This document discusses several techniques for evaluating capital budgeting projects: payback period, net present value, discounted payback period, internal rate of return, profitability index, and average rate of return. It provides examples of calculating each of these metrics for two potential projects, Projects X and Y, and compares the results to determine which project has more favorable financial indicators based on the given criteria.

1. The Task of Mathematical Finance
(Capital Budgeting Evaluation Techniques)
Payback Period
Net Present Value (NPV)
Discounted Payback Period
Internal Rate of Return (IRR)
Profitability Index (PI)
Average Rate of Return (ARR)
WrittenBy
Muhamad Ahadi Satyaguna
C Class

2. 2nd
Semester
Sekolah Tinggi Perpajakan Indonesia
 Payback Period (Pembayaran Kembali)
Year
(t)
Expected After-Tax Net Cash Flow
(CFt)
Project X Arus Kas
Kumulatif
Project Y Arus Kas
Kumulatif
0 - $ 1,000 - $ 1,000 - $ 1,000 - $ 1,000
1 $ 500 - $ 500 $ 100 - $ 900
2 $ 400 - $ 100 $ 300 - $ 600
3 $ 300 $ 200 $ 400 - $ 200
4 $ 100 $ 300 $ 600 $ 400
The Formula of Payback Period
Answer !!!
𝐏𝐚𝐲𝐛𝐚𝐜𝐤 𝐏𝐞𝐫𝐢𝐨𝐝 𝐏𝐫𝐨𝐣𝐞𝐜𝐭 𝐗 = 2 +
100
300
= 2.33 → 2 𝑡𝑎ℎ𝑢𝑛 3𝑏𝑢𝑙𝑎𝑛 28ℎ𝑎𝑟𝑖
𝐏𝐚𝐲𝐛𝐚𝐜𝐤 𝐏𝐞𝐫𝐢𝐨𝐝 𝐏𝐫𝐨𝐣𝐞𝐜𝐭 𝐘 = 3 +
200
600
= 3.33 → 3 𝑡𝑎ℎ𝑢𝑛 3𝑏𝑢𝑙𝑎𝑛 28ℎ𝑎𝑟𝑖
 NPV (Net Present Value)
Year
(t)
Expected After-Tax Net Cash Flow
(CFt)
Project X Project Y
0 - $ 1,000 - $ 1,000
1 $ 500 $ 100
2 $ 400 $ 300
3 $ 300 $ 400
Payback Period = Tahun sebelum pengembalian penuh +
Biaya yang belum dikembalikan pada awal tahun
Arus kas selama tahun berjalan

3. 4 $ 100 $ 600
Keterangan: 𝐶𝐹𝑡 = 𝐴𝑟𝑢𝑠 𝑘𝑎𝑠 𝑝𝑎𝑑𝑎 𝑤𝑎𝑘𝑡𝑢 𝑡
𝑘 = 𝐵𝑖𝑎𝑦𝑎 𝑚𝑜𝑑𝑎𝑙 𝑝𝑟𝑜𝑦𝑒𝑘
𝑡 = 𝑃𝑒𝑟𝑖𝑜𝑑𝑒 𝑤𝑎𝑘𝑡𝑢
𝑛 = 𝑈𝑠𝑖𝑎 𝑝𝑟𝑜𝑦𝑒𝑘
Project X
Diketahui:Biaya modaluntukkeduaproyektersebutadalah 10%
𝑁𝑃𝑉 =
𝐶𝐹0
(1 + 𝑘)0
+
𝐶𝐹1
(1 + 𝑘)1
+
𝐶𝐹2
(1 + 𝑘)2
+
𝐶𝐹3
(1 + 𝑘)3
+
𝐶𝐹4
(1 + 𝑘)4
=
−1,000
(1 + 0.1)0
+
500
(1 + 0.1)1
+
400
(1 + 0.1)2
+
300
(1 + 0.1)3
+
100
(1 + 0.1)4
= −1,000 + 454.55 + 330.58 + 225.39 + 68.30
𝑁𝑃𝑉 = 78.82
Project Y
Diketahui:Biaya modaluntukkeduaproyektersebutadalah 10%
𝑁𝑃𝑉 =
𝐶𝐹0
(1 + 𝑘)0
+
𝐶𝐹1
(1 + 𝑘)1
+
𝐶𝐹2
(1 + 𝑘)2
+
𝐶𝐹3
(1 + 𝑘)3
+
𝐶𝐹4
(1 + 𝑘)4
=
−1,000
(1 + 0.1)0
+
100
(1 + 0.1)1
+
300
(1 + 0.1)2
+
400
(1 + 0.1)3
+
600
(1 + 0.1)4
= −1,000 + 90.91 + 247.93 + 300.53 + 409.81
𝑁𝑃𝑉 = 49.18
𝑁𝑃𝑉 = 𝐶𝐹0 +
𝐶𝐹1
(1 + 𝑘)1 +
𝐶𝐹2
(1 + 𝑘)2 + ⋯+
𝐶𝐹𝑛
(1 + 𝑘) 𝑛
𝑁𝑃𝑉 = ∑
𝐶𝐹𝑡
(1 + 𝑘) 𝑡
𝑛
𝑡=0

4.  Discounted Payback Period
(Periode Pembayaran Kembali Terdiskonto)
o Project X
Tahun Arus Kas PVIF 10% Nilai Sekarang Nilai Sekarang Kumulatif
0 - $ 1,000 1 - $ 1,000 - $ 1,000
1 $ 500 0.909 $ 454.5 - $ 545.5
2 $ 400 0.826 $ 330.4 - $ 215.1
3 $ 300 0.751 $ 225.3 $ 10.2
4 $ 100 0.683 $ 68.3 $ 78.5
𝑫𝒊𝒔𝒄𝒐𝒖𝒏𝒕𝒆𝒅 𝑷𝒂𝒚𝒃𝒂𝒄𝒌 𝑷𝒆𝒓𝒊𝒐𝒅 = 2 𝑡𝑎ℎ𝑢𝑛 +
215.1
225.3
= 2 𝑡𝑎ℎ𝑢𝑛 + 0.95 = 2 𝑡𝑎ℎ𝑢𝑛 11 𝑏𝑢𝑙𝑎𝑛 12 ℎ𝑎𝑟𝑖
o Project Y
Tahun Arus Kas PVIF 10% Nilai Sekarang Nilai Sekarang Kumulatif
0 - $ 1,000 1 - $ 1,000 - $ 1,000
1 $ 100 0.909 $ 90.9 - $ 909.1
2 $ 300 0.826 $ 247.8 - $ 661.3
3 $ 400 0.751 $ 300.4 - $ 360.9
4 $ 600 0.683 $ 409.8 $ 48.9
𝑫𝒊𝒔𝒄𝒐𝒖𝒏𝒕𝒆𝒅 𝑷𝒂𝒚𝒃𝒂𝒄𝒌 𝑷𝒆𝒓𝒊𝒐𝒅 = 3 𝑡𝑎ℎ𝑢𝑛 +
360.9
409.8
= 3 𝑡𝑎ℎ𝑢𝑛 + 0.88 = 3 𝑡𝑎ℎ𝑢𝑛 10 𝑏𝑢𝑙𝑎𝑛 16 ℎ𝑎𝑟𝑖
 ProfitabilityIndex (Indeks Profitabilitas)
𝑃𝐼 =
𝑃𝑉 𝐶𝑎𝑠ℎ 𝐼𝑛𝑓𝑙𝑜𝑤𝑠
𝑃𝑉 𝐶𝑎𝑠ℎ 𝑂𝑢𝑡𝑓𝑙𝑜𝑤𝑠
=
∑
𝐶𝐼𝐹𝑡
(1 + 𝑘) 𝑡
𝑛
𝑡=0
∑
𝐶𝑂𝐹𝑡
(1 + 𝑘) 𝑡
𝑛
𝑡=0

5. Keterangan: 𝐶𝐼𝐹𝑡 = 𝐶𝑎𝑠ℎ 𝐼𝑛𝑓𝑙𝑜𝑤𝑠 𝑝𝑎𝑑𝑎 𝑝𝑒𝑟𝑖𝑜𝑑𝑒 𝑡
𝐶𝑂𝐹𝑡 = 𝐶𝑎𝑠ℎ 𝑂𝑢𝑡𝑓𝑙𝑜𝑤𝑠 𝑝𝑎𝑑𝑎 𝑝𝑒𝑟𝑖𝑜𝑑𝑒 𝑡
𝑘 = 𝐵𝑖𝑎𝑦𝑎 𝑚𝑜𝑑𝑎𝑙 𝑝𝑟𝑜𝑦𝑒𝑘
𝑡 = 𝑃𝑒𝑟𝑖𝑜𝑑𝑒 𝑤𝑎𝑘𝑡𝑢
 Project X
Pertanyaan:Jikamodalproyeknya10%, berapakah
ProfitabilityIndex-nya???
Answer!!!
𝑃𝐼 =
500
(1 + 10%)1 +
400
(1 + 10%)2 +
300
(1 + 10%)3 +
100
(1 + 10%)4
1000
(1 + 10%)0
= 1.079
Denganmelihatkriteriapenerimaan,makaseharusnyaproyektersebut“GOPROJECT”karenaPI> 1.
 Project Y
Pertanyaan:Jikamodalproyeknya10%, berapakah
ProfitabilityIndex-nya???
Answer!!!
𝑃𝐼 =
100
(1 + 10%)1 +
300
(1 + 10%)2 +
400
(1 + 10%)3 +
600
(1 + 10%)4
1000
(1 + 10%)0
= 1.049
Tahun Arus Kas Proyek X
0 - $ 1,000
1 $ 500
2 $ 400
3 $ 300
4 $ 100
Tahun Arus Kas Proyek X
0 - $ 1,000
1 $ 100
2 $ 300
3 $ 400
4 $ 600

6. Denganmelihatkriteriapenerimaan,makaseharusnyaproyektersebut“diterima”karenaPI> 1.
 IRR (Internal Rate of Return)
≫ 𝑇𝑖𝑛𝑔𝑘𝑎𝑡 𝑃𝑒𝑛𝑔𝑒𝑚𝑏𝑎𝑙𝑖𝑎𝑛 𝐼𝑛𝑡𝑒𝑟𝑛𝑎𝑙
Rumuss…!!! (2cara )
1. 𝐼𝑅𝑅 = 𝑘1 + [
𝑁𝑃𝑉1
𝑁𝑃𝑉1−𝑁𝑃𝑉2
× ( 𝑘2 − 𝑘1)]
2. 𝐼𝑅𝑅 = 𝑘1 − [𝑁𝑃𝑉1 ×
𝑘2−𝑘1
𝑁𝑃𝑉1−𝑁𝑃𝑉2
]
 Project X
Tahun Perkiraan Arus Kas PVIF 10% Nilai Sekarang PVIF 15% Nilai Sekarang
0 - $ 1,000 1 - $ 1,000 1 - $ 1,000
1 $ 500 0.909 $ 454.5 0.870 $ 435
2 $ 400 0.826 $ 330.4 0.756 $ 302.4
3 $ 300 0.751 $ 225.3 0.658 $ 197.4
4 $ 100 0.683 $ 68.3 0.572 $ 57.2
NPV $ 78.5 $ -8
𝐶𝑎𝑟𝑎 1 → 𝐼𝑅𝑅 = 10%+ [
78.5
78.5− (−8)
× (15%− 10%)] = 0.1453 = 14.53%
𝐶𝑎𝑟𝑎 2 → 𝐼𝑅𝑅 = 10% −[78.5×
(15%−10%)
(−8)− 78.5
] = 0.1453= 14.53%

7.  Project Y
Tahun Perkiraan Arus Kas PVIF 10% Nilai Sekarang PVIF 15% Nilai Sekarang
0 - $ 1,000 1 - $ 1,000 1 - $ 1,000
1 $ 100 0.909 $ 90.9 0.870 $ 87
2 $ 300 0.826 $ 247.8 0.756 $ 226.8
3 $ 400 0.751 $ 300.4 0.658 $ 263.2
4 $ 600 0.683 $ 409.8 0.572 $ 409.8
NPV $ 48.9 $ -13.2
𝐶𝑎𝑟𝑎 1 → 𝐼𝑅𝑅 = 10%+ [
48.9
48.9− (−13.2)
× (15% −10%)] = 0.1394 = 13.94%
𝐶𝑎𝑟𝑎 2 → 𝐼𝑅𝑅 = 10%− [48.9×
(15%−10%)
(−13.2) −48.9
] = 0.1394 = 13.94%
 ARR (Average Rate of Return)
Rumuuss !
𝐴𝑅𝑅 =
𝑅𝑎𝑡𝑎 𝑙𝑎𝑏𝑎 𝑠𝑒𝑡𝑒𝑙𝑎ℎ 𝑝𝑎𝑗𝑎𝑘
𝑅𝑎𝑡𝑎 − 𝑟𝑎𝑡𝑎 𝑖𝑛𝑣𝑒𝑠𝑡𝑎𝑠𝑖
× 100
𝑅𝑎𝑡𝑎 − 𝑟𝑎𝑡𝑎 𝑖𝑛𝑣𝑒𝑠𝑡𝑎𝑠𝑖 =
𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑣𝑒𝑠𝑡𝑎𝑠𝑖
2
Berikut iniadalah rencana investasiproyek X danY

8. Investasi ProyekX =$ 1,000 (usiainvestasi4tahun)
Investasi Proyek Y =$ 1,000 (usiainvestasi4tahun)
Profit after taxesuntukproyek X danY adalahsebagaiberikut :
Tahun Proyek X Proyek Y
1 $ 500 $ 100
2 $ 400 $ 300
3 $ 300 $ 400
4 $ 100 $ 600
Proyekmanakah yang dipilihberdasarkan metode ARR ???
Penyelesaian :
Depresiasi: 𝐷𝑒𝑝𝑟𝑒𝑠𝑖𝑎𝑠𝑖 𝑃𝑟𝑜𝑦𝑒𝑘 𝑿 =
𝐼𝑛𝑣𝑒𝑠𝑡𝑎𝑠𝑖 𝑃𝑟𝑜𝑦𝑒𝑘 𝑿
𝑈𝑠𝑖𝑎 𝐼𝑛𝑣𝑒𝑠𝑡𝑎𝑠𝑖 𝑿
=
$ 𝟏,𝟎𝟎𝟎
𝟒
= $𝟐𝟓𝟎
𝐷𝑒𝑝𝑟𝑒𝑠𝑖𝑎𝑠𝑖 𝑃𝑟𝑜𝑦𝑒𝑘 𝒀 =
𝐼𝑛𝑣𝑒𝑠𝑡𝑎𝑠𝑖 𝑃𝑟𝑜𝑦𝑒𝑘 𝒀
𝑈𝑠𝑖𝑎 𝐼𝑛𝑣𝑒𝑠𝑡𝑎𝑠𝑖 𝒀
=
$ 𝟏, 𝟎𝟎𝟎
𝟒
= $𝟐𝟓𝟎
Keterangan Proyek X Proyek Y
Investasi $ 1,000 $ 1,000
Tahun
1
2
3
4
$ 250
$ 150
$ 50
- $ 150
$ 300
- $ 150
$ 50
$ 150
$ 350
$ 400
Rata laba setelah pajak
$ 300
4
= $ 75
$ 400
4
= $ 100
Rata-rata Investasi
$ 1,000
2
= $ 500
$ 1,000
2
= $ 500
ARR
$ 75
$ 500
× 100 = 15%
$ 100
$ 500
× 100 = 20%
Proyekyang dipilihberdasarkanmetodeARR adalah proyek Y karenamenghasilkan ARR TERBESAR.
(ARR Proyek X < ARR Proyek Y)