# A Review of the Main Issues on the Loan Contracts: Asymmetric Information, Poor Transparency, and Hidden Costs

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## Abstract

**:**

## 1. Introduction

## 2. The Main Loan Amortization Schedules

- Straight-line;
- Annuity;
- Bullet;
- Balloon (amortization payments and large end payment);

#### 2.1. Straight-Line

_{k}) represents the annual percentage interest rate (APR) over the period t

_{k}. The term D

_{cc}represents the so called day count convention, which determines how interest accrues over time for a variety of investments, including loans (Ross et al. 2000). It is clear from (1) that the principal amount during the repayment schedule is fixed and constant, while the accrued interest I

_{k}may be computed as follow:

_{k}for each period k. As it is evident from Equation (5), the installment determined by this amortization scheme is not constant and therefore, it is not often used by credit institutions—also due to the fact that it presents a rate of reduction of the outstanding balance, which overall involves—for the same interest rate—lower interest payments with respect to the other amortization schemes mentioned below.

#### 2.2. Annuity

_{m}) constant for the entire duration of the amortization (a particularly frequent case in banking practice), by applying simplifications referred to as the geometric progression of reason included in the compact [0,1], the model becomes:

_{1/m}represents the compounded infra-annual interest rate as the payments are made with m periodicity (monthly or quarterly etc.). In Equation (7), we determine the corresponding installment R

_{m}. The full loan duration has been denoted with n. The outstanding loan balance D

_{k}is computed in Equation (8), while the portion of interest “I

_{k}” and principal “Q

_{k}” for each payment is reported in Equations (9) and (10), respectively.

#### 2.3. Bullet

#### 2.4. Balloon

## 3. The Loan Interest Rate Framework and the Issue of Asymmetric Information and Hidden Costs

_{1}” and “t

_{2}”, the future value increases to equal initial principal; while Equation (24) states that if there is no investment, which is to say that the duration of the investment is zero, the initial capital remains as it is.

_{i}” the corresponding amount of accrued simple interest, and “M” the related future value of loaned principal amount, respectively. On the hand, the compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the next period is then earned on the principal sum plus previously accumulated interest.

**applied**price and condition including, for credit agreements, any higher charges …”. Well, through the covert adoption of an annuity-type amortization scheme, the bank does not in fact indicate the applied interest rate, as the one indicated in the contract—as specified in the next sections—is the nominal rate not the actual/applied interest rate, therefore, the actual loan contract costs remain undetermined or at least not clearly indicated. We report an instance of loan contract with annuity amortization scheme. We agreed the initial loaned principal amount C. This information is then combined with the periodicity of payments (monthly, quarterly, half-yearly) and sometimes the value of the infra-annual interest rate and installment amount. The following Table 1 reports a classical instance of the so structured loan (in Euro currency: C = 100,000,00; i = 5.00%; m = 2 (half-yearly); n = 20; i

_{1/m}= 5/2%).

_{eff}will obviously be different from the rate i’ (initial APR) and I’’ (variable dynamic APR defined in the contract) and will be the solution of the equation indicated (34).

_{eff}” differs in one important respect from the annual percentage rate (APR)—the APR method converts this weekly/monthly/quarterly interest rate into what would be called an annual rate, which does not take into account the effect of compounding. By contrast, in the EIR, the periodic rate is annualized using compounding. It is the standard in the European Union and a large number of countries around the world. As previously described, this is also required by the banking regulations in force in Italy.

_{cc}) on the cost of the loan analyzed.

_{d}) with respect to the number of actual days of the year (N

_{y}), the calculation of the effective rate can thus be re-determined:

- Principal amount “C”;
- APR interest “i”;
- Periodicity payments “m”;
- Loan duration (in years) “n”;
- Indexing policy “f” of the interest rate (in case of variable interest rate);
- Adopted amortization algorithm: Annuity;

#### 3.1. Amortization Schedule Number 1 (Usually Adopted by the Banks)

_{1/m}” equal to the portion of the APR rate specified in the contract according to the adopted periodicity “m”:

_{i}for a single year, the following inequality is obtained:

#### 3.2. Amortization Schedule Number 2

_{1/m}” the infra-annual interest rate as per agreed periodicity “m”. As for the previous case, if we replace the above equation to compute the amount of accrued interest Q

_{i}for a single year, the following identity is obtained:

_{i}during each of the amortized years corresponds exactly to the agreed interest “i”. No issues of hidden costs or transparency are present in this possible amortization scheme which, unfortunately, the banks almost never adopt, at least in the Italian context.

#### 3.3. Amortization Schedule Number 3

_{m}” (including both principal and interest) from (43) to (44) through the simple interest algorithm. After that, we proceed to develop an annuity amortization algorithm (see Equation (45)) by using the installment value computed from Equation (43). In this way, we overcome the limitation of simple interest framework and at the same time we are able to provide an amortization schedule computed at the agreed interest APR, but with reduced amount of accrued interests due to usage of a simple interest-based algorithm. Specifically, it is possible to calculate the value of the outstanding paid principal at time k, for a generic amortization schedule structuring with simple interest:

## 4. The EAPR: The Effective Annual Percentage Interest Rate

_{0}the date of stipulation of the loan agreement and C

_{s}is the amount of drawdown, while t

_{s}and t

_{j}represent the related time interval, expressed in years and dates of the first drawdown (in t

_{0}), and finally D

_{j}denoted the amount of a repayment or payment of charges.

^{j}

_{ut}that the borrower must pay at the time of obtaining the loan, and with s

^{i}

_{k}the recurrent costs that the borrower must pay for each payment R

_{k}during the whole amortization.

_{0}:

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Dynamic of the the effective annual percentage interest rate (EAPR) indicator with the variation of the parameter m’ of the Equation (48).

**Figure 2.**Amortization schedule of the loan reported in Table 6: (

**a**) Cumulative interests; (

**b**) amount of principal for each payment; (

**c**) amount of interests for each payment; (

**d**) decreasing outstanding balance.

**Figure 3.**Benchmark in terms of paid interests for the loan reported in Table 6: The blue curve represents the cumulative paid interests for amortization schedule number 1 (Table 8); the red curve represents the cumulative paid interests for amortization schedule number 2—APR = EAPR—(Table 10); the black curve represents the cumulative paid interests for amortization schedule number 3—simple interest—(Table 12).

**Figure 4.**Amortization schedule of the loan reported in Table 13: (

**a**) Cumulative interests; (

**b**) amount of principal for each payment; (

**c**) amount of interests for each payment; (

**d**) decreasing outstanding balance.

**Figure 5.**Benchmark in terms of paid interests for the loan reported in Table 14: The blue curve represents the cumulative paid interests for amortization schedule number 1 (Table 15); the red curve represents the cumulative paid interests for amortization schedule number 2—APR = EAPR—(Table 17); the black curve represents the cumulative paid interests for amortization schedule number 3—simple interest—(Table 19).

Progress | Installment | Principal | Interest | Decreasing Balance | Cumulative Interest |
---|---|---|---|---|---|

1 | 6414.713 | 3914.713 | 2500 | 100,000.00 | 0 |

2 | 6414.713 | 4012.581 | 2402.132 | 96,085.29 | 2500 |

3 | 6414.713 | 4112.895 | 2301.818 | 92,072.71 | 4902.132 |

4 | 6414.713 | 4215.718 | 2198.995 | 87,959.81 | 7203.95 |

5 | 6414.713 | 4321.111 | 2093.602 | 83,744.09 | 9402.945 |

6 | 6414.713 | 4429.138 | 1985.575 | 79,422.98 | 11,496.55 |

7 | 6414.713 | 4539.867 | 1874.846 | 74,993.84 | 13,482.12 |

8 | 6414.713 | 4653.363 | 1761.349 | 70,453.98 | 15,356.97 |

9 | 6414.713 | 4769.698 | 1645.015 | 65,800.61 | 17,118.32 |

10 | 6414.713 | 4888.94 | 1525.773 | 61,030.92 | 18,763.33 |

11 | 6414.713 | 5011.163 | 1403.549 | 56,141.98 | 20,289.11 |

12 | 6414.713 | 5136.443 | 1278.27 | 51,130.81 | 21,692.66 |

13 | 6414.713 | 5264.854 | 1149.859 | 45,994.37 | 22,970.93 |

14 | 6414.713 | 5396.475 | 1018.238 | 40,729.52 | 24,120.78 |

15 | 6414.713 | 5531.387 | 883.3261 | 35,333.04 | 25,139.02 |

16 | 6414.713 | 5669.671 | 745.0414 | 29,801.66 | 26,022.35 |

17 | 6414.713 | 5811.413 | 603.2996 | 24,131.98 | 26,767.39 |

18 | 6414.713 | 5956.699 | 458.0143 | 18,320.57 | 27,370.69 |

19 | 6414.713 | 6105.616 | 309.0968 | 12,363.87 | 27,828.7 |

20 | 6414.713 | 6258.256 | 156.4564 | 0 | 28,137.8 |

APR (%) | EIR (%) | Periodicity | Day Count Convention |
---|---|---|---|

5.00 | 5.11619 | Monthly | 360/360 |

5.00 | 5.09453 | Quarterly | 360/360 |

5.00 | 5.06250 | Half-yearly | 360/360 |

3.00 | 3.04159 | Monthly | 360/360 |

3.00 | 3.03391 | Quarterly | 360/360 |

3.00 | 3.02250 | Half-yearly | 360/360 |

**Table 3.**Amortization schedule day count convention (D

_{cc}) = 360/360 − (C = 250,000.00; I = 3.00%; m = 2 (half-yearly); n = 20; i1/m = 3/2%).

Progress | Installment | Principal | Interest | Balance | Cumulative Interest |
---|---|---|---|---|---|

1 | 14,561.43 | 10,811.43 | 3750 | 250,000.00 | 0 |

2 | 14,561.43 | 10,973.61 | 3587.83 | 239,188.57 | 3750.00 |

3 | 14,561.43 | 11,138.21 | 3423.22 | 228,214.96 | 7337.83 |

4 | 14,561.43 | 11,305.28 | 3256.15 | 217,076.75 | 10,761.05 |

5 | 14,561.43 | 11,474.86 | 3086.57 | 205,771.47 | 14,017.20 |

6 | 14,561.43 | 11,646.98 | 2914.45 | 194,296.61 | 17,103.78 |

7 | 14,561.43 | 11,821.69 | 2739.74 | 182,649.62 | 20,018.23 |

8 | 14,561.43 | 11,999.01 | 2562.42 | 170,827.93 | 22,757.97 |

9 | 14,561.43 | 12,179.00 | 2382.43 | 158,828.92 | 25,320.39 |

10 | 14,561.43 | 12,361.69 | 2199.75 | 146,649.92 | 27,702.82 |

11 | 14,561.43 | 12,547.11 | 2014.32 | 134,288.23 | 29,902.57 |

12 | 14,561.43 | 12,735.32 | 1826.12 | 121,741.12 | 31,916.89 |

13 | 14,561.43 | 12,926.35 | 1635.09 | 109,005.80 | 33,743.01 |

14 | 14,561.43 | 13,120.24 | 1441.19 | 96,079.46 | 35,378.10 |

15 | 14,561.43 | 13,317.05 | 1244.39 | 82,959.21 | 36,819.29 |

16 | 14,561.43 | 13,516.80 | 1044.63 | 69,642.17 | 38,063.68 |

17 | 14,561.43 | 13,719.55 | 841.88 | 56,125.37 | 39,108.31 |

18 | 14,561.43 | 13,925.35 | 636.09 | 42,405.81 | 39,950.19 |

19 | 14,561.43 | 14,134.23 | 427.21 | 28,480.47 | 40,586.28 |

20 | 14,561.43 | 14,346.24 | 215.19 | 0.00 | 41,013.49 |

**Table 4.**Amortization schedule D

_{cc}= 360/365 − (C = 250,000.00; I = 3.00%; m = 2 (half-yearly); n = 20; i1/m = 3/2%).

Progress | Installment | Principal | Interest | Balance | Cumulative Interest |
---|---|---|---|---|---|

1 | 14,531.96 | 10,833.33 | 3698.63 | 250,000.00 | 0.00 |

2 | 14,531.96 | 10,993.60 | 3538.36 | 239,166.67 | 3698.63 |

3 | 14,531.96 | 11,156.24 | 3375.71 | 228,173.08 | 7236.99 |

4 | 14,531.96 | 11,321.30 | 3210.66 | 217,016.83 | 10,612.70 |

5 | 14,531.96 | 11,488.79 | 3043.17 | 205,695.54 | 13,823.36 |

6 | 14,531.96 | 11,658.76 | 2873.20 | 194,206.75 | 16,866.52 |

7 | 14,531.96 | 11,831.25 | 2700.71 | 182,547.99 | 19,739.72 |

8 | 14,531.96 | 12,006.28 | 2525.67 | 170,716.74 | 22,440.43 |

9 | 14,531.96 | 12,183.91 | 2348.05 | 158,710.46 | 24,966.10 |

10 | 14,531.96 | 12,364.17 | 2167.79 | 146,526.55 | 27,314.15 |

11 | 14,531.96 | 12,547.09 | 1984.87 | 134,162.38 | 29,481.94 |

12 | 14,531.96 | 12,732.72 | 1799.24 | 121,615.29 | 31,466.81 |

13 | 14,531.96 | 12,921.09 | 1610.87 | 108,882.58 | 33,266.05 |

14 | 14,531.96 | 13,112.25 | 1419.70 | 95,961.49 | 34,876.91 |

15 | 14,531.96 | 13,306.24 | 1225.71 | 82,849.24 | 36,296.62 |

16 | 14,531.96 | 13,503.10 | 1028.86 | 69,543.00 | 37,522.33 |

17 | 14,531.96 | 13,702.87 | 829.08 | 56,039.90 | 38,551.19 |

18 | 14,531.96 | 13,905.60 | 626.36 | 42,337.02 | 39,380.27 |

19 | 14,531.96 | 14,111.33 | 420.63 | 28,431.42 | 40,006.62 |

20 | 14,531.96 | 14,320.10 | 211.86 | 0.00 | 40,427.25 |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | Total Amount of Included “One-Off” Fees (Euro Currency) | Total Amount of Included Recurring Fees (Euro Currency) |
---|---|---|---|---|---|

250,000.00 | 3.00 | Half-yearly | 360/360 | 2400.00 | 2.00/installment |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | Total Amount of Included “One-Off” Fees (Euro Currency) | Total Amount of Included Recurring Fees (Euro Currency) |
---|---|---|---|---|---|

100,000.00 | 5.00 | Monthly (m = 12) | 360/360 | 1500.00 | 5.00/installment |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | APR/Monthly (%) | EIR (%) |
---|---|---|---|---|---|

100,000.00 | 5.00 | Monthly (m = 12) | 360/360 | APR/12 | 5.11619 |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | Total amount of Paid Interests (Euro Currency) | EIR (%) |
---|---|---|---|---|---|

100,000.00 | 5.00 | Monthly (m = 12) | 360/360 | 27,278.62 | 5.11619 |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | APR/monthly (%) | EIR (%) |
---|---|---|---|---|---|

100,000.00 | 5.00 | Monthly (m = 12) | 360/360 | ${i}_{1/m}={\left(1+APR\right)}^{\frac{1}{m}}-1$ | 5.00 |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | Total Amount of Paid Interests (Euro Currency) | EIR (%) |
---|---|---|---|---|---|

100,000.00 | 5.00 | Monthly (m = 12) | 360/360 | 26,628.24 | 5.00 |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | APR/Monthly (%) | EIR (%) |
---|---|---|---|---|---|

100,000.00 | 5.00 | Monthly (m = 12) | 360/360 | Simple Interest | 5.00 |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | Total Amount of Paid Interests (Euro Currency) | EIR (%) |
---|---|---|---|---|---|

100,000.00 | 5.00 | Monthly (m = 12) | 360/360 | 24,034.62 | 5,00 |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | Total Amount of Included “One-Off” Fees (Euro Currency) | Total Amount of Included Recurring Fees (Euro Currency) |
---|---|---|---|---|---|

250,000.00 | 3.00 | Quarterly (m = 4) | 360/365 | 2400.00 | 2.00/installment |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | APR/Quarterly (%) | EIR (%) |
---|---|---|---|---|---|

250,000.00 | 3.00 | Quarterly (m = 4) | 360/365 | APR/4 | 3.034076 |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | Total Amount of Paid Interests (Euro Currency) | EIR (%) |
---|---|---|---|---|---|

250,000.00 | 3.00 | Quarterly (m = 4) | 360/365 | 39,724.47 | 3.034076 |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | APR/Quarterly (%) | EIR (%) |
---|---|---|---|---|---|

250,000.00 | 3.00 | Quarterly (m = 4) | 360/365 | ${i}_{1/m}={\left(1+APR\right)}^{\frac{1}{m}}-1$ | 3.00 |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | Total amount of Paid Interests (Euro Currency) | EIR (%) |
---|---|---|---|---|---|

250,000.00 | 3.00 | Quarterly | 360/365 | 39,263.45 | 3.00 |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | APR/Quarterly (%) | EIR (%) |
---|---|---|---|---|---|

250,000.00 | 3.00 | Quarterly (m = 4) | 360/365 | Simple Interest | 3.00 |

Principal Amount (Euro Currency) | APR (%) | Periodicity | Day Count Convention | Total Amount of Paid Interests (Euro Currency) | EIR (%) |
---|---|---|---|---|---|

250,000.00 | 3.00 | Quarterly (m = 4) | 360/365 | 36,186.37 | 3.00 |

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## Share and Cite

**MDPI and ACS Style**

Rundo, F.; Di Stallo, A.L. A Review of the Main Issues on the Loan Contracts: Asymmetric Information, Poor Transparency, and Hidden Costs. *Economies* **2019**, *7*, 91.
https://doi.org/10.3390/economies7030091

**AMA Style**

Rundo F, Di Stallo AL. A Review of the Main Issues on the Loan Contracts: Asymmetric Information, Poor Transparency, and Hidden Costs. *Economies*. 2019; 7(3):91.
https://doi.org/10.3390/economies7030091

**Chicago/Turabian Style**

Rundo, Francesco, and Agatino Luigi Di Stallo. 2019. "A Review of the Main Issues on the Loan Contracts: Asymmetric Information, Poor Transparency, and Hidden Costs" *Economies* 7, no. 3: 91.
https://doi.org/10.3390/economies7030091