Discount

A discount bond is a bond whose price $P$ is below its redemption value $C$ . The shortfall $C - P$ is the discount.

$P < C$

$C - P = (Cj - Fr)\,a_{\overline{n}|j}$

$F$ is the face value, $r$ the coupon rate per period, $j$ the yield rate per period and $n$ the number of coupons left. The discount is positive exactly when $Fr < Cj$ , that is, when the modified coupon rate $g = Fr/C$ is below the yield rate: $C - P = C(j - g)\,a_{\overline{n}|j}$ .
Unless an exam question says otherwise, the redemption value equals the face value (SOA's FM notation note). With $C = F$ the test reduces to the familiar one: the coupon rate is below the yield rate, and the price is below par. When $C \neq F$ , compare the price with $C$ and the yield with $g$ , not with $F$ and $r$ .
The discount $C - P$ is earned gradually through the accumulation of discount as the book value rises to $C$ at redemption.
The opposite case, a price above the redemption value, is a bond bought at a Premium.

Identifying a Discount Bond A $ $1{,}000$ par bond, redeemable at par, with 4% annual coupons is priced to yield 6%. Is it a discount or premium bond?

Answer Here $C = F = 1{,}000$ , so $g$ equals the 4% coupon rate, which is below the 6% yield. The price falls below the redemption value to compensate the buyer: $P < C = \$1{,}000$ The bond sells at a discount; the buyer's return combines the coupons with the accretion of the $ $(1{,}000 - P)$ discount up to the redemption value at maturity.

A Discount Bond Priced Above Its Face Value A 10-year bond with face value $1{,}000$ and 5% annual coupons will be redeemed for $1{,}100$ . It is bought to yield 5.2% annually. Find the price, and decide whether the bond is bought at a premium or a discount.
Answer The coupon is $Fr = 50$ and $Cj = 1{,}100(0.052) = 57.20$ . The modified coupon rate is $g = \frac{Fr}{C} = \frac{50}{1{,}100} = 4.545\% < 5.2\%$ so the bond is bought at a discount:
 $\begin{align*} C - P &= (Cj - Fr)\,a_{\overline{10}|5.2\%} \\ &= (57.20 - 50)(7.6473) \\ &= 55.06 \\ P &= 1{,}100 - 55.06 \\ &= 1{,}044.94 \end{align*}$ 
The price is above the $1{,}000$ face value, yet the bond is bought at a discount of $ $55.06$ : it is redeemed for $1{,}100$ , so its book value is written up from $1{,}044.94$ to $1{,}100$ .