2.7 Fixed Income Securities: Dollar Price and Yield-to-Maturity Calculation (Solution)
It is important not only to solve problems, but to interpret them as well.
Discount Rates
| .08 | .10 | .12 | |
| Coupon (PVAF) | 8.1109 | 7.7217 | 7.3601 |
| Par Value (PVF) | 0.6756 | 0.6139 | 0.5584 |
Dollar Values
| Coupon ($50) | 405.55 | 386.10 | 368.01 |
| Par ($1,000) | 675.60 | 613.90 | 558.40 |
| Total/Price | $1,081.15 | $1,000.00 | 558.40 |
| Quoted Price | 108.115% | 100% or “Par” | 92.641% |
| Premium | Par | Discount | |
| Cpn > YTM | Cpn > YTM | Cpn < TYM |
- Note that this illustration uses ordinary annuity factors, rather than simple present value factors.
- The annuity factors represent the sums of the simple present value factors.
- Premium/(Discount):
- You get more (less), you pay more (less)!
- You get more (less) coupon than yield, you pay more (less)!
Thus, the present value – or price – of the par bond represents that, for say $1,000, the investor will receive payments at various future times totaling $1,500 (i.e., ten $50 coupon payments plus the one-time principal payment). In fact, the future value of $1,000 at a semi-annual compound rate of 10% equals $1,000 (1.05)10 = $1,628.90.
Alternatively, the future value of the $50 annuity plus the face value have a combined future value of ($50) (12.578) + ($1,000) (1.05)0 = $1,628.90. This shows that the present value, or price, as calculated is equivalent to the future value that the bond’s cash flows produce. Actually, in mathematical terms, this is a kind of “truism.”
The same reasoning and calculations hold true for premium and discount bonds. For instance, using the discount bond above, the future value of its price equals the future value of its cash flows. 926.41 (1.7908) = $50 (13.181) + 1,000.
