## Write-up/Write-down formula

I received the following question from a student:

I saw from another study manual and learned that the formula for the bond’s write-up in the t-th year is:

(Ci – Fr) * v ^(n-t+1)

For example, the write up for the 15-th year of a 20 years term bond is: (Ci – Fr) * v ^(20-15+1) = (Ci – Fr) * v ^6

I went through the lessons on your website but could not find this formula. Can you explain what are write-up and write-down of a bond and show the formula for each? Or can you point me to the appropriate lesson on your website so I can go back and review?

My response was:

You are correct. I don’t explicitly cover this formula in the lesson (I think I do in a video solution but I’m not 100% positive). It really is just an extension of the idea from loan amortization. Remember if we borrow L and make a level payment of R for n periods we have:

and the principal repaid at time t is

For bonds, we make level payments and a ballon payment of C at time n. Start with the Premium/Discount formula for bonds:

If we isolate the a-angle-n on the right side (like in the basic loan amortization) we have:

If P > C then we have a premium bond. In other words, each coupon payment (Fr) is greater than the interest due (Ci). Because the bond investor receives a coupon larger than the interest due he must pay a premium for the bond. P – C is the amount of the premium and it “funded” by the difference in each coupon minus interest due (Fr – Ci). So you can treat as the basic loan case where L = P – C and R = Fr – Ci. So the amount of principal repaid (or in this case amount of write-down) for the t-th payment is

You can do the same analysis for a discount bond (coupon payment is less than interest due) and will get the amount of the write-up is