Using compound interest rate in wrong way

Question

I will explain the problem with an example.

Today (14/03/2021) y agree a Zero-Coupon Mortgage with a nominal of a milion dolars an with an annual interest rate compounded annualy and with an ACT/360 basis of 3% and with a repayment at 31/07/2022.

As between the two dates there are 504 days what we will do is the following to compute the final payment value:

F.V. = P.V * (1 + r/m)^(m*t) 

And as m = 1:

F.V. = P.V * (1 + r)^(t) = 1M * (1 + 0.03)^(504/360) = 1.042.250,5059

But the think is that, the interest is compounded only once (only compounded at 14/03/2022) and from that date until the end as no more interest is reinvested the interest earn untill maturity (for me) should be based on a normal interest rate so it should be divided intro first period 14/03/2021 until 14/03/2022 (annual compounding)+ period from 15/02/2022 to end (simple compounding) so:

F.V. = 1M * (1 + 0.03)^(365/360) * (1 + 0.03*(139/360)) = 1.042.358,67431

If it is not in this way how I have to threat this broken period as in this case the compounding for the second period will not occur until 14/03/2023.

Thank you in advance

Answer 1

I respond myself using the principle of no-arbitrage (The absence of opportunities to earn a risk-free profit with no investment).

Imagine we have 2 risk free bonds from the same country with the same maturity and all the same characteristics same calculus basis (ACT/360)... except for the compounding: one with annual compounding and another one with continious compounding.

The first bond with annual compounding have an ACT/360 basis and an interest rate of 3%.

As both bonds are exactly the same (except for the compounding) they have to provide the same future cash flow and must be sell for the same price.

Having that we can convert the annual interest rate to have the continious compounding rate that gives the same interest (That is: What i'm 100% shure is that if the bond expires in exactly 1 year (14/03/2022) the interest payed for both bonds should be the same so:

P.V = 1M F.V.(continious) = 1M * e^(r(c)*365/360) F.V.(annual) = 1M * (1+r(a))^(365/360) = 1M * (1+0.03)^(365/360)

So having F.V.(continious) = F.V.(annual) we have:

e^(r(c)*365/360) = (1+0.03)^(365/360)

Applying ln to each side we have:

r(c) *365/360 = 365/360 * ln(1.03) -> r(c) = ln(1.03)

Now we have the equivalent continious compounding for the bond, as compounding is continious we can be shure that the price for the bond at 31/07/2022 should be:

F.V. = 1M * e^(504/360 * ln(1.03)) = 1.042.250,51 eur

As it is compounded infinitesimaly we are shure that we can apply this. As this have to be the same value of the annual compounding due to no-arbitrage rule, we prove that ¡the formula that we could and should use is:

F.V. = P.V * (1 + r/m)^(m*t)

For all t in reals (not only in discrete times depending on the compounding)