# Depreciation (finance)

Amortization of debt ( bank or bond ) is the part of capital which is paid every due periodic (eg monthly).

This payment is made at the same time as the interest due for the same period. The total payment (amortization + interest) at each maturity is called, according to its frequency, the monthly payment, quarterly or annuity . There are two main possible depreciation formulas: constant amortization or constant annuity.

In the event of constant depreciation, the payment made will decrease each time by an amount equivalent to the amount of interest on the capital repaid at the time of the previous payment.

In case of constant annuity, the repaid capital will increase in the same proportions as before.

Depreciation can be in fine , which consists of paying interest throughout the loan and then repaying the principal when due. This can be used to finance the production of a special order from a customer, the settlement of the sale price then allowing to repay the loan.

## Amortization of a fixed rate loan with constant monthly payments

### Amount of monthly payments

It is possible to establish the formula giving the monthly payment {\ displaystyle M}repayment of a loan , an amount (or capital) noted{\ displaystyle K}, carried out at an annual fixed interest rate{\ displaystyle T}, while {\ displaystyle n} monthly payments, proceeding as follows.

The monthly rate is given by {\ displaystyle r = {\ sqrt [{12}] {(T + 1)}} – 1}.

The remaining capital due after {\ displaystyle i} monthly payments is noted {\ displaystyle K_ {i}}.

By definition, {\ displaystyle K_ {n} = 0}since after the payment of the n- th and last monthly installments the loan is totally refunded.

After the first monthly payment {\ displaystyle K_ {1} = K (1 + r) -M}.

After the second monthly payment {\ displaystyle K_ {2} = K_ {1} (1 + r) -M} is {\ displaystyle K_ {2} = K (1 + r) ^ {2} -M (1 + r) -M}.

After the third monthly payment {\ displaystyle K_ {3} = K_ {2} (1 + r) -M = K (1 + r) ^ {3} -M (1 + r) ^ {2} -M (1 + r) -M }.

And this until the nth and last monthly payment.

{\ displaystyle K_ {n} = K (1 + r) ^ {n} -M \ sum _ {k = 0} ^ {n-1} {(1 + r) ^ {k}}}.

As {\ displaystyle K_ {n} = 0} and {\ displaystyle \ sum _ {k = 0} ^ {n-1} {(1 + r) ^ {k}} = {\ frac {(1 + r) ^ {n} -1} {r}}} so {\ displaystyle M = K {\ frac {r (1 + r) ^ {n}} {(1 + r) ^ {n} -1}}} is {\ displaystyle M = K {\ frac {r} {1- (1 + r) ^ {- n}}}}.

By using the rule of The Hospital it is easy to verify that{\ displaystyle \ lim _ {r \ rightarrow 0 ^ {+}} {\ frac {r} {1- (1 + r) ^ {- n}}} = \ lim _ {r \ rightarrow 0 ^ {+} } {\ frac {1} {n (1 + r) ^ {- n-1}}} = {\ frac {1} {n}}}, and therefore in the case of a zero interest rate the previous formula is reduced to {\ displaystyle M = K / n}, what is expected.

### Applications

The previous formula is actually valid regardless of the rate of repayment, where r represents the interest rate over the period, n the number of periods, M the amount of the periodic repayment. If the loan agreement provides for incidental expenses, such as disability-death insurance , the amount of the monthly (or periodic) premium should be added to M to obtain the monthly installment actually paid by the borrower.

Moreover, it is possible to reverse the previous formula the borrowed capital for a given monthly payment, depending on the rate and duration of the credit.

It is easy once M calculated to establish the credit amortization table , the capital remaining due{\ displaystyle K_ {k}} and the amount of interest paid {\ displaystyle I_ {k}}after the kth monthly payment being given by the relations:

{\ displaystyle K_ {i} = K_ {i-1} (1 + r) -M} and {\ displaystyle I_ {i} = K_ {i-1} r}with {\ displaystyle i = 1, …, n} and the convention {\ displaystyle K = K_ {0}}.

The total amount of interest, noted I is obtained easily by subtraction between the total amount of the monthly payments and that of the capital borrowed:{\ displaystyle I = nM-K}. In the absence of accessories (expenses of files, insurance …) this amount will correspond to the total cost of the credit .

Examples:

• Capital K = 1000 € borrowed over one year, without insurance or file fees, at an annual rate T = 12% . In this case n = 12 and r = 1% , ie r = 0.01 in decimal form, the application of the preceding formula shows that the monthly payment is then M = 88.84 € . The total amount of interest is I = 66.08 € and here corresponds to the total cost of the credit given the absence of incidental expenses.
• Loanable capital K over a period of n = 24 months, for a monthly payment M = € 200 , and an annual rate T = 12% , ie r = 0.01 . The inversion of the preceding formula makes it possible to calculate that in this case K = 4248.68 €.

## Credit smoothing

In some financial transactions, such as home loans, it is common to use several loans, such as loans with low rates, different durations. Typically one of the loans, which it is possible to qualify as “principal”, will correspond to a borrowed capital and a longer duration than all the other loans, called “secondary” loans: in the case of a real estate purchase this loan will be the real estate loan, with a typical term of 15 to 25 years, as opposed to traditional assisted or financial loans, whose duration does not exceed 5 to 10 years at most.

It is possible to repay all these different loans independently, and the calculation of the different monthly payments is easy using the previous formula, and taking into account any additional costs (credit insurance, normally required to obtain a loan immovable). However, this way of doing things is generally not very interesting, because on the one hand the sum of the monthly payments of the different loans, calculated independently, may exceed the repayment capacities of the borrower (often set at 33% of the net income in France), and on the other hand, it is not possible to increase the monthly payment to benefit from the newly released repayment capacity at the maturity of short-term subordinated loans.

It is thus often interesting to smooth between them the different loans, that is to say, to adjust the monthly repayment of the main loan as and when the repayment of various secondary loans. It is therefore necessary to distinguish several phases of repayment of the main loan, the first having a smaller monthly installment, which increases in the following phases of the amount of the monthly payments of the various secondary loans when they come to maturity.

The calculation of “adjusted” monthly payments becomes more complicated, but uses the previous formula. For example, for a principal loan of borrowed capital{\ displaystyle K_ {p}}, at the fixed annual rate {\ displaystyle T_ {p}} corresponding to the monthly rate {\ displaystyle r_ {p}}, of a total duration of N months, repayable in two phases: the first of duration{\ displaystyle n_ {1}} month during which the loan is smoothed with a secondary capital loan {\ displaystyle K_ {s}}fixed annual rate {\ displaystyle T_ {s}} corresponding to the monthly rate {\ displaystyle r_ {s}}, and the second of duration {\ displaystyle n_ {2} = N-n_ {1}}, the calculation is carried out as follows.

The repayment repayment of the secondary loan is calculated directly from the previous formula:

{\ displaystyle M_ {s} = K_ {s} {\ frac {r_ {S}} {1- (1 + r_ {s}) ^ {- n_ {1}}}}} (In the case of a zero-rate assisted loan, this relationship is reduced to {\ displaystyle M_ {s} = K_ {s} / n_ {1}}).

Repayment payments for each phase of the main loan are noted respectively {\ displaystyle M_ {p1}} and {\ displaystyle M_ {p2}}, respectively. By definition of smoothing, the following relationship must be checked between the three monthly payments:

{\ displaystyle M_ {p1} + M_ {s} = M_ {p2}}.

Moreover, if {\ displaystyle K_ {p2}} is the remaining capital due at the end of the first phase, the amount {\ displaystyle M_ {p2}} of the repayment monthly of the second phase is also given by the previous calculation formula:

{\ displaystyle M_ {p2} = K_ {p2} {\ frac {r_ {p}} {1- (1 + r_ {p}) ^ {- n_ {2}}}}}.

It remains therefore to express the outstanding capital {\ displaystyle K_ {p2}}at the beginning of the second phase. Noting{\ displaystyle K_ {p, j}}the capital remaining due after payment of the jth monthly payment, with{\ displaystyle j = 1, …, N}, it’s obvious that {\ displaystyle K_ {p2} = K_ {p, n_ {1}}} and that as in the demonstration of the previous part it is possible to write for the different deadlines of the first phase of repayment of the principal loan, monthly payment {\ displaystyle M_ {p1}} :

{\ displaystyle K_ {p, 1} = K_ {p} (1 + r_ {p}) – M_ {p1}},
{\ displaystyle K_ {p, 2} = K_ {p, 1} (1 + r_ {p}) – M_ {p1} = K_ {p} (1 + r_ {p}) ^ {2} -M_ {p1 } (1 + r_ {p}) – M_ {p1}},

etc., then it comes easily by recurrence:

{\ displaystyle K_ {p, n_ {1}} = K_ {p2} = K_ {p} (1 + r_ {p}) ^ {n_ {1}} – M_ {p1} \ sum _ {j = 0} ^ {n_ {1} 1} {(1 + r_ {p}) ^ {j}} = K_ {p} (1 + r_ {p}) ^ {n_ {1}} + M_ {p1} {\ frac {1- (1 + r_ {p}) ^ {n_ {1}}} {r_ {p}}}}.

It is then possible to substitute this expression in that giving {\ displaystyle M_ {p2}} in terms of {\ displaystyle K_ {p2}} and eliminating {\ displaystyle M_ {p1}} by the relation between the different monthly payments, it comes all calculated computation:

{\ displaystyle M_ {p2} = {\ frac {K_ {p} r_ {p} (1 + r_ {p}) ^ {n_ {1}} + M_ {s} \ left ((1 + r_ {p} ) ^ {n_ {1}} – 1 \ right)} {(1 + r_ {p}) ^ {n_ {1}} – (1 + r_ {p}) ^ {- n_ {2}}}}}.

Example: a household whose borrowing capacity (excluding insurance costs and annexes) is limited to approximately  1,000 . This household requires a loan of 120 000  divided into two loans: a loan with 0% of capital s = 20000 € on n = 60 months, monthly payment M = 333,34 €, and a classic mortgage loan of nominal annual fixed rate T = 3.6%, ie a monthly rate r = 0.30%, of an amount K = 100 000 €, over n months.If the two loans are not smoothed, with the constraint of limiting the total of the two monthly payments to around  1,000 , the maximum amount that can be used to repay the mortgage is therefore around  670 . According to the formula linking capital and monthly payments, it will be necessary to extend the duration of the mortgage loan to n= 198 months (ie 16 years and a half, for a monthly payment of € 670.55, ie a total monthly payment of € 1003 with the loan assisted) to make the financing, with a total cost of credit excluding insurance of € 32,768. By performing smoothing of the two credits, involving two phases of repayment for the main mortgage, it is possible by using the above formulas to obtain a fixed monthly payment over the entire period, of the order of 1012 €, by reducing the total duration of the principal credit at n = 142 months, ie 12 years. It comes p1 = 679.41 € and p2 = 1012.74 €, the total cost of credit excluding insurance is then € 25,834.79. This example shows that the smoothing of credits makes it possible to appreciably reduce the duration of the main loan and thus its cost, for an equal amount of financing and an almost identical total monthly payment, compared to independent credits.