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 ?

Answer 1

I am only taking a stab at this so please do not consider what I have to say as authoritative.

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 $25$.

But this would depend on how profit/growth on the instrument is recognised. Is it seen as capital gains or income? If it is seen as income, what proportion of it is recognised in each period?

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.

Would be great if other members could help me out on this one as well... As I am not particularly sure of myself on this one.