7.3 Examination of YTC – YTM Relationships
1. For a Par Bond, YTM = YTC. (This assumes no call penalty.)
- Let’s use the example of a 10%, five-year semi-annual bond, discounted at 10%, and therefore, priced at par.
| 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 |
- What if the bond were called? We will assume that there is no call penalty and that the bond is expected to be called at Par in three years. At the same discount rate as the YTM, or 10% (semi-annually), the bond price is still Par. This is illustrated below and reflects that YTM = YTC.
| Period | Coupon | YTC 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 |
| 1,000.00 | .7462 | 746.21 | |
| 1,300.00 | 1,000.00 |
- The “disappearance” of the coupons is equally offset by the gain in present value of the final, principal payment.
- If there were a call penalty, the YTC would exceed the YTM.
2. For a Discount Bond, YTC > YTM.
- We will use the same bond example as before. However, the discount rate is now 12% and the bond is priced at a discount.
| 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 | |
| 926.43 |
|||
If the bond is callable at par in three years (with two years to go to final maturity), but is not expected to be called, the bond’s price will be 950.88 – due only to maturity pull. We see that in the table below. If we use the same discount rate as the YTM, which in this instance is 12%, the bond’s price should be higher due to maturity pull.
- In three years, the bond will either be called or not! If not, the bond’s price, using the same YTM, will be 95.088% due to maturity pull. If called, the bondholder will receive 100% earlier. Thus, for a discount bond, YTC > YTM.
| 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 |
| 1,000.00 | .7050 | 705.00 | |
| 950.88 | |||
- In other words, this bond will naturally experience maturity pull over the following years. In addition, if the bond is called and paid off at or above par, the yield-to-call will be higher than the YTM. This results from a large incremental cash flow which comes into play, adding substantial PV to Price.
- In reality, what are the prospects that an issuer will call away a discount bond? Why call the bond at Par if the bond is trading at 92% in the open market?
- Assuming the bond will be called away at Par or above Par, it would be necessary to re-calculate the new YTC iteratively or use a financial calculator.
Below is another template that you may use in order to – by trial and error – arrive at the YTC. Without a bond calculator, you are in for a lot of number-crunching work.
| Period | Coupon | PVF @ YTC | PVCF |
|---|---|---|---|
| 1 | $50.00 | ||
| 2 | 50.00 | ||
| 3 | 50.00 | ||
| 4 | 50.00 | ||
| 5 | 50.00 | ||
| 6 | 50.00 | ||
| 1,000.00 | |||
| 926.43 | |||
These examples have been worked out the “long way,” but you can certainly do them the “short way,” by using annuity tables – or a financial calculator.
3. For a Premium Bond, YTC < YTM. Maybe.
- What does this assume about the relationship between the size of the call penalty and bond premium?
- These relationships are not so simple – as ranked above – and bear some discussion. The size of the call penalty is relevant.
