Premium

A premium bond is a bond whose price $P$ is above its redemption value $C$ . The excess $P - C$ is the premium.

$P > C$

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

$F$ is the face value, $r$ the coupon rate per period, $j$ the yield rate per period and $n$ the number of coupons left. The premium is positive exactly when $Fr > Cj$ , that is, when the modified coupon rate $g = Fr/C$ is above the yield rate: $P - C = C(g - j)\,a_{\overline{n}|j}$ .
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 above the yield rate, and the price is above par. When $C \neq F$ , compare the price with $C$ and the yield with $g$ , not with $F$ and $r$ .
The premium $P - C$ is written off gradually through the amortization of premium as the book value falls to $C$ at redemption.
The opposite case, a price below the redemption value, is a bond bought at a Discount.

Identifying a Premium Bond A $ $1{,}000$ par bond, redeemable at par, with 7% annual coupons is priced to yield 5%. Is it a discount or premium bond?

Answer Here $C = F = 1{,}000$ , so $g$ equals the 7% coupon rate, which exceeds the 5% yield. The price rises above the redemption value: $P > C = \$1{,}000$ The bond sells at a premium; the $ $(P - 1{,}000)$ premium is amortized against the above-market coupons over the bond's life.

A Premium Bond Priced Below Its Face Value A 10-year bond with face value $1{,}000$ and 5% annual coupons will be redeemed for $950$ . 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 $Cj = 950(0.052) = 49.40$ . The modified coupon rate is $g = \frac{Fr}{C} = \frac{50}{950} = 5.263\% > 5.2\%$ so the bond is bought at a premium:
 $\begin{align*} P - C &= (Fr - Cj)\,a_{\overline{10}|5.2\%} \\ &= (50 - 49.40)(7.6473) \\ &= 4.59 \\ P &= 950 + 4.59 \\ &= 954.59 \end{align*}$ 
The price is below the $1{,}000$ face value and the 5% coupon rate is below the 5.2% yield, yet the bond is bought at a premium of $ $4.59$ : it is redeemed for only $950$ , so its book value is written down from $954.59$ to $950$ .