Base rate
5. Empirical results
5.2 The optimum cut-off rate of interest-rate markups
Based on the statistical analysis results, we identify the borrowers with relatively high default probability, which can reduce the loan risk and improve the operational performance of the bank. However, determining the optimum interest-rate markup is the key factor for banks’ loan business. After excluding the defaulting customers from the samples, we take 456 regular customers as the sample to calculate profits. We use the benchmark interest rate of 1.155% of the sample bank as the loan interest rate. Based on the current distribution of personal loan interest rates for the sample bank, the interest rates are divided into five intervals (𝐼𝐼1, 𝐼𝐼2, 𝐼𝐼3, 𝐼𝐼4, and 𝐼𝐼5) with the markup rates of 1%, 2%, 2.5%, 3%, and 4.5%. We employ the previous determined linear formula as follows:
𝑍𝑍𝑗𝑗 = 𝑊𝑊𝑥𝑥 = 𝑤𝑤0+ 𝑤𝑤1𝑥𝑥𝑗𝑗1+ 𝑤𝑤2𝑥𝑥𝑗𝑗2+ ⋯ + 𝑤𝑤𝑘𝑘𝑥𝑥𝑗𝑗𝑘𝑘 and 𝜋𝜋𝑗𝑗 = 1
1+𝑒𝑒−�𝑊𝑊𝑥𝑥′�
We calculate the π value of each borrower to determine the pricing of interest-rate markups (Table 10).
Based on the markup intervals, we substitute the profit calculation models for the profit functions 1 to 4 to calculate the profit maximization cut-off rates -
Table 10
Markup Intervals and Interest Rates
Unit: % Cut-off Rate C1 C2 C3 C4
Intervals I1 I2 I3 I4 I5
Markups 1% 2% 2.5% 3% .5%
Interest rates 2.155% 3.155% 3.655% 4.155% 5.655%
that is, the optimum cut-off rates: 𝐶𝐶4~𝐶𝐶1. Table 11 shows the increase in bank profits from the cut-off rates for each interest-rate markup interval.
The interest-rate markups of the samples are more concentrated in certain interest rate ranges, which mean most of the interest-rates markups of the samples fall within certain spreads, resulting in less samples of higher and lower interest rates. The main purpose of this paper is to verify the feasibility of our interest-rate markup model. The four quadrants of the profit function in this model are composed of profit and opportunity cost in order to make easier analysis of cut-off rates and more equal cut-off samples in each quadrant.
Therefore, the probability value in the profit function can be changed accordingly in the four quadrants of this model, so that the profit function changes in order to achieve maximum profit. We then find the cut-off rate for this study and use the markup rates of 1%, 2%, 2.5%, 3%, and 4.5%, in order to obtain the existences of real samples in each quadrant in their corresponding cute-off rate and hence observe the evolution from the change of interest intervals.
The monotonicity and changing direction basically do not change. The higher the risk is, the higher the interest rate. However, the interest-rate markup in the risk premium is different under different risk levels. Therefore, setting different ranges of interest-rate markups does not change the monotonicity that higher risk means higher interest-rate markups.
To collate the bank’s comprehensive loan data, we divide the customers into credit-granted customers and denied customers. However, because the proportion of defaulting customers in the samples is excessive, the cut-off rates of the various interest-rate markups concentrate in certain areas for credit-granted
ate Management Review Vol. 37 No. 1, 2017149 Table 11 The optimum cut-off rates le presents the profits of the bank within each cut-off rate. 𝐶𝐶1, 𝐶𝐶2, 𝐶𝐶3, and 𝐶𝐶
4are the cut-off rates of the five markup intervals. 𝐿𝐿𝑙𝑙is the number of t had equivalent low forecast and actual interest rates. 𝐿𝐿
ℎis the number of loans that had a forecast interest rate greater than the actual interest rate. e number of loans that had a forecast interest rate lower than the actual interest rate. 𝐻𝐻
ℎis the number of loans that had equivalent high forecast and nterest rates. Profit is the sum of the profits of each cut-off rate. Probability is the number of loans in the various conditions divided by the total f loans. 𝑪𝑪𝟏𝟏𝑪𝑪𝟐𝟐𝑪𝑪𝟑𝟑𝑪𝑪𝟒𝟒 𝐿𝐿𝑙𝑙 𝐿𝐿ℎ𝐻𝐻𝑙𝑙 𝐻𝐻ℎProfit 𝐿𝐿𝑙𝑙 𝐿𝐿ℎ𝐻𝐻𝑙𝑙 𝐻𝐻ℎProfit 𝐿𝐿𝑙𝑙 𝐿𝐿ℎ𝐻𝐻𝑙𝑙 𝐻𝐻ℎProfit 𝐿𝐿𝑙𝑙 𝐿𝐿ℎ𝐻𝐻𝑙𝑙 𝐻𝐻ℎProfit (Probability)(Probability)(Probability)(Probability) 023600.0056320013600.014838806800.022442403200.0260 48) (0.00)(0.52) (0.00)(0.70)(0.00)(0.30)(0.00)(0.85) (0.00)(0.15) (0.00)(0.93) (0.00)(0.07) (0.00) 023330.0058320013150.015438806440.023042403020.0264 48) (0.00)(0.51) (0.01)(0.70)(0.00)(0.29)(0.01)(0.85) (0.00)(0.14) (0.01)(0.93) (0.00)(0.06) (0.01) 0226100.00653200124120.0146385059120.0237421222110.0277 48) (0.00)(0.50) (0.02)(0.70)(0.00)(0.27) (0.03) (0.85) (0.00)(0.13) (0.02)(0.92) (0.01) (0.05) (0.02) 1212250.00783185114190.0167379847220.02424052018130.0260 48)(0.00)(0.47) (0.05) (0.70)(0.01) (0.25) (0.04) (0.83) (0.02) (0.10) (0.05) (0.89) (0.04) (0.04) (0.03) 13188450.008830517101330.01683652538280.02313824712150.0234 46) (0.03) (0.41) (0.10) (0.67)(0.04) (0.22) (0.07) (0.80) (0.06) (0.08) (0.06) (0.84) (0.10) (0.03) (0.03) 13186490.00913041895390.01733207828300.01753805011150.0231 45) (0.03) (0.41) (0.11) (0.66) (0.04) (0.21) (0.09) (0.70) (0.17) (0.06) (0.07) (0.84) (0.11) (0.002) (0.03) 532159750.00982665582530.014528013014320.012332410610160.0159 43) (0.07) (0.34) (0.16) (0.58) (0.12) (0.18) (0.12) (0.61) (0.29) (0.03) (0.07) (0.71) (0.23) (0.02) (0.04) 601201120.010721710564700.010825515011400.01362551759170.0069 36) (0.13)(0.26) (0.25) (0.48) (0.23) (0.14) (0.15) (0.56) (0.33) (0.02) (0.09) (0.56) (0.38) (0.02) (0.04) 71991260.011520011967700.008918522010410.0016155280417-0.0063 35) (0.15) (0.22) (0.28) (0.44) (0.26) (0.15) (0.15) (0.41) (0.48) (0.02) (0.09) (0.34) (0.61) (0.01) (0.04) 103971390.008715815661810.0054138268842-0.0042145290318-0.0074 26)(0.23) (0.21) (0.30) (0.35) (0.34) (0.14) (0.17) (0.30) (0.59) (0.02) (0.09) (0.32) (0.63) (0.01) (0.04) 131761580.007812519546900.0024136268646-0.003999337218-0.0135 20) (0.29) (0.17) (0.34) (0.27) (0.43) (0.10) (0.20) (0.30) (0.59) (0.01) (0.10) (0.21) (0.74) (0.01) (0.04) 157621750.0067952382598-0.000595309448-0.008880356119-0.0158 14) (0.34)(0.14) (0.38) (0.21) (0.52) (0.05) (0.22) (0.21) (0.68) (0.01) (0.10) (0.17) (0.78) (0.01) (0.04) 180252130.00805128016109-0.004350351055-0.013542394020-0.0207 08) (0.39) (0.05) (0.48) (0.11) (0.61) (0.04) (0.24) (0.11) (0.77) (0.00) (0.12) (0.09) (0.87) (0.00) (0.004)
customers under the condition of normal risk. This prevents the empirical model of this study from determining the optimum interest-rate markups under various risk types. Therefore, Procedure 1 is not performed. We then move directly to Procedure 2 and classify credit-granted customers to calculate the optimum cut-off rates.
Procedure 1: Calculate the optimum cut-off rate 𝑪𝑪𝟒𝟒
We divide the customers into two categories, 𝐼𝐼5 (with a markup of 4.5%) and 𝐼𝐼1~𝐼𝐼4 (with a markup of 1% to 3%), to calculate the optimum cut-off rate: 𝐶𝐶4 (Table 12).
Profit function 1
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃＝ �𝑅𝑅𝐼𝐼1~4×𝐿𝐿𝑁𝑁𝑙𝑙𝐶𝐶4� − ��𝑅𝑅𝐼𝐼5− 𝑅𝑅𝐼𝐼1~4� ×𝐻𝐻𝑁𝑁𝑙𝑙𝐶𝐶4� − �𝑅𝑅𝐼𝐼1~4×𝐿𝐿𝑁𝑁ℎ𝐶𝐶4� + �𝑅𝑅𝐼𝐼5× 𝐻𝐻𝑁𝑁ℎ𝐶𝐶4�. (1) The profits from accurate forecasts minus the losses of incorrect forecasts are the bank profits.
N = 456
Table 12
Calculate the optimum cut-off rate 𝑪𝑪𝟒𝟒
Forecast π values
The actual credit value of borrowers
(l) 𝐼𝐼1~𝐼𝐼4 (h) 𝐼𝐼5 (L) 𝐼𝐼1~𝐼𝐼4 𝐿𝐿𝑙𝑙𝐶𝐶4 𝐿𝐿ℎ𝐶𝐶4
(H) 𝐼𝐼5 𝐻𝐻𝑙𝑙𝐶𝐶4 𝐻𝐻ℎ𝐶𝐶4 L and H: The actual credit value of borrowers.
(L) 𝐼𝐼1~𝐼𝐼4: Based on the actual credit value of the borrowers, the interest-rate markups of 𝐼𝐼1~𝐼𝐼4 (an interest rate ranging between 2.155% and 4.155%) are offered.
(H) 𝐼𝐼5: Based on the actual credit value of the borrowers, the interest-rate markups of 𝐼𝐼5 (the interest rate 5.655%) are offered.
l and h: forecast π values.
(l) 𝐼𝐼1~𝐼𝐼4: when π ≤ 𝐶𝐶4, the interest-rate markups 𝐼𝐼1~𝐼𝐼4 (an interest rate ranging between 2.155% and 4.155%) are offered to borrowers.
(h) 𝐼𝐼5 when π > 𝐶𝐶4, the interest-rate markups 𝐼𝐼5 (the interest rate 5.655%) are offered to borrowers.
𝐿𝐿𝐶𝐶𝑙𝑙4 = The number of loans when the forecast interest rate of the π value and the interest rate based on the actual credit value of borrowers are 𝐼𝐼1~𝐼𝐼4 (2.155% ~ 4.155%).
𝐻𝐻𝑙𝑙𝐶𝐶4 = The number of loans when the forecast interest rates of the π value are 𝐼𝐼1~𝐼𝐼4 (2.155%~4.155%) and the interest rate based on the actual credit value of borrowers is 𝐼𝐼5(5.655%).
𝐿𝐿𝐶𝐶ℎ4 = The number of loans when the forecast interest rates of the π value are 𝐼𝐼5(5.655%) and the interest rate based on the actual credit value of borrowers is 𝐼𝐼1~𝐼𝐼4 (2.155% ~ 4.155%).
𝐻𝐻ℎ𝐶𝐶4 = The number of loans when the forecast interest rate of the π value and the interest rate based on the actual credit value of borrowers are 𝐼𝐼5(5.655%).
𝑅𝑅𝐼𝐼1~4= The average interest rate (2.834%) of loans 𝐼𝐼1~𝐼𝐼4. 𝑅𝑅𝐼𝐼5= The interest rate (2.834%) of loans 𝐼𝐼5.
The profit calculation formula can be rewritten as follows:
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃＝ �(2.834%) × 𝐿𝐿𝐶𝐶𝑙𝑙4
456� − �(2.821%) ×𝐻𝐻𝑙𝑙𝐶𝐶4
456� − �(2.834%) × 𝐿𝐿ℎ𝐶𝐶4
456� + �(5.655%) ×𝐻𝐻ℎ𝐶𝐶4 456�
The results indicate that when 𝐿𝐿𝑙𝑙𝐶𝐶4, 𝐿𝐿ℎ𝐶𝐶4, 𝐻𝐻𝑙𝑙𝐶𝐶4, and 𝐻𝐻ℎ𝐶𝐶4 are (421, 2, 22, 11), the maximum profit is 0.0277. Therefore, a cut-off rate (0.80) is the optimum cut-off rate: 𝐶𝐶4 (Table 11).
Procedure 2: Calculate the optimum cut-off rate 𝑪𝑪𝟑𝟑
We divide the customers into (𝐼𝐼1~𝐼𝐼3 with the markups of 1%, 2%, and 2.5%) and (𝐼𝐼4~𝐼𝐼5 with the markups of 3% and 4.5%) to calculate the optimum cut-off rate, 𝐶𝐶3 (Table 13).
Profit function 2
Profit = �𝑅𝑅𝐼𝐼1~3×𝐿𝐿𝑁𝑁𝑙𝑙𝐶𝐶3� − ��𝑅𝑅𝐼𝐼5− 𝑅𝑅𝐼𝐼1~4� ×𝐻𝐻𝑁𝑁𝑙𝑙𝐶𝐶4+ �𝑅𝑅𝐼𝐼4− 𝑅𝑅𝐼𝐼1~3� ×𝐻𝐻𝑙𝑙𝐶𝐶3𝑁𝑁−𝐻𝐻𝑙𝑙𝐶𝐶4� − �𝑅𝑅𝐼𝐼1~4×𝐿𝐿𝑁𝑁𝐶𝐶4ℎ +
𝑅𝑅𝐼𝐼1~3×𝐿𝐿𝐶𝐶3ℎ−𝐿𝐿𝑁𝑁𝐶𝐶4ℎ � + �𝑅𝑅𝐼𝐼5 ×𝐻𝐻𝑁𝑁ℎ𝐶𝐶4+ 𝑅𝑅𝐼𝐼4×𝐻𝐻ℎ𝐶𝐶3−𝐻𝐻𝑁𝑁 ℎ𝐶𝐶4� (2) 𝐿𝐿𝐶𝐶𝑙𝑙3 = The number of loans when the forecast interest rate of the π value and the
interest rate based on the actual credit value of borrowers are 𝐼𝐼1~𝐼𝐼3 (2.155% ~ 3.655%).
Table 13
Calculate the optimum cut-off rate 𝑪𝑪𝟑𝟑
Forecast π values (l) 𝐼𝐼1~𝐼𝐼3 (h) 𝐼𝐼4~𝐼𝐼5
The actual credit value of borrowers
(L) 𝐼𝐼1~𝐼𝐼3 𝐿𝐿𝑙𝑙𝐶𝐶3 𝐿𝐿ℎ𝐶𝐶3 (H) 𝐼𝐼4~𝐼𝐼5 𝐻𝐻𝑙𝑙𝐶𝐶3 𝐻𝐻ℎ𝐶𝐶3 L and H: The actual credit value of borrowers.
(L) 𝐼𝐼1~𝐼𝐼3: Based on the actual credit value of the borrowers, the interest-rate markups of 𝐼𝐼1~𝐼𝐼3 (an interest rate ranging between 2.155% and 3.655%) are offered.
(H) 𝐼𝐼4~𝐼𝐼5: Based on the actual credit value of the borrowers, the interest-rate markups of 𝐼𝐼4 and 𝐼𝐼5 (the interest rate 4.155%、5.655% ) are offered.
l and h: forecast π values.
(l) 𝐼𝐼1~𝐼𝐼3: when π ≤ 𝐶𝐶3, the interest-rate markups 𝐼𝐼1~𝐼𝐼3 (an interest rate ranging between 2.155% and 3.655%) are offered to borrowers.
(h) 𝐼𝐼4~𝐼𝐼5: when π > 𝐶𝐶3, the interest-rate markups 𝐼𝐼4 and 𝐼𝐼5 (the interest rates 4.155% and 5.655%) are offered to borrowers.
𝐻𝐻𝑙𝑙𝐶𝐶3 = The number of loans when the forecast interest rates of the π value are 𝐼𝐼1~𝐼𝐼3 (2.155%~3.655%) and the interest rates based on the actual credit value of borrowers are 𝐼𝐼4 and 𝐼𝐼5 (4.155%, 5.655% ).
𝐿𝐿𝐶𝐶ℎ3 = The number of loans when the forecast interest rates of the π value are 𝐼𝐼4
and 𝐼𝐼5 (4.155% and 5.655%) and the interest rate based on the actual credit value of borrowers is 𝐼𝐼1~𝐼𝐼3 (2.155% ~ 3.655%).
𝐻𝐻ℎ𝐶𝐶3 = The number of loans when the forecast interest rate of the π value and the interest rate based on the actual credit value of borrowers are 𝐼𝐼4 and 𝐼𝐼5 (4.155%, 5.655%).
𝑅𝑅𝐼𝐼1~3 = The average interest rate (2.710%) of loans 𝐼𝐼1~𝐼𝐼3. 𝑅𝑅𝐼𝐼4 = The interest rate (4.155%) of loans 𝐼𝐼4.
In this phase, the numbers of 𝐼𝐼5 and 𝐼𝐼4 are classified and categorized. The profit calculation formula can be rewritten as follows:
Profit = �(2.710%) ×𝐿𝐿456𝐶𝐶3𝑙𝑙 � − �(2.821%) ×45622 + (1.445%) ×𝐻𝐻𝑙𝑙𝐶𝐶3456−22� − �(2.834%) ×4562 + (2.710%) ×𝐿𝐿ℎ𝐶𝐶3456−2� + �(5.655%) ×45611 + (4.155%) ×𝐻𝐻ℎ𝐶𝐶3456−11�
The profit of accurate forecasts (𝐿𝐿𝑙𝑙𝐶𝐶3 and 𝐻𝐻ℎ𝐶𝐶3) minus the premium losses of underestimated loan risk (𝐻𝐻𝑙𝑙𝐶𝐶3) and the profit losses of overestimated loan risk is the sum total of profit. When 𝐿𝐿𝑙𝑙𝐶𝐶3, 𝐿𝐿ℎ𝐶𝐶3, 𝐻𝐻𝑙𝑙𝐶𝐶3, and 𝐻𝐻ℎ𝐶𝐶3 are (379, 8, 47, 22), the maximum profit is 0.0242, and the cut-off rate is 0.70 - that is, the optimum cut-off rate 𝐶𝐶3 (Table 11).
Procedure 3: Calculate the optimum cut-off rate 𝑪𝑪𝟐𝟐
We divide the customers into (𝐼𝐼1 and 𝐼𝐼2 with the markups of 1% and 2%) and (𝐼𝐼3, 𝐼𝐼4, and 𝐼𝐼5 with the markups of 2.5%, 3%, and 4.5%) to calculate the optimum cut-off rate 𝐶𝐶2 (Table 14).
Profit function 3
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �𝑅𝑅𝐼𝐼1~2×𝐿𝐿𝑁𝑁𝑙𝑙𝐶𝐶2� − ��𝑅𝑅𝐼𝐼5− 𝑅𝑅𝐼𝐼1~4� ×𝐻𝐻𝑁𝑁𝑙𝑙𝐶𝐶4+ �𝑅𝑅𝐼𝐼4− 𝑅𝑅𝐼𝐼1~3� ×𝐻𝐻𝑙𝑙𝐶𝐶3𝑁𝑁−𝐻𝐻𝑙𝑙𝐶𝐶4+
�𝑅𝑅𝐼𝐼3− 𝑅𝑅𝐼𝐼1~2� ×𝐻𝐻𝑙𝑙𝐶𝐶2𝑁𝑁−𝐻𝐻𝑙𝑙𝐶𝐶3� − �𝑅𝑅𝐼𝐼1~4×𝐿𝐿𝑁𝑁𝐶𝐶4ℎ + 𝑅𝑅𝐼𝐼1~3×𝐿𝐿ℎ𝐶𝐶3−𝐿𝐿𝑁𝑁 𝐶𝐶4ℎ + 𝑅𝑅𝐼𝐼1~2×𝐿𝐿ℎ𝐶𝐶2−𝐿𝐿𝑁𝑁 ℎ𝐶𝐶3� +
�𝑅𝑅𝐼𝐼5×𝐻𝐻𝑁𝑁ℎ𝐶𝐶4+ 𝑅𝑅𝐼𝐼4×𝐻𝐻ℎ𝐶𝐶3−𝐻𝐻𝑁𝑁 ℎ𝐶𝐶4+ 𝑅𝑅𝐼𝐼3×𝐻𝐻ℎ𝐶𝐶2−𝐻𝐻𝑁𝑁 ℎ𝐶𝐶3� (3)
Table 14
Calculate the optimum cut-off rate 𝑪𝑪𝟐𝟐
Forecast π values
The actual credit value of borrowers
(l) 𝐼𝐼1~𝐼𝐼2 (h) 𝐼𝐼3~𝐼𝐼5 (L) 𝐼𝐼1~𝐼𝐼2 𝐿𝐿𝑙𝑙𝐶𝐶2 𝐿𝐿ℎ𝐶𝐶2 (H) 𝐼𝐼3~𝐼𝐼5 𝐻𝐻𝑙𝑙𝐶𝐶2 𝐻𝐻ℎ𝐶𝐶2 L and H: The actual credit value of borrowers.
(L) 𝐼𝐼1~𝐼𝐼2: Based on the actual credit value of the borrowers, the interest-rate markups of 𝐼𝐼1~𝐼𝐼2(an interest rate ranging between 2.155% and 3.155%) are offered.
(H)𝐼𝐼3~𝐼𝐼5: Based on the actual credit value of the borrowers, the interest-rate markups of 𝐼𝐼3, 𝐼𝐼4, and 𝐼𝐼5
(the interest rate 3.655%, 4.155%, 5.655%) are offered.
l and h: forecast π values.
(l) 𝐼𝐼1~𝐼𝐼2: when π ≤ 𝐶𝐶2, the interest-rate markups 𝐼𝐼1~𝐼𝐼2 (an interest rate ranging between 2.155% and 3.155%) are offered to borrowers.
(h)𝐼𝐼3~𝐼𝐼5: when π > 𝐶𝐶2, the interest-rate markups 𝐼𝐼3, 𝐼𝐼4, and 𝐼𝐼5 (the interest rate 3.655%, 4.155%, 5.655%) are offered.
𝐿𝐿𝑙𝑙𝐶𝐶2 = The number of loans when the forecast interest rate of the π value and the interest rate based on the actual credit value of borrowers are 𝐼𝐼1~𝐼𝐼2 (2.155% ~ 3.155%).
𝐻𝐻𝑙𝑙𝐶𝐶2 = The number of loans when the forecast interest rates of the π value are 𝐼𝐼1~𝐼𝐼2 (2.155%~3.155%) and the interest rate based on the actual credit value of borrowers are 𝐼𝐼3, 𝐼𝐼4, and 𝐼𝐼5 (the interest rate 3.655%, 4.155%, 5.655% )
𝐿𝐿ℎ𝐶𝐶2 = The number of loans when the forecast interest rates of the π value are 𝐼𝐼3, 𝐼𝐼4, and 𝐼𝐼5 (the interest rate 3.655%, 4.155%, 5.655%) and the interest rate based on the actual credit value of borrowers is 𝐼𝐼1~𝐼𝐼2 (2.155% ~ 3.155%).
𝐻𝐻ℎ𝐶𝐶2 = The number of loans when the forecast interest rate of the π value and the interest rate based on the actual credit value of borrowers are 𝐼𝐼3, 𝐼𝐼4, and 𝐼𝐼5 (3.655%, 4.155%, 5.655%).
𝑅𝑅𝐼𝐼1~2 = The average interest rate (2.476%) of loans 𝐼𝐼1~𝐼𝐼2. 𝑅𝑅𝐼𝐼3 = The interest rate (3.655%) of loans 𝐼𝐼3.
In this phase, the numbers of 𝐼𝐼5, 𝐼𝐼4, and 𝐼𝐼3 are classified and categorized.
The profit calculation formula can be rewritten as follows:
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �(2.476%) ×𝐿𝐿456𝑙𝑙𝐶𝐶2� − �(2.821%) ×45622 + (1.445%) ×47−22456 + (1.176%) ×𝐻𝐻𝑙𝑙𝐶𝐶2456−47� −
�(2.834%) ×4562 + (2.71%) ×8−2456+ (2.476%) ×𝐿𝐿ℎ𝐶𝐶2456−8� + �(5.655%) ×45611 + (4.155%) ×
22−11
456 + (3.655%) ×𝐻𝐻ℎ𝐶𝐶2456−22�
The profit of accurate forecasts on loans (𝐿𝐿𝑙𝑙𝐶𝐶2 and 𝐻𝐻ℎ𝐶𝐶2) minus the premium losses of underestimated loan risk (𝐻𝐻𝑙𝑙𝐶𝐶2) and the customer losses of overestimated loan risk (𝐿𝐿ℎ𝐶𝐶2) is the sum total of profit. When 𝐿𝐿𝑙𝑙𝐶𝐶2, 𝐻𝐻𝑙𝑙𝐶𝐶2, 𝐿𝐿ℎ𝐶𝐶2 and 𝐻𝐻ℎ𝐶𝐶2 are (304, 18, 95, 39), the maximum profit is obtained (0.0173), and the cut-off rate is 0.59 - that is, the optimum cut-off rate, 𝐶𝐶2 (Table 11).
Procedure 4: Calculate the optimum cut-off rate 𝑪𝑪𝟏𝟏
We finally divide the customers into (𝐼𝐼1 with the markups of 1%) and (𝐼𝐼2, 𝐼𝐼3,
𝐼𝐼4, and 𝐼𝐼5 with the markups of 2%, 2.5%, 3%, and 4.5%) to calculate the optimum cut-off rate 𝐶𝐶1 (Table 15).
Table 15
Calculate the optimum cut-off rate 𝑪𝑪𝟏𝟏
Forecast π values
The actual credit value of borrowers
(l) 𝐼𝐼1 (h) 𝐼𝐼2+ 𝐼𝐼3+ 𝐼𝐼4+ 𝐼𝐼5 (L) 𝐼𝐼1 𝐿𝐿𝑙𝑙𝐶𝐶1 𝐿𝐿ℎ𝐶𝐶1
(H) 𝐼𝐼2+ 𝐼𝐼3+ 𝐼𝐼4+ 𝐼𝐼5 𝐻𝐻𝑙𝑙𝐶𝐶1 𝐻𝐻ℎ𝐶𝐶1 L and H: The actual credit value of borrowers.
(L) 𝐼𝐼1: Based on the actual credit value of the borrowers, the interest-rate markups of 𝐼𝐼1(the interest rate ranging 2.155%) are offered.
(H) 𝐼𝐼2+ 𝐼𝐼3+ 𝐼𝐼4+ 𝐼𝐼5: Based on the actual credit value of the borrowers, the interest-rate markups of 𝐼𝐼2, 𝐼𝐼3, 𝐼𝐼4, and 𝐼𝐼5 (the interest rate 3.155%, 3.655%, 4.155%, 5.655%) are offered.
l and h: forecast π values.
(l) 𝐼𝐼1: when π ≤ 𝐶𝐶1, the interest-rate markups 𝐼𝐼1 (the interest rate 2.155%) are offered to borrowers.
(h) 𝐼𝐼2+ 𝐼𝐼3+ 𝐼𝐼4+ 𝐼𝐼5: when π > 𝐶𝐶1, the interest-rate markups 𝐼𝐼2, 𝐼𝐼3, 𝐼𝐼4, and 𝐼𝐼5 (the interest rate 3.155%, 3.655%, 4.155%, 5.655%) are offered.
Profit function 4
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 = �𝑅𝑅𝐼𝐼1×𝐿𝐿𝑁𝑁𝐶𝐶1𝑙𝑙 � − ��𝑅𝑅𝐼𝐼5− 𝑅𝑅𝐼𝐼1~4� ×𝐻𝐻𝑁𝑁𝑙𝑙𝐶𝐶4+ �𝑅𝑅𝐼𝐼4− 𝑅𝑅𝐼𝐼1~3� ×𝐻𝐻𝑙𝑙𝐶𝐶3−𝐻𝐻𝑁𝑁 𝑙𝑙𝐶𝐶4+ �𝑅𝑅𝐼𝐼3− 𝑅𝑅𝐼𝐼1~2� ×
𝐻𝐻𝑙𝑙𝐶𝐶2−𝐻𝐻𝑙𝑙𝐶𝐶3
𝑁𝑁 + �𝑅𝑅𝐼𝐼2− 𝑅𝑅𝐼𝐼1� ×𝐻𝐻𝑙𝑙𝐶𝐶1𝑁𝑁−𝐻𝐻𝑙𝑙𝐶𝐶2� − �𝑅𝑅𝐼𝐼1~4×𝐿𝐿𝑁𝑁𝐶𝐶4ℎ + 𝑅𝑅𝐼𝐼1~3×𝐿𝐿ℎ𝐶𝐶3𝑁𝑁−𝐿𝐿𝐶𝐶4ℎ + 𝑅𝑅𝐼𝐼1~2×𝐿𝐿ℎ𝐶𝐶2𝑁𝑁−𝐿𝐿ℎ𝐶𝐶3+ 𝑅𝑅𝐼𝐼1×
𝐿𝐿ℎ𝐶𝐶1−𝐿𝐿𝐶𝐶2ℎ
𝑁𝑁 � + �𝑅𝑅𝐼𝐼5×𝐻𝐻𝑁𝑁ℎ𝐶𝐶4+ 𝑅𝑅𝐼𝐼4×𝐻𝐻ℎ𝐶𝐶3𝑁𝑁−𝐻𝐻ℎ𝐶𝐶4+ 𝑅𝑅𝐼𝐼3×𝐻𝐻ℎ𝐶𝐶2𝑁𝑁−𝐻𝐻ℎ𝐶𝐶3+ 𝑅𝑅𝐼𝐼2×𝐻𝐻ℎ𝐶𝐶1−𝐻𝐻𝑁𝑁 ℎ𝐶𝐶2� (4)
𝐿𝐿𝑙𝑙𝐶𝐶1 = The number of loans when the forecast interest rate of the π value and the interest rate based on the actual credit value of borrowers are 𝐼𝐼1 (2.155%).
𝐻𝐻𝑙𝑙𝐶𝐶1 = The number of loans when the forecast interest rates of the π value are 𝐼𝐼1 (2.155%) and the interest rates based on the actual credit value of borrowers are 𝐼𝐼2, 𝐼𝐼3, 𝐼𝐼4, and 𝐼𝐼5 (the interest rate 3.155%, 3.655%, 4.155%, and 5.655%).
𝐿𝐿ℎ𝐶𝐶1 = The number of loans when the forecast interest rates of the π value are 𝐼𝐼2, 𝐼𝐼3, 𝐼𝐼4, and 𝐼𝐼5 (the interest rate 3.155%, 3.655%, 4.155%, and 5.655%) and the interest rate based on the actual credit value of borrowers is 𝐼𝐼1 (2.155%).
𝐻𝐻ℎ𝐶𝐶1 = The number of loans when the forecast interest rate of the π value and the interest rate based on the actual credit value of borrowers are 𝐼𝐼2, 𝐼𝐼3, 𝐼𝐼4, and 𝐼𝐼5 (the interest rate
3.155%, 3.655%, 4.155%, and 5.655%).
In this phase, the numbers of 𝐼𝐼5, 𝐼𝐼4, 𝐼𝐼3, 𝐼𝐼2, and 𝐼𝐼1 are completely classified and categorized. Because the numbers of 𝐼𝐼5, 𝐼𝐼4, 𝐼𝐼3, and 𝐼𝐼2 are classified, we use the interest rates 𝑅𝑅𝐼𝐼5, 𝑅𝑅𝐼𝐼4, 𝑅𝑅𝐼𝐼3, and 𝑅𝑅𝐼𝐼2 (5.655%, 4.155%, 3.655%, and 3.155%) for calculation. When 𝐿𝐿𝑙𝑙𝐶𝐶1, 𝐿𝐿ℎ𝐶𝐶1, 𝐻𝐻𝑙𝑙𝐶𝐶1, and 𝐻𝐻ℎ𝐶𝐶1 are (160, 71, 99, 126), the maximum profit is obtained (0.0115), and the cut-off rate of 0.37 is the optimum cut-off rate 𝐶𝐶1 (Table 11).
The results show that the optimum cut-off rates are 0.80, 0.70, 0.59, and 0.37. This indicates that when the value of a borrower is 0.80, the bank offers the benchmark interest-rate markup of 4.5%; when the value is between 0.80 and 0.70, the markup is 3%; when the value is between 0.70 and 0.59, the markup is 2.5%; when the value is between 0.59 and 0.37, the markup is 2%; and when the value is less than 0.37, 1% is added to the benchmark interest rate as the interest rate (see Figure 2).