity. There, the “loss” or “disappearance” of the coupons when they are paid (in PV terms) is greater than the increase in the PV of the par value due to the face value’s getting closer to its maturity. Time “pulls” the price downward until maturity. Once again, we observe Maturity Pull. Let’s concoct “Case 3” to demonstrate the premium bond instance.
First, for a premium to occur the coupon (still set at 10% semi-annually) must exceed the market yield. You get more coupon than yield, you pay more. We will assume an 8% semi-annual market yield (YTM). Remember, as in the earlier cases, the bond carries a 10% coupon, paid semi-annually. We are still analyzing the same bond.
| Case 3 | |||
|---|---|---|---|
| Period | Coupon | PVF @ .04 | PVCF |
| 1 | $50.00 | .9615 | 48.08 |
| 2 | 50.00 | .9246 | 46.23 |
| 3 | 50.00 | .8890 | 44.45 |
| 4 | 50.00 | .8548 | 42.74 |
| 5 | 50.00 | .8219 | 41.10 |
| 6 | 50.00 | .7903 | 39.52 |
| 7 | 50.00 | .7599 | 37.00 |
| 8 | 50.00 | .7307 | 36.54 |
| 9 | 50.00 | .7026 | 35.13 |
| 10 | 50.00 | .6756 | 33.78 |
| 1,000.00 | 675.60 | ||
| 1,500.00 | 1,080.27 | ||
Here is a summary of the price change, i.e. the bond’s Maturity Pull.
| Period | Cash Flow | Present Value Factor | Present Value | Difference |
|---|---|---|---|---|
| 9 | 50 | .7026 | (35.13) | (68.91) |
| 10 | 50 | .6756 | (33.78) | |
| 1000 | (.6756) | (675.60) | 55.10 | |
| 1000 | .7307 | 730.70 | ||
| ($13.81) | ($13.81) | |||
After one year’s time (or two half-year periods), the premium will be reduced as follows:
1,080.27 – $13.81 = $1,066.06 = 106.646%.
It will continue its downward path until it hits Par at maturity.
