# What is the difference between a book value and a market value?

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I would like to understand the following problem.

A 2yr zero-coupon bond has an annual yield rate of 11% per year. A 4yr zero-coupon bond has an annual yield rate of 19% per year. Find the difference between the book value and the market value of a par value 100 4yr zero-coupon bond in two years.

Which one represents the present (at time 2) value of the bond that can be calculated as 400(1.19)−2 ?

If we buy a two year zero coupon bond at time zero (at the above mentioned rates assumed to be annual effective rates), we would pay $\frac{100}{1.11^2}=81.16224$ and $\frac{100}{1.19^4}=49.86688$ respectively. The book value of an instrument in this context, I suspect is cost price plus amortization/accretion. In other words, when is “profit” from the instrument recognised? If profit is allocated to each year linearly, we can say that $\frac{100-81.16224}{2}$ is realised in each year for the 2-year-zero. And that $\frac{100-49.86688}{4}$ Is realised each year on the 4-year-zero. So after two years (at maturity) the two year zero would have market value and book value of $100$. The 4-year-zero would have a book value of $49.86688+2\times12.53328=74.93344$. So the difference would be approximately $25$.
If all return are seen as capital gains and only realized at maturity. Then the book value of the two-year-zero would be $100$ at time $2$. and the book value of the four-year-zero would $49.86688$ (Purchase Price) at every date before maturity.