Let's consider two examples to illustrate
what we mean by debt securities issued at
a discount or a premium.
First, a zero-coupon bond with a face value
of 100 is issued for 80.
The discount is 20.
Second, a zero-coupon bond with a face value
of 100 is issued at 110.
The premium is 10.
The discount of 20 and the premium of 10 represent
the total amount of interest
to be amortized over the life of the bond
on an accrual basis.
A discount would result in interest of 20
being accrued over the life of the bond.
The interest expense accruing in each period
is recorded as being
reinvested in the debt security.
So, it increases the principal amount outstanding.
This is consistent with the accrual of interest.
It is not an other economic flow in the form
of a revaluation.
The premium of 10 would result in accruing
a reduction in interest
over the life of the bond.
When issued at a premium, the negative amount
accruing in each period
reduces the value of the bond.
It also reduces interest expense.
Now, how would we calculate the accrued interest
resulting from the discount or the premium
on a debt security?
From our basic compound interest formula,
we derive the interest rate to accrue
the discount or premium in each period
where p equals the number of times 
interest accrues per year
and n equals the number of accounting periods
in which interest accrues 
over the entire period.
In other words, n is the entire period over
which interest accrues.
Based on this formula, we can say that
the maturity value divided by the issue price
to the power of 1 divided by n multiplied by p
equals 1 plus the interest rate divided by p.
Let's apply this formula now to our earlier
example.
We have a zero-coupon bond with a face value
and maturity value of 100.
Its maturity is 5 years, so n is equal to
5.
This bond is issued at a price of 80.
The interest accrued from the discount on
the debt security
is calculated monthly.
So, p is equal to 12.
The interest rate can therefore be calculated
as follows:
100 divided by 80 to the power of 1 divided
by the result of 5
multiplied by 12.
This means the interest rate that we will
use is 0.3726% per month.
We can now calculate the outstanding principal
of the bond and the accrued interest.
At the beginning of month 1, the issue price
is equal to 80.
During the first month, interest of 0.298
accrues.
This amount is reinvested in the bond, which
means, at the end of the first month,
the value of the bond is 80.298.
At the end of the second month, the total
outstanding principal
is 80.597 and so on.
At the end of month 60 
— in other words, after 5 years —
the value of the outstanding principal is 100.
This value is equal to the par or face value.
The total interest accrued from the discount
on the debt security
is 20, which is 100 minus 80.
Note that this accrued interest of 20
excludes any possible coupon interest payments.
Those would be an additional interest component
of the bond
if it was not a zero-coupon bond.
Interest accrued but not yet paid is one of
the factors that determine
the nominal value of a debt instrument.
It does not matter whether this interest is
from a discount or a premium
and/or from a coupon.
