Book Value

Study guide: Exam FM-2

Book value of a bond is the value assigned to the bond on the issuer's or investor's books, typically equal to the present value of future cash flows at the original yield rate.

Prospective formula (present value of remaining cash flows):

$BV_k = Fr \cdot a_{\overline{n-k}|j} + C \cdot v^{n-k}$

Retrospective formula (price accumulated minus coupons accumulated):

$BV_k = P(1+j)^k - Fr \cdot s_{\overline{k}|j}$

The book value of a bond at time $k$ (immediately after the $k$ -th coupon is paid) is the amortized value of the bond at that point, using the original Yield Rate $j$ .
It equals the present value of all remaining cash flows (coupons and redemption) discounted at $j$ , and also equals the price accumulated at $j$ minus the accumulated coupons.
At time 0, $BV_0 = P$ (the purchase price). At time $n$ , $BV_n = C$ (the redemption value).
For a premium bond ( $P > C$ ), the book value decreases from $P$ toward $C$ over time (amortization of premium). For a discount bond ( $P < C$ ), it increases from $P$ toward $C$ (accumulation of discount).
Book value is used in the Amortization Schedule for bonds: each coupon period the investor earns $j \cdot BV_{k-1}$ in yield, and the difference between the coupon $Fr$ and the yield $j \cdot BV_{k-1}$ is the write-down (premium) or write-up (discount) of the book value.

Book Value of a Premium Bond A $ $1{,}000$ face value 3-year annual-coupon bond has coupon rate $8\%$ and yield rate $6\%$ . Find the book value after the 2nd coupon.

Answer First find the price: $P = 80\,a_{\overline{3}|6\%} + 1000\,v^3 = 80(2.6730) + 1000(0.839619) = 213.84 + 839.62 = \$1{,}053.46$ .

Prospective: $BV_2 = 80\,a_{\overline{1}|6\%} + 1000\,v^1 = 80(0.9434) + 1000(0.9434) = 75.47 + 943.40 = \$1{,}018.87$

Retrospective: $BV_2 = 1053.46(1.06)^2 - 80\,s_{\overline{2}|6\%} = 1053.46(1.1236) - 80(2.0600) = 1183.67 - 164.80 = \$1{,}018.87$

Both methods give $ $1{,}018.87$ . The book value has moved from $ $1{,}053.46$ at time 0 toward $ $1{,}000$ at maturity, as expected for a premium bond.