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Rate of Interest.

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Presentation on theme: "Rate of Interest."β€” Presentation transcript:

1 Rate of Interest

2 Effective VS. Nominal 𝟏+ π’Š 𝒑 𝒑 𝒑 = 𝟏+π’Š

3 𝑾𝒉𝒆𝒓𝒆 π’Š π’Šπ’” 𝒕𝒉𝒆 𝒂𝒏𝒏𝒖𝒂𝒍 π’†π’‡π’‡π’†π’„π’•π’Šπ’—π’† π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆
Effective VS. Nominal 𝟏+ π’Š 𝒑 𝒑 𝒑 = 𝟏+π’Š 𝑾𝒉𝒆𝒓𝒆 π’Š π’Šπ’” 𝒕𝒉𝒆 𝒂𝒏𝒏𝒖𝒂𝒍 π’†π’‡π’‡π’†π’„π’•π’Šπ’—π’† π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆

4 𝑾𝒉𝒆𝒓𝒆 π’Š π’Šπ’” 𝒕𝒉𝒆 𝒂𝒏𝒏𝒖𝒂𝒍 π’†π’‡π’‡π’†π’„π’•π’Šπ’—π’† π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆
Effective VS. Nominal 𝟏+ π’Š 𝒑 𝒑 𝒑 = 𝟏+π’Š 𝑾𝒉𝒆𝒓𝒆 π’Š π’Šπ’” 𝒕𝒉𝒆 𝒂𝒏𝒏𝒖𝒂𝒍 π’†π’‡π’‡π’†π’„π’•π’Šπ’—π’† π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆

5 Effective VS. Nominal Interest is charged on the same basis as the payment

6 Explanation π‘¨π’”π’”π’–π’Žπ’† π’’π’–π’‚π’“π’•π’†π’“π’π’š π’‘π’‚π’šπ’Žπ’†π’π’•π’” 𝒐𝒇 𝟏 𝒇𝒐𝒓 𝒂 π’šπ’†π’‚π’“.
π‘΅π’π’Žπ’Šπ’π’‚π’ 𝒂𝒏𝒏𝒖𝒂𝒍 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒐𝒇 𝟏𝟐% π’‘π’‚π’Šπ’… π’’π’–π’‚π’“π’•π’†π’“π’π’š. π‘‡β„Žπ‘’ π‘€π‘œπ‘Ÿπ‘‘ π‘›π‘œπ‘šπ‘–π‘›π‘Žπ‘™ π‘šπ‘’π‘Žπ‘›π‘  : β€œπ‘–π‘› π‘›π‘Žπ‘šπ‘’ π‘œπ‘›π‘™π‘¦β€ 𝑀𝑒 π‘π‘Žπ‘› 𝑁𝑂𝑇 𝑒𝑠𝑒 π‘‘β„Žπ‘’ 𝑔𝑖𝑣𝑒𝑛 π‘Ÿπ‘Žπ‘‘π‘’ π‘‘π‘–π‘Ÿπ‘’π‘π‘‘π‘™π‘¦. πΌπ‘›π‘ π‘‘π‘’π‘Žπ‘‘ π‘¦π‘œπ‘’ π‘π‘Žπ‘› 𝑓𝑖𝑛𝑑 π‘‘β„Žπ‘’ 𝑒𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘Ÿπ‘Žπ‘‘π‘’ π‘‘β„Žπ‘Žπ‘‘ 𝑠𝑒𝑖𝑑𝑠 π‘‘β„Žπ‘’ π‘π‘Žπ‘¦π‘šπ‘’π‘›π‘‘ π‘π‘’π‘Ÿπ‘–π‘œπ‘‘

7 Explanation π‘¨π’”π’”π’–π’Žπ’† π’’π’–π’‚π’“π’•π’†π’“π’π’š π’‘π’‚π’šπ’Žπ’†π’π’•π’” 𝒐𝒇 𝟏 𝒇𝒐𝒓 𝒂 π’šπ’†π’‚π’“.
π‘΅π’π’Žπ’Šπ’π’‚π’ 𝒂𝒏𝒏𝒖𝒂𝒍 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒐𝒇 𝟏𝟐% π’‘π’‚π’Šπ’… π’’π’–π’‚π’“π’•π’†π’“π’π’š. π‘‡β„Žπ‘’ π‘€π‘œπ‘Ÿπ‘‘ π‘›π‘œπ‘šπ‘–π‘›π‘Žπ‘™ π‘šπ‘’π‘Žπ‘›π‘  : β€œπ‘–π‘› π‘›π‘Žπ‘šπ‘’ π‘œπ‘›π‘™π‘¦β€ 𝑀𝑒 π‘π‘Žπ‘› 𝑁𝑂𝑇 𝑒𝑠𝑒 π‘‘β„Žπ‘’ 𝑔𝑖𝑣𝑒𝑛 π‘Ÿπ‘Žπ‘‘π‘’ π‘‘π‘–π‘Ÿπ‘’π‘π‘‘π‘™π‘¦. πΌπ‘›π‘ π‘‘π‘’π‘Žπ‘‘ π‘¦π‘œπ‘’ π‘π‘Žπ‘› 𝑓𝑖𝑛𝑑 π‘‘β„Žπ‘’ 𝑒𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘Ÿπ‘Žπ‘‘π‘’ π‘‘β„Žπ‘Žπ‘‘ 𝑠𝑒𝑖𝑑𝑠 π‘‘β„Žπ‘’ π‘π‘Žπ‘¦π‘šπ‘’π‘›π‘‘ π‘π‘’π‘Ÿπ‘–π‘œπ‘‘. π‘†π‘œ , 𝐸𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘π‘’π‘Ÿ π‘žπ‘’π‘Žπ‘Ÿπ‘‘π‘’π‘Ÿ= 𝑖 (4) 4 = 12 4 %=3%= 𝑖 (4)

8 Example-1 π‘¨π’”π’”π’–π’Žπ’† π’Žπ’π’π’•π’‰π’π’š π’‘π’‚π’šπ’Žπ’†π’π’•π’” 𝒐𝒇 𝟏 𝒇𝒐𝒓 𝒂 π’šπ’†π’‚π’“.
π‘΅π’π’Žπ’Šπ’π’‚π’ 𝒂𝒏𝒏𝒖𝒂𝒍 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒐𝒇 𝟏𝟐% π’‘π’‚π’Šπ’… π’Žπ’π’π’•π’‰π’π’š. π‘Šβ„Žπ‘Žπ‘‘ 𝑖𝑠 π‘‘β„Žπ‘’ 𝐸𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘π‘’π‘Ÿ π‘šπ‘œπ‘›π‘‘β„Ž ?

9 Example-1 π‘¨π’”π’”π’–π’Žπ’† π’Žπ’π’π’•π’‰π’π’š π’‘π’‚π’šπ’Žπ’†π’π’•π’” 𝒐𝒇 𝟏 𝒇𝒐𝒓 𝒂 π’šπ’†π’‚π’“.
π‘΅π’π’Žπ’Šπ’π’‚π’ 𝒂𝒏𝒏𝒖𝒂𝒍 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒐𝒇 𝟏𝟐% π’‘π’‚π’Šπ’… π’Žπ’π’π’•π’‰π’π’š. π‘‡β„Žπ‘’ 𝐸𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘π‘’π‘Ÿ π‘šπ‘œπ‘›π‘‘β„Ž= 𝑖 (12) = 𝑖 (12) 12 =1%

10 Example-2 π‘Šβ„Žπ‘Žπ‘‘ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘ β„Žπ‘œπ‘’π‘™π‘‘ π‘¦π‘œπ‘’ 𝑓𝑖𝑛𝑑 ?
π‘¨π’”π’”π’–π’Žπ’† 𝒂 π’šπ’†π’‚π’“π’π’š π’‘π’‚π’šπ’Žπ’†π’π’• 𝒐𝒇 𝟏 𝒇𝒐𝒓 𝒂 π’šπ’†π’‚π’“. π‘΅π’π’Žπ’Šπ’π’‚π’ 𝒂𝒏𝒏𝒖𝒂𝒍 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒐𝒇 𝟏𝟐% π’‘π’‚π’Šπ’… π’Žπ’π’π’•π’‰π’π’š. π‘Šβ„Žπ‘Žπ‘‘ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘ β„Žπ‘œπ‘’π‘™π‘‘ π‘¦π‘œπ‘’ 𝑓𝑖𝑛𝑑 ?

11 Example-2 π‘Šβ„Žπ‘Žπ‘‘ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘ β„Žπ‘œπ‘’π‘™π‘‘ π‘¦π‘œπ‘’ 𝑓𝑖𝑛𝑑 ?
π‘¨π’”π’”π’–π’Žπ’† 𝒂 π’šπ’†π’‚π’“π’π’š π’‘π’‚π’šπ’Žπ’†π’π’• 𝒐𝒇 𝟏 𝒇𝒐𝒓 𝒂 π’šπ’†π’‚π’“. π‘΅π’π’Žπ’Šπ’π’‚π’ 𝒂𝒏𝒏𝒖𝒂𝒍 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒐𝒇 𝟏𝟐% π’‘π’‚π’Šπ’… π’Žπ’π’π’•π’‰π’π’š. π‘Šβ„Žπ‘Žπ‘‘ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘ β„Žπ‘œπ‘’π‘™π‘‘ π‘¦π‘œπ‘’ 𝑓𝑖𝑛𝑑 ? π΄π‘›π‘›π‘’π‘Žπ‘™ 𝐸𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’

12 Example-2 π‘Šβ„Žπ‘Žπ‘‘ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘ β„Žπ‘œπ‘’π‘™π‘‘ π‘¦π‘œπ‘’ 𝑓𝑖𝑛𝑑 ?
π‘¨π’”π’”π’–π’Žπ’† 𝒂 π’šπ’†π’‚π’“π’π’š π’‘π’‚π’šπ’Žπ’†π’π’• 𝒐𝒇 𝟏 𝒇𝒐𝒓 𝒂 π’šπ’†π’‚π’“. π‘΅π’π’Žπ’Šπ’π’‚π’ 𝒂𝒏𝒏𝒖𝒂𝒍 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒐𝒇 𝟏𝟐% π’‘π’‚π’Šπ’… π’’π’–π’‚π’“π’•π’†π’“π’π’š. π‘Šβ„Žπ‘Žπ‘‘ π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘ β„Žπ‘œπ‘’π‘™π‘‘ π‘¦π‘œπ‘’ 𝑓𝑖𝑛𝑑 ? π΄π‘›π‘›π‘’π‘Žπ‘™ 𝐸𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’=𝑖= 1+ 𝑖 βˆ’1=

13 Example-3 π‘¨π’”π’”π’–π’Žπ’† 𝒂 π’’π’–π’‚π’“π’•π’†π’“π’π’š π’‘π’‚π’šπ’Žπ’†π’π’• 𝒐𝒇 𝟏 𝒇𝒐𝒓 𝒂 π’šπ’†π’‚π’“.
π‘΅π’π’Žπ’Šπ’π’‚π’ 𝒂𝒏𝒏𝒖𝒂𝒍 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒐𝒇 𝟏𝟐% π’‘π’‚π’Šπ’… π’Žπ’π’π’•π’‰π’π’š. 𝑾𝒉𝒂𝒕 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒔𝒉𝒐𝒖𝒍𝒅 π’šπ’π’– 𝒖𝒔𝒆 ?π‘ͺ𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒆 π’Šπ’• ?

14 Example-3 π‘¨π’”π’”π’–π’Žπ’† 𝒂 π’’π’–π’‚π’“π’•π’†π’“π’π’š π’‘π’‚π’šπ’Žπ’†π’π’• 𝒐𝒇 𝟏 𝒇𝒐𝒓 𝒂 π’šπ’†π’‚π’“.
π‘΅π’π’Žπ’Šπ’π’‚π’ 𝒂𝒏𝒏𝒖𝒂𝒍 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒐𝒇 𝟏𝟐% π’‘π’‚π’Šπ’… π’Žπ’π’π’•π’‰π’π’š. 𝑾𝒉𝒂𝒕 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒔𝒉𝒐𝒖𝒍𝒅 π’šπ’π’– 𝒖𝒔𝒆 ?𝑻𝒉𝒆 π‘¬π’‡π’‡π’†π’„π’•π’Šπ’—π’† π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒑𝒆𝒓 𝒒𝒖𝒂𝒓𝒕𝒆𝒓 π‘―π’Šπ’π’• : 𝒖𝒔𝒆 𝒕𝒉𝒆 π’‡π’π’π’π’˜π’Šπ’π’ˆ π’“π’†π’π’‚π’•π’Šπ’π’π’” 1+ 𝑖 (4) = 1+𝑖 1+ 𝑖 (12) = 1+𝑖

15 Example-3 π‘¨π’”π’”π’–π’Žπ’† 𝒂 π’’π’–π’‚π’“π’•π’†π’“π’π’š π’‘π’‚π’šπ’Žπ’†π’π’• 𝒐𝒇 𝟏 𝒇𝒐𝒓 𝒂 π’šπ’†π’‚π’“.
π‘΅π’π’Žπ’Šπ’π’‚π’ 𝒂𝒏𝒏𝒖𝒂𝒍 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒐𝒇 𝟏𝟐% π’‘π’‚π’Šπ’… π’Žπ’π’π’•π’‰π’π’š. 𝑾𝒉𝒂𝒕 π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒔𝒉𝒐𝒖𝒍𝒅 π’šπ’π’– 𝒖𝒔𝒆 ?𝑻𝒉𝒆 π‘¬π’‡π’‡π’†π’„π’•π’Šπ’—π’† π’Šπ’π’•π’†π’“π’†π’”π’• 𝒓𝒂𝒕𝒆 𝒑𝒆𝒓 𝒒𝒖𝒂𝒓𝒕𝒆𝒓 π‘―π’Šπ’π’• : 𝒖𝒔𝒆 𝒕𝒉𝒆 π’‡π’π’π’π’˜π’Šπ’π’ˆ π’“π’†π’π’‚π’•π’Šπ’π’π’” 1+ 𝑖 (4) = 1+ 𝑖 (12) 1+ 𝑖 (4) = % 𝑖 = % βˆ’1= = 𝑖 (4)

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18 Rate of Discount π‘…π‘’π‘šπ‘’π‘šπ‘π‘’π‘Ÿ π‘‘β„Žπ‘’ π‘šπ‘Žπ‘‘β„Žπ‘’π‘šπ‘Žπ‘‘π‘–π‘π‘Žπ‘™ π‘Ÿπ‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘›π‘ β„Žπ‘–π‘ 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 𝑣 π‘Žπ‘›π‘‘ 𝑑 ? π΄π‘ π‘ π‘’π‘šπ‘’ 𝑀𝑒 π‘€π‘Žπ‘›π‘‘ π‘‘π‘œ π‘‘π‘–π‘ π‘π‘œπ‘’π‘›π‘‘ π‘Žπ‘› π‘Žπ‘šπ‘œπ‘’π‘›π‘‘ π‘œπ‘“ 100 π‘π‘Žπ‘–π‘‘ π‘œπ‘›π‘’ π‘¦π‘’π‘Žπ‘Ÿ π‘“π‘Ÿπ‘œπ‘š π‘›π‘œπ‘€. π‘‡β„Žπ‘’π‘› π‘‘β„Žπ‘’π‘ π‘’ π‘‘π‘€π‘œ π‘Žπ‘π‘π‘Ÿπ‘œπ‘Žπ‘β„Žπ‘’π‘  π‘Žπ‘Ÿπ‘’ π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ : 𝑃𝑉=100 𝑣 𝑃𝑉=100 (1βˆ’π‘‘) 𝑾𝒉𝒂𝒕 𝒅𝒐 π’šπ’π’– π’•π’‰π’Šπ’π’Œ ?𝑰𝒔 𝒕𝒉𝒂𝒕 𝒄𝒐𝒓𝒓𝒆𝒄𝒕 ?

19 Rate of Discount π‘‡β„Žπ‘’ π‘šπ‘Žπ‘‘β„Žπ‘’π‘šπ‘Žπ‘‘π‘–π‘π‘Žπ‘™ π‘Ÿπ‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘›π‘ β„Žπ‘–π‘ 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 𝑣 π‘Žπ‘›π‘‘ 𝑑 𝑣=1βˆ’π‘‘ π‘‡β„Žπ‘’π‘Ÿπ‘’π‘“π‘œπ‘Ÿπ‘’β€¦ 𝑃𝑉=100 𝑣 is equivalent to 𝑃𝑉=100 (1βˆ’π‘‘)

20 Nominal and Effective rate of Discount
1βˆ’ 𝑑 𝑝 𝑝 𝑝 = 1βˆ’π‘‘

21 Nominal and Effective rate of Discount
π‘π‘œπ‘šπ‘–π‘›π‘Žπ‘™ π·π‘–π‘ π‘π‘œπ‘’π‘›π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘π‘œπ‘šπ‘π‘œπ‘’π‘›π‘‘π‘’π‘‘ π‘žπ‘’π‘Žπ‘Ÿπ‘‘π‘’π‘Ÿπ‘™π‘¦ 𝑑 (4) =12% 𝐹𝑖𝑛𝑑 π‘‘β„Žπ‘’ 𝐸𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘‘π‘–π‘ π‘π‘œπ‘’π‘›π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘π‘’π‘Ÿ π‘žπ‘’π‘Žπ‘Ÿπ‘‘π‘’π‘Ÿ . 𝐹𝑖𝑛𝑑 π‘‘β„Žπ‘’ 𝑒𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘Žπ‘›π‘›π‘’π‘Žπ‘™ π·π‘–π‘ π‘π‘œπ‘’π‘›π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ .

22 Nominal and Effective rate of Discount
π‘π‘œπ‘šπ‘–π‘›π‘Žπ‘™ π·π‘–π‘ π‘π‘œπ‘’π‘›π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘π‘œπ‘šπ‘π‘œπ‘’π‘›π‘‘π‘’π‘‘ π‘žπ‘’π‘Žπ‘Ÿπ‘‘π‘’π‘Ÿπ‘™π‘¦ 𝑑 (4) =12% π‘‡β„Žπ‘’ 𝐸𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘‘π‘–π‘ π‘π‘œπ‘’π‘›π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘π‘’π‘Ÿ π‘žπ‘’π‘Žπ‘Ÿπ‘‘π‘’π‘Ÿ= 12 4 %=3%. π‘‡β„Žπ‘’ 𝑒𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘Žπ‘›π‘›π‘’π‘Žπ‘™ π·π‘–π‘ π‘π‘œπ‘’π‘›π‘‘ π‘Ÿπ‘Žπ‘‘π‘’= 𝑑=1βˆ’ 1βˆ’ 𝑑 (12) =

23

24 Discounting using the Annual Effective
For a payment of 100 made in the middle of the year : π‘‡β„Žπ‘’ π‘π‘Ÿπ‘’π‘ π‘’π‘›π‘‘ π‘£π‘Žπ‘™π‘’π‘’= 𝑖 2 2 βˆ’1 Is mathematically equivalent to : π‘‡β„Žπ‘’ π‘π‘Ÿπ‘’π‘ π‘’π‘›π‘‘ π‘£π‘Žπ‘™π‘’π‘’=100 1+𝑖 βˆ’1 2 Remember 1+ 𝑖 =(1+𝑖)

25 Two-year Effective interest rate
𝐿𝑒𝑑 𝑒𝑠 π‘ π‘Žπ‘¦ π‘‘β„Žπ‘Žπ‘‘ π‘‘β„Žπ‘’π‘Ÿπ‘’ 𝑀𝑖𝑙𝑙 𝑏𝑒 π‘Žπ‘› π‘Žπ‘šπ‘œπ‘’π‘›π‘‘ π‘œπ‘“ 100 π‘π‘Žπ‘–π‘‘ π‘Žπ‘“π‘‘π‘’π‘Ÿ π‘‘π‘€π‘œ π‘¦π‘’π‘Žπ‘Ÿπ‘  π‘€β„Žπ‘’π‘Ÿπ‘’ 𝑖=10% : π‘Šπ‘’ π‘π‘œπ‘’π‘™π‘‘ πΆπ‘Žπ‘™π‘π‘’π‘™π‘Žπ‘‘π‘’ π‘‘β„Žπ‘’ π‘π‘Ÿπ‘’π‘ π‘’π‘›π‘‘ π‘£π‘Žπ‘™π‘’π‘’ π‘Žπ‘  : βˆ’2 = π‘‚π‘Ÿ 𝑀𝑒 π‘π‘œπ‘’π‘™π‘‘ 𝑓𝑖𝑛𝑑 π‘‘β„Žπ‘’ π‘‘π‘€π‘œβˆ’π‘¦π‘’π‘Žπ‘Ÿ 𝑒𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘–π‘›π‘‘π‘’π‘Ÿπ‘’π‘ π‘‘ π‘Ÿπ‘Žπ‘‘π‘’ π‘π‘Žπ‘™π‘™ 𝑖𝑑 𝑗 = 1+𝑗 𝑗= βˆ’1=0.21 𝐻𝑒𝑛𝑐𝑒 βˆ’1 =82.645

26 πΆπ‘œπ‘›π‘ π‘–π‘‘π‘’π‘Ÿ π‘Ž π‘π‘’π‘Ÿπ‘π‘’π‘‘π‘’π‘–π‘‘π‘¦ π‘‘β„Žπ‘Žπ‘‘ π‘π‘Žπ‘¦π‘  10 π‘’π‘£π‘’π‘Ÿπ‘¦ π‘‘β„Žπ‘Ÿπ‘’π‘’ π‘¦π‘’π‘Žπ‘Ÿπ‘  𝑔𝑖𝑣𝑒𝑛 π‘‘β„Žπ‘’ π‘Žπ‘›π‘›π‘’π‘Žπ‘™ 𝑒𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘Ÿπ‘Žπ‘‘π‘’ 𝑖𝑠 8%.
πΆπ‘Žπ‘™π‘π‘’π‘™π‘Žπ‘‘π‘’ π‘‘β„Žπ‘’ π‘π‘Ÿπ‘’π‘ π‘’π‘›π‘‘ π‘£π‘Žπ‘™π‘’π‘’. 𝐻𝑖𝑛𝑑 : π‘Ž ∞| = 1 𝑖

27 πΆπ‘Žπ‘™π‘π‘’π‘™π‘Žπ‘‘π‘’ π‘‘β„Žπ‘’ π‘π‘Ÿπ‘’π‘ π‘’π‘›π‘‘ π‘£π‘Žπ‘™π‘’π‘’. 𝐻𝑖𝑛𝑑 : π‘Ž ∞| = 1 𝑖
πΆπ‘œπ‘›π‘ π‘–π‘‘π‘’π‘Ÿ π‘Ž π‘π‘’π‘Ÿπ‘π‘’π‘‘π‘’π‘–π‘‘π‘¦ π‘‘β„Žπ‘Žπ‘‘ π‘π‘Žπ‘¦π‘  10 π‘’π‘£π‘’π‘Ÿπ‘¦ π‘‘β„Žπ‘Ÿπ‘’π‘’ π‘¦π‘’π‘Žπ‘Ÿπ‘  𝑔𝑖𝑣𝑒𝑛 π‘‘β„Žπ‘’ π‘Žπ‘›π‘›π‘’π‘Žπ‘™ 𝑒𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒 π‘Ÿπ‘Žπ‘‘π‘’ 𝑖𝑠 8%. πΆπ‘Žπ‘™π‘π‘’π‘™π‘Žπ‘‘π‘’ π‘‘β„Žπ‘’ π‘π‘Ÿπ‘’π‘ π‘’π‘›π‘‘ π‘£π‘Žπ‘™π‘’π‘’. 𝐻𝑖𝑛𝑑 : π‘Ž ∞| = 1 𝑖 1+𝑗 = 1+𝑖 3 𝑗= βˆ’1= 𝑃𝑉=10 π‘Ž ∞| 𝑗 = =

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