❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
❖ ❊♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛
❋✐♥❛♥❝❡✐r❛ ❯t✐❧✐③❛♥❞♦ ❛
❈❛❧❝✉❧❛❞♦r❛ ❍P ✶✷❈
♣♦r▼❛②❛♥❛ ❈②❜❡❧❡ ❉❛♥t❛s ❞❡ ❖❧✐✈❡✐r❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦✲ ❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛✲ t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚✲ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❖ ❊♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛
❋✐♥❛♥❝❡✐r❛ ❯t✐❧✐③❛♥❞♦ ❛
❈❛❧❝✉❧❛❞♦r❛ ❍P ✶✷❈
♣♦r▼❛②❛♥❛ ❈②❜❡❧❡ ❉❛♥t❛s ❞❡ ❖❧✐✈❡✐r❛
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛✳ ❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛ ✲ ❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❆❧❡①❛♥❞r❡ ❞❡ ❇✉st❛♠❛♥t❡ ❙✐♠❛s ✲ ❯❋P❇
Pr♦❢✳ ❉r✳ ❚✉rí❜✐♦ ❏♦sé ●♦♠❡s ❞♦s ❙❛♥t♦s ✲ ❯◆■P✃
❆❣r❛❞❡❝✐♠❡♥t♦s
❆ ❉❡✉s ♣♦r t❡r ♠❡ ❞❛❞♦ s❛ú❞❡ ♣❛r❛ ❝✉♠♣r✐r ♠❛✐s ✉♠❛ ❡t❛♣❛ ♥❛ ♠✐♥❤❛ ❝❛rr❡✐r❛ ♣r♦✜ss✐♦♥❛❧❀
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛✱ ♣❡❧❛ ❛t❡♥çã♦ ❡ ❝♦♥✜✲ ❛♥ç❛ ♥❡st❡ ♠♦♠❡♥t♦ tã♦ ✐♠♣♦rt❛♥t❡ ❞❛ ♠✐♥❤❛ ✈✐❞❛❀
➚ ♠✐♥❤❛ ♠ã❡✱ ❙ô♥✐❛✱ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ♣♦r t❡r s✐❞♦ ♠❡✉ ♣♦rt♦ s❡❣✉r♦❀
❆♦ ♠❡✉ ♠❛r✐❞♦✱ ▼❛r①✱ ❡ ♠✐♥❤❛s ✜❧❤❛s✱ ▼❛r②ê✈❛ ❡ ▼❛r❛②❛✱ ♦s ♠❛✐♦r❡s ♣r❡s❡♥t❡s q✉❡ ❉❡✉s ♠❡ ❞❡✉❀
➚ ❝♦♦r❞❡♥❛çã♦ ❡ ♣r♦❢❡ss♦r❡s ❞♦ ♠❡str❛❞♦ ♣r♦✜ss✐♦♥❛❧✱ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❡ s❛❜❡❞♦r✐❛ ❝♦♠♣❛rt✐❧❤❛❞❛❀
➚ ❈❆P❊❙ ❡ ❛ ❯❋P❇✱ ♣❡❧♦s ❛♣♦✐♦s ✜♥❛♥❝❡✐r♦s✱ ❛❝❛❞ê♠✐❝♦ ❡ ❡st✉❞❛♥t✐❧❀
❉❡❞✐❝❛tór✐❛
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ ❞❡ ♣❡sq✉✐s❛ t❡♠ ❝♦♠♦ ❡s❝♦♣♦ ♦ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛✱ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞❡ ✉♠❛ ❢❡rr❛♠❡♥t❛✿ ❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛ ❍P ✶✷❈✱ ♣r❡♣❛r❛♥❞♦ ❝✐❞❛❞ã♦s ❝❛♣❛③❡s ❞❡ ❛❞♠✐♥✐str❛r s✉❛s ♣ró♣r✐❛s ✜♥❛♥ç❛s ❡ ♣❛r❛ ♦ ♠❡r❝❛❞♦ ❞❡ tr❛❜❛❧❤♦✳ ❆♣r❡s❡♥t❛ ❝♦♥❝❡✐t♦s ♣r❡❧✐♠✐♥❛r❡s ❛♦ t❡♠❛✱ s❡❣✉✐❞♦ ❞❡ ❡①❡♠♣❧♦s ❝♦♥t❡①t✉❛❧✐③❛❞♦s✱ ❛♣♦♥t❛♥❞♦ ♦s ♣r✐♥❝✐♣❛✐s ❢❛t♦r❡s✱ q✉❡ ❢❛❝✐❧✐t❛♠ ❛ ❝♦♠♣r❡❡♥sã♦ ❡♠ ❝❛❞❛ ❝♦♥t❡ú❞♦✳ ❙❡♥❞♦ tr❛t❛❞♦ ❛tr❛✈és ❞❡ ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s✱ q✉❡ ❢❛❝✐❧✐t❛ ❛ ❛♣r❡♥❞✐③❛❣❡♠✱ ♠♦t✐✈❛çã♦ ❡ ✐♥t❡r❡ss❡ ❞♦ ❛❧✉♥♦✳
P❛❧❛✈r❛s✲❝❤❛✈❡s✿ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛❀ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ❍P ✶✷❈❀ ❊♥✲ s✐♥♦ ❇ás✐❝♦✳
❆❜str❛❝t
❚❤✐s r❡s❡❛r❝❤ ❤❛s ❛s ♦❜❥❡❝t ♦❢ st✉❞② t❤❡ t❡❛❝❤✐♥❣ ♦❢ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s✱ ✇✐t❤ t❤❡ ❛✐❞ ♦❢ ❛ t♦♦❧✿ t❤❡ ❋✐♥❛♥❝✐❛❧ ❈❛❧❝✉❧❛t♦r ❍P ✶✷❈✱ ♣r❡♣❛r✐♥❣ ❝✐t✐③❡♥s ❛❜❧❡ t♦ ♠❛♥❛❣❡ t❤❡✐r ♦✇♥ ❜✉❞❣❡ts ❛♥❞ t❤❡ ❧❛❜♦r ♠❛r❦❡t✳ Pr❡❧✐♠✐♥❛r② ❝♦♥❝❡♣ts ❝♦♥❝❡r♥✐♥❣ t❤❡ t❤❡♠❡ ❛r❡ ♣r❡s❡♥t❡❞✱ ❢♦❧❧♦✇❡❞ ❜② ❡①❛♠♣❧❡s ✐♥ ❝♦♥t❡①t✱ ♣♦✐♥t✐♥❣ ♦✉t t❤❡ ♠❛✐♥ ❢❛❝t♦rs t❤❛t ❢❛❝✐❧✐t❛t❡ t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ ❡❛❝❤ ❝♦♥t❡♥t✳ ❇❡✐♥❣ ❛♣♣r♦❛❝❤❡❞ t❤r♦✉❣❤ ♣r❛❝t✐❝❛❧ ❛♣♣❧✐❝❛t✐♦♥s✱ ✇❤✐❝❤ ♠❛❦❡s ❧❡❛r♥✐♥❣ ❡❛s✐❡r✱ ❛♥❞ ✐♥❝r❡❛s❡s ♠♦t✐✈❛t✐♦♥ ❛♥❞ st✉❞❡♥ts✬ ✐♥t❡r❡st✳
❑❡②✇♦r❞s✿ ❋✐♥❛♥❝✐❛❧ ▼❛t❤❡♠❛t✐❝s❀ ❍P ✶✷❈ ❋✐♥❛♥❝✐❛❧ ❈❛❧❝✉❧❛t♦r❀ ❇❛s✐❝ t❡❛✲ ❝❤✐♥❣✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✶✳✶ ❈❛❧❝✉❧❛❞♦r❛ ❍P ✶✷❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ ❋✉♥çõ❡s ❋✐♥❛♥❝❡✐r❛s ❞❛ ❈❛❧❝✉❧❛❞♦r❛ ❍P ✶✷❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸ ■❞❡♥t✐✜❝❛çã♦ ❞❛s ❋✉♥çõ❡s ❋✐♥❛♥❝❡✐r❛s ♥❛ ❈❛❧❝✉❧❛❞♦r❛ ❍P ✶✷❈ ✳ ✳ ✳ ✳ ✶✷ ✶✳✹ ❈♦♠♦ ▲✐♠♣❛r ♦ ❱✐s♦r ❞❛ ❈❛❧❝✉❧❛❞♦r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✺ ❋❧✉①♦ ❞❡ ❈❛✐①❛ ♥♦ P♦♥t♦ ❞❡ ❱✐st❛ ❞♦ ❊♠♣r❡st❛❞♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✻ ❋❧✉①♦ ❞❡ ❈❛✐①❛ ♥♦ P♦♥t♦ ❞❡ ❱✐st❛ ❞♦ ❚♦♠❛❞♦r ❞♦ ❊♠♣rést✐♠♦ ✳ ✳ ✳ ✶✸ ✶✳✼ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✽ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✾ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✶✵ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✶✶ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✶✷ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✶ ❋❧✉①♦ ❞❡ ❈❛✐①❛ ♣❛r❛ ❙ér✐❡s ❞❡ P❛❣✳ P♦st❡❝✐♣❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷ ❇♦tõ❡s ❯s❛❞♦s ❡♠ ❙ér✐❡s ❞❡ P❛❣❛♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✸ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✹ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✺ ❋❧✉①♦ ❞❡ ❈❛✐①❛ ♣❛r❛ ❙ér✐❡s ❞❡ P❛❣✳ ❆♥t❡❝✐♣❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✻ ❆t✐✈❛r ❋✉♥çã♦ ❇❡❣✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✼ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✽ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
❙✉♠ár✐♦
✶ ❈❛♣✐t❛❧✐③❛çã♦ ❙✐♠♣❧❡s ❡ ❈♦♠♣♦st❛ ✶
✶✳✶ ❖ ❈❛♣✐t❛❧ ❡ ♦s ❏✉r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✶✳✷ ❏✉r♦s ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✸ ❏✉r♦s ❈♦♠♣♦st♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✸✳✶ ❯t✐❧✐③❛♥❞♦ ❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛ ❍P ✶✷❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✸✳✷ ❆♣r❡s❡♥t❛♥❞♦ ❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛ ❍P ✶✷❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✹ ❚❛①❛s ❊q✉✐✈❛❧❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✷ ❙ér✐❡s ❞❡ P❛❣❛♠❡♥t♦s ✷✹
✷✳✶ ❱❛❧♦r Pr❡s❡♥t❡ ♦✉ ❋❛t♦r ❞❡ ❱❛❧♦r ❆t✉❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✶✳✶ ❙ér✐❡ ❞❡ P❛❣❛♠❡♥t♦s P♦st❡❝✐♣❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✶✳✷ ❙ér✐❡ ❞❡ P❛❣❛♠❡♥t♦s ❆♥t❡❝✐♣❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷ ❙ér✐❡s ❞❡ P❛❣❛♠❡♥t♦s ❉✐❢❡r✐❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✸ ❱❛❧♦r ❋✉t✉r♦ ♦✉ ❋❛t♦r ❞❡ ❆❝✉♠✉❧❛çã♦ ❞❡ ❈❛♣✐t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✸✳✶ ❙ér✐❡s ❞❡ P❛❣❛♠❡♥t♦s P♦st❡❝✐♣❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✸✳✷ ❙ér✐❡s ❞❡ P❛❣❛♠❡♥t♦s ❆♥t❡❝✐♣❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✸ ❙✐st❡♠❛s ❞❡ ❆♠♦rt✐③❛çõ❡s ✹✼
✸✳✶ ❙✐st❡♠❛ ❞❡ ❆♠♦rt✐③❛çã♦ ❋r❛♥❝ês ✭P❘■❈❊✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✷ ❙✐st❡♠❛ ❞❡ ❆♠♦rt✐③❛çã♦ ❈♦♥st❛♥t❡ ✭❙❆❈✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✸ ❙✐st❡♠❛ ❞❡ ❆♠♦rt✐③❛çã♦ ▼✐st♦ ✭❙❆▼✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✽
■♥tr♦❞✉çã♦
❊st❡ tr❛❜❛❧❤♦ tr❛t❛ ❞♦ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛✱ ✉t✐❧✐③❛♥❞♦ ❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛ ❍P ✶✷❈✱ ♦♥❞❡ ✈❛♠♦s ❡st✉❞❛r ❛ ✈❛r✐❛çã♦ ❞♦ ✈❛❧♦r ❞♦ ❞✐♥❤❡✐r♦ ♥♦ t❡♠♣♦✱ ♥❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❞✐♥❤❡✐r♦ ❡ ♥♦s ♣❛❣❛♠❡♥t♦s ❞❡ ❡♠♣rést✐♠♦s✳ ❚❛❧ ✈❛r✐❛çã♦ ♦❝♦rr❡ ❡♠ ❢✉♥çã♦ ❞♦s ❡❢❡✐t♦s ❞❛ ✐♥✢❛çã♦ s♦❜r❡ ♦ ♣♦❞❡r ❞❡ ❝♦♠♣r❛ ❞❛ ♠♦❡❞❛ ♦✉ ♣❡❧❛ ✐♥❝✐❞ê♥❝✐❛ ❞❛ t❛①❛ ❞❡ ❥✉r♦s s♦❜r❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✈❛❧♦r ♠♦♥❡tár✐♦✳ ❋♦r♥❡❝❡♥❞♦ ✐♥str✉♠❡♥t♦s ♣❛r❛ ♦ ❡st✉❞♦ ❡ ❛ ❛✈❛❧✐❛çã♦ ❞❛s ❢♦r♠❛s ❞❡ ❛♣❧✐❝❛çã♦ ❞❡ ❞✐♥❤❡✐r♦✱ ❜❡♠ ❝♦♠♦ ❞❡ ♣❛❣❛♠❡♥t♦ ❞❡ ❡♠♣rést✐♠♦✳
❖s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✲ P❈◆✬s✱ ♣❛r❛ ♠❛t❡♠át✐❝❛ ♣r♦♣õ❡♠ ✐♥✲ ✈❡st✐❣❛r✱ ❝♦♠♣r❡❡♥❞❡r ❡ ❝♦♥t❡①t✉❛❧✐③❛r ♣r♦❜❧❡♠❛s✱ ❧❡✈❛♥t❛r ❤✐♣ót❡s❡s✱ r❡❧❛❝✐♦♥❛r ❛ ❞✐s❝✐♣❧✐♥❛ ❛ ❢❛t♦s ❝♦♥❤❡❝✐❞♦s✱ ❞❡s❡♥✈♦❧✈❡r ❡ ✉t✐❧✐③❛r ❛ ♠❛t❡♠át✐❝❛ ♥❛ ✐♥t❡r♣r❡t❛çã♦ ❡ ✐♥t❡r✈❡♥çã♦ ❞❛ r❡❛❧✐❞❛❞❡ ❡ ❛♣❧✐❝❛r ❡♠ s✐t✉❛çõ❡s r❡❛✐s✳
❇❘❆❙■▲ ✭✷✵✵✻✱ ♣✳✼✶✮✱ q✉❛♥❞♦ ❢❛❧❛ ❞❛ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ❢♦❝❛ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❡♠✿
✑❖ tr❛❜❛❧❤♦ ❝♦♠ ❡ss❡ ❜❧♦❝♦ ❞❡ ❝♦♥t❡ú❞♦s ❞❡✈❡ t♦r♥❛r ♦ ❛❧✉♥♦✱ ❛♦ ✜♥❛❧ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❝❛♣❛③ ❞❡ ❞❡❝✐❞✐r s♦❜r❡ ❛s ✈❛♥t❛✲ ❣❡♥s✴❞❡s✈❛♥t❛❣❡♥s ❞❡ ✉♠❛ ❝♦♠♣r❛ à ✈✐st❛ ♦✉ ❛ ♣r❛③♦❀ ❛✈❛❧✐❛r ♦ ❝✉st♦ ❞❡ ✉♠ ♣r♦❞✉t♦ ❡♠ ❢✉♥çã♦ ❞❛ q✉❛♥t✐❞❛❞❡❀ ❝♦♥❢❡r✐r s❡ ❡stã♦ ❝♦rr❡t❛s ❛s ✐♥❢♦r♠❛çõ❡s ❡♠ ❡♠❜❛❧❛❣❡♥s ❞❡ ♣r♦❞✉t♦s q✉❛♥t♦ ❛♦ ✈♦❧✉♠❡❀ ❝❛❧❝✉❧❛r ✐♠♣♦st♦s ❡ ❝♦♥tr✐❜✉✐çõ❡s ♣r❡✈✐❞❡♥✲ ❝✐ár✐❛s❀ ❛✈❛❧✐❛r ♠♦❞❛❧✐❞❛❞❡s ❞❡ ❥✉r♦s ❜❛♥❝ár✐♦s✳✑
♣♦r ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ s❡r✈✐♥❞♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ❝♦♠ s❡✉s r❡❝✉rs♦s t❡❝♥♦❧ó❣✐❝♦s✳
❆s ✐♥♦✈❛çõ❡s t❡❝♥♦❧ó❣✐❝❛s s❡r✈❡♠ ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ ✈✐❞❛ ❞♦ ❤♦♠❡♠✱ ❡ ❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛ ❍P ✶✷❈ ♣❡r♠✐t❡ ✉♠❛ ❡♥tr❛❞❛ ♠❛✐s rá♣✐❞❛ ❞❡ ❞❛❞♦s ❡ ❛ ❡①❡❝✉çã♦ ♠❛✐s ❡✜❝✐❡♥t❡ ❞♦s ❝á❧❝✉❧♦s✱ tr❛③❡♥❞♦ ❝♦♠♦❞✐❞❛❞❡ ❛♦ ✉t✐❧✐③❛r♠♦s ❢✉♥çõ❡s ♣ré✲❡st❛❜❡❧❡❝✐❞❛s ❛♦ ✐♥✈és ❞❡ ❢ór♠✉❧❛s tr❛❜❛❧❤♦s❛s ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ✜♥❛♥❝❡✐r♦s✳
❆t✉❛❧♠❡♥t❡ ♦s ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ❡♥❝♦♥tr❛♠ ❣r❛♥❞❡s ❞✐✜❝✉❧❞❛❞❡s ❝♦♠ ♦ ❞❡s✐♥t❡r❡ss❡ ❡ ❛s ❢r❡q✉❡♥t❡s ♣❡r❣✉♥t❛s ❞♦s ❛❧✉♥♦s✿ ✏P❛r❛ q✉❡ s❡r✈❡ ✐ss♦✑❄✱ ✏❖♥❞❡ ✈♦✉ ✉t✐❧✐③❛r ✐ss♦ ♥❛ ♠✐♥❤❛ ✈✐❞❛❄✑✳ ❙❡❣✉♥❞♦ ❙❆◆❚❖❙ ✭✷✵✶✷✱♣✳✹✮✿
✑➱ s❛❜✐❞♦ ❞❡ t♦❞♦s q✉❡ ❛ ▼❛t❡♠át✐❝❛ ♦r✐❣✐♥♦✉ ❡ s❡ ❞❡s❡♥✲ ✈♦❧✈❡✉ ❡♠ ❢✉♥çã♦ ❞❛s ♥❡❝❡ss✐❞❛❞❡s ❡♥❢r❡♥t❛❞❛s ♣❡❧♦ ❤♦♠❡♠ ♥❛s s✉❛s r❡❧❛çõ❡s s♦❝✐❛✐s ❡ ♥♦ ❡♥❢r❡♥t❛♠❡♥t♦ ❞❛s ❞✐✜❝✉❧❞❛✲ ❞❡s ✐♠♣♦st❛s ♣❡❧❛ ♥❛t✉r❡③❛✳ ❆♣❡s❛r ❞✐ss♦✱ ❞❡✈✐❞♦ às ❞✐✈❡rs❛s tr❛♥s❢♦r♠❛çõ❡s ♦❝♦rr✐❞❛s ♣❡❧❛s ♣♦❧ít✐❝❛s ❡❞✉❝❛❝✐♦♥❛✐s✱ ♦ q✉❡ s❡ ✈ê ❤♦❥❡ ❡♠ ❞✐❛ é ✉♠ ❡♥s✐♥♦ ❞❛ ▼❛t❡♠át✐❝❛ ♣♦✉❝♦ ❝♦♥✲ t❡①t✉❛❧✐③❛❞❛✱ ❝♦♥tr✐❜✉✐♥❞♦ ♣❛r❛ ❛ ❢❛❧t❛ ❞❡ ❡stí♠✉❧♦ ❞♦s ♥♦ss♦s ❛❧✉♥♦s✳✑
❆ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛ ❝♦♠ ♠❛✐s ❢r❡q✉ê♥❝✐❛ ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦✱ às q✉❡stõ❡s sã♦ ❝♦♥t❡①t✉❛❧✐③❛❞❛s✱ ❛❥✉❞❛♥❞♦ ♥❛ ❢♦r♠❛çã♦ ❞❡ ✉♠ ✐♥❞✐✈í❞✉♦ ❝rít✐❝♦ ❡ ♥❛ t♦♠❛❞❛ ❞❡ ❞❡❝✐sõ❡s ♥❛ s✉❛ ✈✐❞❛✳ ❱✐✈❡♠♦s ♥✉♠ ♣❛ís ♦♥❞❡ ❛s ♣❡ss♦❛s ❝❛❞❛ ✈❡③ ♠❛✐s ❡stã♦ s❡ ❡♥❞✐✈✐❞❛♥❞♦ ❡ ❝♦♠ ✐ss♦ ❝r❡s❝❡ ❛ ♦❢❡rt❛ ❞❡ ❝ré❞✐t♦s✳
❆ ✐♥t❡♥çã♦ ❞❡ss❡ tr❛❜❛❧❤♦ é ❢❛❝✐❧✐t❛r ❛ ❞❡❝✐sã♦ ❞♦ ✐♥❞✐✈í❞✉♦ ♥❛s ❛t✐✈✐❞❛❞❡s ✜♥❛♥✲ ❝❡✐r❛s✳ ❊♠ ❝❛❞❛ ❝❛♣ít✉❧♦✱ ♠♦str❛♠♦s ♦s ❝♦♥t❡ú❞♦s ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ tr❛❜❛❧❤❛❞♦s✱ ❝♦♠♦✿ P♦t❡♥❝✐❛çã♦✱ ❘❛❞✐❝✐❛çã♦✱ ❘❛③ã♦✱ Pr♦♣♦rçã♦✱ P♦r❝❡♥t❛❣❡♠✱ ❋✉♥çã♦ ❞♦ ✶♦❣r❛✉✱
♦✉tr♦s✱ ♣♦❞❡♠ s❡r ❛♣❧✐❝❛❞♦s✳ ❆ s❡❣✉✐r ❞❡s❝r❡✈❡♠♦s ❝♦♠♦ ❡stá ❞✐✈✐❞✐❞♦ ♦ tr❛❜❛❧❤♦✳ ◆♦ ❈❛♣ít✉❧♦ ✶✱ ❝❤❛♠❛❞♦ ❞❡ ❈❛♣✐t❛❧✐③❛çã♦ ❙✐♠♣❧❡s ❡ ❈♦♠♣♦st❛✱ ❛♣r❡s❡♥t❛♠♦s ✉♠ ❜r❡✈❡ ❝♦♥❝❡✐t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛✱ ❡ ❞♦s ❡❧❡♠❡♥t♦s ✉t✐❧✐③❛❞♦s ❡♠ ❥✉r♦s s✐♠♣❧❡s ❡ ❝♦♠♣♦st♦s✳ ❘❡❛❧✐③❛♥❞♦ ✉♠ ❡st✉❞♦ ❝♦♠ ❝♦♥❝❡✐t♦s ❡ ❞❡♠♦♥str❛çõ❡s ❞❛s ❢ór♠✉❧❛s ❞❡ ❥✉r♦s s✐♠♣❧❡s ❡ ❝♦♠♣♦st♦✱ ❜✉s❝❛♥❞♦ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❛tr❛✈és ❞❡ ❡①❡♠♣❧♦s✳ ■♥❝❧✉✐ ✉♠❛ ❜r❡✈❡ ❤✐stór✐❛ ❞❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛ ❍P ✶✷❈✱ ❡ ❝♦♠♦ ✉t✐❧✐③á✲❧❛ ❡♠ ❥✉r♦s ❝♦♠♣♦st♦s✳
◆♦ ❈❛♣ít✉❧♦ ✷✱ ✐♥t✐t✉❧❛❞♦ ❞❡ ❙ér✐❡s ❞❡ P❛❣❛♠❡♥t♦s✱ tr❛❜❛❧❤❛♠♦s ❝♦♠ sér✐❡s ♣♦st❡✲ ❝✐♣❛❞❛s✱ ❛♥t❡❝✐♣❛❞❛s ❡ ❞✐❢❡r✐❞❛s✱ ♦♥❞❡ sã♦ ❛♥❛❧✐s❛❞♦s ❞♦✐s ♣r♦❝❡ss♦s ❞❡ ✐♥✈❡st✐♠❡♥t♦✱ ❱❛❧♦r Pr❡s❡♥t❡ ❡ ❱❛❧♦r ❋✉t✉r♦✱ ❞❡ ❢♦r♠❛ ❝♦♥❝❡✐t✉❛❧✱ ♠♦str❛♥❞♦ ❡①❡♠♣❧♦s t❛♠❜é♠ r❡s♦❧✈✐❞♦s ♥❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛ ❍P ✶✷❈✳
❏á ♦ ❈❛♣ít✉❧♦ ✸✱ ❙✐st❡♠❛s ❞❡ ❆♠♦rt✐③❛çã♦✱ tr❛t❛r❡♠♦s ❞♦s s✐st❡♠❛s ❞❡ ❛♠♦rt✐③❛✲ çã♦ ♠❛✐s ✉t✐❧✐③❛❞♦s ♥♦ ♠❡r❝❛❞♦✱ ❝♦♠♦ s✐st❡♠❛ ❢r❛♥❝ês✱ ❝♦♥st❛♥t❡✱ ♠✐st♦ ❡ ❛♠❡r✐❝❛♥♦✱ ❢❛❧❛♥❞♦ ❞♦s ❝♦♥❝❡✐t♦s ❡ ❡①❡♠♣❧♦s✱ ❝♦♥str✉✐♥❞♦ ♣❧❛♥✐❧❤❛s ✜♥❛♥❝❡✐r❛s ❡ ✉t✐❧✐③❛♥❞♦ ❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛ ❍P ✶✷❈✳
❈❛♣ít✉❧♦ ✶
❈❛♣✐t❛❧✐③❛çã♦ ❙✐♠♣❧❡s ❡ ❈♦♠♣♦st❛
❙❡❣✉♥❞♦ ❏❯❊❘ ✭✷✵✵✾✱ ♣✳✾✮✱ q✉❛♥❞♦ ❢❛③❡♠♦s ✉♠ ❡♠♣rést✐♠♦ ♦✉ ✐♥✈❡st✐♠❡♥t♦ ♥♦ ♣r❡s❡♥t❡✱ ♦ s❡✉ ✈❛❧♦r é ❛✉♠❡♥t❛❞♦ ♥♦ ❢✉t✉r♦✳ ❊ q✉❛♥t✐❛s ❞✐s♣♦♥í✈❡✐s ♥♦ ❢✉t✉r♦✱ t❡♠ s❡✉ ✈❛❧♦r r❡❞✉③✐❞♦ ♥♦ ♣r❡s❡♥t❡✳
❱♦❝ê ♣r❡❢❡r❡ r❡❝❡❜❡r ❘✩ ✶✵✳✵✵✵✱✵✵ ❤♦❥❡ ♦✉ ❘✩✶✺✳✵✵✵✱✵✵✱ ❞❛q✉✐ ❛ ❞♦✐s ❛♥♦s❄ Pr♦✲ ✈❛✈❡❧♠❡♥t❡ ❛ s✉❛ r❡s♣♦st❛ s❡r✐❛ ❤♦❥❡✱ ♠❛s ❡♥tr❡ ❘✩ ✶✵✳✵✵✵✱✵✵ ❤♦❥❡ ❡ ❘✩ ✶✺✳✵✵✵✱✵✵ ❞❛q✉✐ ❛ ❞♦✐s ❛♥♦s✱ ❛ r❡s♣♦st❛ ♠❛✐s ❝♦❡r❡♥t❡ é✿ ❞❡♣❡♥❞❡✱ ♣♦✐s ✈❛✐ ❞❡♣❡♥❞❡r ❞❛s ❛❧t❡r♥❛t✐✈❛s ✜♥❛♥❝❡✐r❛s✱ ♥♦ ♠♦♠❡♥t♦ ❞❛ ❞❡❝✐sã♦✳
✶✳✶ ❖ ❈❛♣✐t❛❧ ❡ ♦s ❏✉r♦s
❈❛♣✐t❛❧✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❍❆❩❩❆◆ ❡ P❖▼P❊❖ ✭✷✵✵✼✱ ♣✳✶✮✱ é q✉❛❧q✉❡r ✈❛❧♦r ♠♦♥❡tár✐♦ q✉❡ ✉♠❛ ♣❡ss♦❛ ✭❢ís✐❝❛ ♦✉ ❥✉rí❞✐❝❛✮ ❡♠♣r❡st❛ ♣❛r❛ ♦✉tr❛ ❞✉r❛♥t❡ ❝❡rt♦ t❡♠♣♦✳ ❈♦♠♦ ♦ ❡♠♣r❡st❛❞♦r ♥ã♦ t❡♠ ♠❛✐s ♣♦ss❡ ❞♦ ✈❛❧♦r ❡♠♣r❡st❛❞♦✱ ❡ ❛✐♥❞❛ ❡♠ ❢✉♥çã♦ ❞♦ r✐s❝♦ ❞❡ ♥ã♦ ♣❛❣❛♠❡♥t♦ ❡ ❞❛ ♣❡r❞❛ ❞❡ ♣♦❞❡r ❛q✉✐s✐t✐✈♦ ❞♦ ❞✐♥❤❡✐r♦ ♣❡❧❛ ✐♥✢❛çã♦✱ s✉r❣❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❏✉r♦s✱ q✉❡ é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ❝✉st♦ ❞♦ ❡♠♣rést✐♠♦ ✭♣❛r❛ ♦ t♦♠❛❞♦r✮ ♦✉ ❛ r❡♠✉♥❡r❛çã♦ ♣❡❧♦ ✉s♦ ❞♦ ❝❛♣✐t❛❧ ✭❡♠♣r❡st❛❞♦r✮✳
❈❤❛♠❛♠♦s ❞❡ t❛①❛ ❞❡ ❥✉r♦s ♦ ✈❛❧♦r ❞♦s ❥✉r♦s ❡♠ ❝❡rt❛ ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦✱ ❡①✲
♣r❡ss♦ ❝♦♠♦ ✉♠❛ ♣♦r❝❡♥t❛❣❡♠ ❞♦ ❝❛♣✐t❛❧✳
❊①❡♠♣❧♦ ✶✳✶✳✶ ❙❡ ✉♠ ❝❛♣✐t❛❧ ❞❡ ❘✩ ✶✳✵✵✵✱✵✵ é ❡♠♣r❡st❛❞♦ ♣♦r ✉♠ ❛♥♦ à t❛①❛ ❞❡
10% ❛✳❛✳✭✶✵✪ ❛♦ ❛♥♦✮ ♦ ❥✉r♦s s❡rá ✐❣✉❛❧ ❛ ✶✵✪ ❞❡ ❘✩ ✶✳✵✵✵✱✵✵✱ ❡ ♣❛r❛ r❡s♦❧✈❡r✱
♠✉❧t✐♣❧✐❝❛♠♦s ✶✳✵✵✵ ♣♦r ✵✱✶✱ q✉❡ é ❛ ❢♦r♠❛ ❞❡❝✐♠❛❧ ❞❡ ✶✵✪ ✭✶✵✪ ❂ 10
100 ❂ ✵✱✶✮✱ ❧♦❣♦
♦s ❥✉r♦s ❝♦❜r❛❞♦ é ❞❡ ❘✩ ✶✵✵✱✵✵✳
❉❡♥♦t❛♠♦s ❞❡ ❈ ♦ ❝❛♣✐t❛❧✱ ▼ ♦ ♠♦♥t❛♥t❡✱ ❏ ♦s ❥✉r♦s✱ ✐ é ❛ t❛①❛ ✭❞♦ ✐♥❣❧ês✱ ✐♥t❡r❡st✱ q✉❡ s✐❣♥✐✜❝❛ ❥✉r♦s✮ ❡ ♥ ♦ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦✳ ❖ ♠♦♥t❛♥t❡ é ♦ ✈❛❧♦r ❞♦ ❝❛♣✐t❛❧ ♠❛✐s ♦s ❥✉r♦s✳ ❊♥tã♦ t❡♠♦s✿
J =C×i,
❡
M =C+J.
✶✳✷ ❏✉r♦s ❙✐♠♣❧❡s
❉❡✜♥✐çã♦ ✶✳✷✳✶ ❏✉r♦s s✐♠♣❧❡s é q✉❛♥❞♦ ❛ t❛①❛ ❞❡ ❥✉r♦s ✐♥❝✐❞❡ s❡♠♣r❡ ❡♠ ❝✐♠❛ ❞♦ ❝❛♣✐t❛❧ ✐♥✐❝✐❛❧✳
❙ã♦ r❛r❛s ❛s ♦♣❡r❛çõ❡s ✜♥❛♥❝❡✐r❛s ❡ ❝♦♠❡r❝✐❛✐s q✉❡ ✉t✐❧✐③❛♠ ❡ss❡ t✐♣♦ ❞❡ ❝❛♣✐t❛❧✐③❛✲ çã♦✳
❊①❡♠♣❧♦ ✶✳✷✳✷ ❯♠ ❝❛♣✐t❛❧ ❞❡ ❘✩ ✶✵✳✵✵✵✱✵✵✱ ❢♦✐ ❛♣❧✐❝❛❞♦ ❞✉r❛♥t❡ três ♠❡s❡s ❛ ✉♠❛ t❛①❛ ❞❡ ✶✪ ❛✳♠✳✭❛♦ ♠ês✮✱ ❡♠ r❡❣✐♠❡ ❞❡ ❥✉r♦s s✐♠♣❧❡s✳ ◗✉❛❧ ♦ ♠♦♥t❛♥t❡❄
Pr✐♠❡✐r♦ ✈❛♠♦s tr❛♥s❢♦r♠❛r ❛ t❛①❛ ♣❡r❝❡♥t✉❛❧ ❡♠ ❞❡❝✐♠❛❧✱ ♦✉ s❡❥❛✿
1% = 1
100 = 0,01.
J =C×i.
❆ss✐♠ t❡♠♦s✿ ✶♦ ♠ês✿
J1 = 10.000×0,01 = 100,00.
✷♦ ♠ês✿
J2 = 10.000×0,01 = 100,00.
✸♦ ♠ês✿
J3 = 10.000×0,01 = 100,00.
❈♦♠♦ ♦ r❡❣✐♠❡ ❞❡ ❝❛♣✐t❛❧✐③❛çã♦ é s✐♠♣❧❡s ♦ ❥✉r♦s é ❛♣❧✐❝❛❞♦ s❡♠♣r❡ ❡♠ r❡❧❛çã♦ ❛♦ ❝❛♣✐t❛❧✱ ♦ ❥✉r♦s r❡❢❡r❡♥t❡ ❛ ❡ss❡ ♣❡rí♦❞♦ ❞❡ ✸ ♠❡s❡s s❡rá✿
J1+J2+J3 = 100,00 + 100,00 + 100,00 = 300,00.
▲♦❣♦ ♦ ♠♦♥t❛♥t❡✱ ❛♣ós três ♠❡s❡s é✿
M =C+J.
M = 10.000 + 300 = 10.300,00.
❯s❛♥❞♦ ♦ r❛❝✐♦❝í♥✐♦ ❛❝✐♠❛ ❞❡❞✉③✐♠♦s ✉♠❛ ❢ór♠✉❧❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ❥✉r♦s s✐♠♣❧❡s✿
• ❏✉r♦s ❛♣ós ✶ ♣❡rí♦❞♦✿ J1 =C×i;
• ❏✉r♦s ❛♣ós ✷ ♣❡rí♦❞♦s✿ J2 =C×i+C×i=C×i×2;
• ❏✉r♦s ❛♣ós ✸ ♣❡rí♦❞♦s✿ J3 =C×i+C×i+C×i=C×i×3;
❯t✐❧✐③❛♥❞♦ ✉♠ ❛r❣✉♠❡♥t♦ ❞❡ ✐♥❞✉ç❛♦✱ ❝♦♥❝❧✉✐♠♦s q✉❡✿
• ❏✉r♦s ❛♣ós n ♣❡rí♦❞♦✿ Jn =C×i+C×i+C×i+...+C×i.
Jn =C×i×n. ✭✶✳✶✮
❖❜s❡r✈❛çã♦ ✶ ❆ t❛①❛ ✐ ❡ ♦ ♣❡rí♦❞♦ ♥ t❡♠ q✉❡ ❡stá ♥❛ ♠❡s♠❛ ✉♥✐❞❛❞❡ ❞❡ t❡♠♣♦✱ ♣♦r ❡①❡♠♣❧♦✱ s❡ ❛ t❛①❛ ✐ ❢♦r ❛♦ ❛♥♦✱ ♥ ❞❡✈❡ ❡stá ❡♠ ❛♥♦✳
❊①❡♠♣❧♦ ✶✳✷✳✸ ❯♠ ❝❛♣✐t❛❧ ❞❡ ❘✩ ✷✳✵✵✵✱✵✵✱ ❛♣❧✐❝❛❞♦ ❛ ✉♠❛ t❛①❛ ❞❡ ❥✉r♦s s✐♠♣❧❡s ❞❡ ✻✪ ❛✳♠✳ ✭❛♦ ♠ês✮✱ ♣♦r ✉♠ ♣❡rí♦❞♦ ❞❡ ✶✵ ♠❡s❡s✱ ✈❛✐ r❡♥❞❡r q✉❛♥t♦ ❞❡ ❥✉r♦s❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✿ ❖ ❝❛♣✐t❛❧ éC = 2.000.
❆ t❛①❛ ❞❡ ❥✉r♦s ❡♠ ♣♦r❝❡♥t❛❣❡♠ éi= 6% a.m. ◆❛ ❢♦r♠❛ ❞❡❝✐♠❛❧ ✜❝❛ i = 6
100 = 0,06, ✈❛♠♦s tr❛❜❛❧❤❛r ❝♦♠ ❛ t❛①❛ ❞❡ ❥✉r♦s ♥❛
❢♦r♠❛ ❞❡❝✐♠❛❧✳
❖ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ é n= 10 ♠❡s❡s✳
❊♥tã♦ ✈❛♠♦s ❡♥❝♦♥tr❛r ♦s ❥✉r♦s J.
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛✱ ✶✳✶ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s✿
J = 2.000×0,06×10 = 1.200.
▲♦❣♦✱ ✈❛✐ r❡♥❞❡r ✉♠ ❥✉r♦s ❞❡ ❘✩ ✶✳✷✵✵✱✵✵✳
❙❛❜❡♠♦s q✉❡M =C+J✱ ❛ss✐♠ t❡♠♦sJ =▼ ✲ ❈✱ s✉❜st✐t✉✐♥❞♦ ♥❛ ❋ór♠✉❧❛ ✶✳✶✱
t❡♠♦s✿
M−C =C×i×n, ❧♦❣♦✱
M =C+C×i×n. ❈♦❧♦❝❛♥❞♦ ♦ ❈ ❡♠ ❡✈✐❞ê♥❝✐❛✱ ♦❜t❡♠♦s✿
M =C(1 +in). ✭✶✳✷✮
❖❜s❡r✈❛çã♦ ✷ ❉❡st❛❝❛♠♦s q✉❡ ❛❧❣✉♥s ❝♦♥t❡ú❞♦s ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ sã♦ ❢r❡q✉❡♥t✐✲ ♠❡♥t❡ ✉t✐❧✐③❛❞♦s ❝♦♠♦✿
• P♦r❝❡♥t❛❣❡♠✿ ♣❛r❛ ❡♥❝♦♥tr❛r ♦s ❥✉r♦s✱ ❝❛❧❝✉❧❛♠♦s ♦ ♣❡r❝❡♥t✉❛❧ ❡♠ ❝✐♠❛ ❞♦
❝❛♣✐t❛❧✱ ❛ t❛①❛ ❞❡ ❥✉r♦s é ❞❛❞❛ ❡♠ ♣♦r❝❡♥t❛❣❡♠✱ ❡ t❡♠♦s q✉❡ tr❛♥s❢♦r♠❛r ❡♠ ❞❡❝✐♠❛❧❀
• Pr♦❣r❡ssã♦ ❆r✐t♠ét✐❝❛ ✭P✳❆✮✿ ♦s ❥✉r♦s sã♦ ❝❛❧❝✉❧❛❞♦s s❡♠♣r❡ ❡♠ ❝✐♠❛ ❞♦ ❝❛✲
♣✐t❛❧ ✐♥✐❝✐❛❧✱ q✉❡ sã♦ ✐❣✉❛✐s ❡♠ t♦❞♦s ♦s ♣❡rí♦❞♦s ❞❡ t❡♠♣♦✱ ❢♦r♠❛♥❞♦ ❛ss✐♠ ✉♠❛ P✳❆✱ ♦♥❞❡ ❛ r❛③ã♦ sã♦ ♦s ❥✉r♦s✱ ♦ ❝❛♣✐t❛❧ ✐♥✐❝✐❛❧ é ♦ ♣r✐♠❡✐r♦ t❡r♠♦ ❡ ♦ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ ♦ ♥ú♠❡r♦ ❞❡ t❡r♠♦s ❞❛ P✳❆❀
• ❋❛t♦r❛çã♦ ✭❝♦❧♦❝❛♥❞♦ ♦ ❢❛t♦r ❝♦♠✉♠ ❡♠ ❡✈✐❞ê♥❝✐❛✮✿ ♣❛r❛ ❛❝❤❛r ❛ ❢ór♠✉❧❛ ❞♦
♠♦♥t❛♥t❡✱ ❝♦❧♦❝❛♠♦s ♦ ❝❛♣✐t❛❧ ✐♥✐❝✐❛❧ ❡♠ ❡✈✐❞ê♥❝✐❛❀
• ❊q✉❛çã♦ ❞♦ ✶♦ ❣r❛✉✿ ♣❛r❛ r❡s♦❧✈❡r ♦s ♣r♦❜❧❡♠❛s✱ r❡s♦❧✈❡♠♦s ✉♠❛ ❡q✉❛çã♦ ❞♦
✶♦ ❣r❛✉✳
❉❡✜♥✐çã♦ ✶✳✷✳✹ ❉✐③❡♠♦s q✉❡ ❞✉❛s t❛①❛s sã♦ ❡q✉✐✈❛❧❡♥t❡s ❡♠ ❥✉r♦s s✐♠♣❧❡s q✉❛♥❞♦✱ ❛♣❧✐❝❛❞❛s ❡♠ ✉♠ ♠❡s♠♦ ❝❛♣✐t❛❧ ❡ ❞✉r❛♥t❡ ✉♠ ♠❡s♠♦ ♣r❛③♦✱ ❞❡r❡♠ ❥✉r♦s ✐❣✉❛✐s✳
❊①❡♠♣❧♦ ✶✳✷✳✺ ❊♠ ❥✉r♦s s✐♠♣❧❡s✱ q✉❛❧ ❛ t❛①❛ ❛♥✉❛❧ ❡q✉✐✈❛❧❡♥t❡ ❛ ✶✪ ❛✳♠❄
i= 1% ×✶✷= 12 ✪a.a. ✭❛♦ ❛♥♦✮✳
❊①❡♠♣❧♦ ✶✳✷✳✻ ❊♠ ❥✉r♦s s✐♠♣❧❡s✱ q✉❛❧ ❛ t❛①❛ ♠❡♥s❛❧ ❡q✉✐✈❛❧❡♥t❡ ❛ ✾✪ ❛✳t✳ ✭❛♦ tr✐♠❡str❡✮❄
i= 9
3 = 3 ✪a.m.
❊①❡♠♣❧♦ ✶✳✷✳✼ ❊♠ ❥✉r♦s s✐♠♣❧❡s✱ q✉❛❧ ❛ t❛①❛ ♠❡♥s❛❧ ❡q✉✐✈❛❧❡♥t❡ ❛ ✹✪ ❛✳❜✳ ✭❛♦ ❜✐♠❡str❡✮ ❡ ✶✷✪ ❛✳s✳ ✭❛♦ s❡♠❡str❡✮❄
i= 42 = 2✪ a.m. ❡i= 126 = 2✪a.m.
❖❜s❡r✈❛çã♦ ✸ ❆ ▼❛t❡♠át✐❝❛ ❋✐♥❛♥❝❡✐r❛ ♣♦❞❡ s❡r ♠✉✐t♦ ✉t✐❧✐③❛❞❛ ♥♦ ❊♥s✐♥♦ ❇á✲ s✐❝♦✱ ♣♦✐s ❡♥✈♦❧✈❡ q✉❡stõ❡s ❝♦♥t❡①t✉❛❧✐③❛❞❛s✱ ❛♣❧✐❝❛❞❛s ❡ ✈✐✈❡♥❝✐❛❞❛s ♣❡❧♦ ❛❧✉♥♦✱ s❡♥❞♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞♦ ✐♥❞✐✈í❞✉♦✱ ❛❥✉❞❛♥❞♦ ♥❛ t♦♠❛❞❛ ❞❡ ❞❡✲ ❝✐sã♦ ♣❛r❛ ❛ s✉❛ ✈✐❞❛✳
❊①❡♠♣❧♦ ✶✳✷✳✽ ❉✉r❛♥t❡ ✺ ♠❡s❡s ✉♠ ❝❛♣✐t❛❧ ❞❡ ❘✩ ✹✳✺✵✵✱✵✵✱ ❢♦✐ ❛♣❧✐❝❛❞♦ ❛ ✉♠ t❛①❛ ❞❡ ❥✉r♦s s✐♠♣❧❡s ❞❡ ✹✪ ❛✳♠✳✱ ❞❡t❡r♠✐♥❡ ♦ ✈❛❧♦r ❞♦ ♠♦♥t❛♥t❡ r❡s❣❛t❛❞♦❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✿ ◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ♦ ♠♦♥t❛♥t❡ M✳ ❖ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ é n= 5 ♠❡s❡s✳
❖ ❝❛♣✐t❛❧ éC = 4.500.
❆ t❛①❛ ❞❡ ❥✉r♦s ♣❡r❝❡♥t✉❛❧ é i= 4%a.m. ◆❛ ❢♦r♠❛ ❞❡❝✐♠❛❧ ♦s ❥✉r♦s ✜❝❛♠i= 4
100 = 0,04.
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ✶✳✷✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s✱
M = 4.500(1 + 0,04×5);
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❞♦ ✶♦ ❣r❛✉❀
M = 4.500(1 + 0,2)
M = 4.500×1,2,
♦❜t❡♠♦s❀
M = 5.400,00. ▲♦❣♦✱ ♦ ♠♦♥t❛♥t❡ r❡s❣❛t❛❞♦ ❢♦✐ ❞❡ ❘✩ ✺✳✹✵✵✱✵✵✳
❊①❡♠♣❧♦ ✶✳✷✳✾ ◗✉❛❧ ♦ ✈❛❧♦r ❞♦ r❡s❣❛t❡ ❞❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡ ❘✩ ✹✵✳✵✵✵✱✵✵ ♣❡❧♦ ♣r❛③♦ ❞❡ ✷ ❛♥♦s ❡ ✻ ♠❡s❡s ❛ t❛①❛ ❞❡ ❥✉r♦s s✐♠♣❧❡s ❞❡ ✶✷✪ ❛✳❛✳❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✿ ◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ♦ ♠♦♥t❛♥t❡ M✳ ❖ ❝❛♣✐t❛❧ éC = 40.000,00✳
❖ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ é n = 2 anos e 6 meses = 2,5 anos ✭❈♦♠♦ ❛ t❛①❛ ❞❡ ❥✉r♦s ❡stá ❛♦ ❛♥♦✱ ❞❡✐①❛♠♦s ♦ ♣❡rí♦❞♦ t❛♠❜é♠ ❡♠ ❛♥♦s✮✳
❆ t❛①❛ ❞❡ ❥✉r♦s ♣❡r❝❡♥t✉❛❧ i= 12% a.a. ◆❛ ❢♦r♠❛ ❞❡❝✐♠❛❧ ✜❝❛i= 10012 = 0,12.
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ✶✳✷✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s✿
M = 40.000(1 + 0,12×2,5),
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❞♦ ✶♦ ❣r❛✉❀
M = 40.000(1 + 0,3)
M = 40.000×1,3 ✱
♦❜t❡♠♦s❀
M = 52.000,00. ▲♦❣♦✱ ♦ ✈❛❧♦r ❞♦ r❡s❣❛t❡ ❢♦✐ ❞❡ ❘✩ ✺✷✳✵✵✵✱✵✵✳
❊①❡♠♣❧♦ ✶✳✷✳✶✵ ❊♠♣r❡st❡✐ ❘✩ ✶✵✵✳✵✵✵✱✵✵ ❡ r❡❝❡❜✐ ❘✩ ✶✸✵✳✵✵✵✱✵✵ ♥♦ ✜♥❛❧ ❞❡ ✽ ♠❡s❡s✱ ❞❡t❡r♠✐♥❡ ❛ t❛①❛ ❞❡ ❥✉r♦s s✐♠♣❧❡s r❡❢❡r❡♥t❡ ❛ ❡ss❡ ❡♠♣rést✐♠♦✿
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✿ ❖ ❝❛♣✐t❛❧ éC = 100.000,00.
❖ ♠♦♥t❛♥t❡ é M = 130.000,00. ❖ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ é n= 8 ♠❡s❡s✳
◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ❛ t❛①❛ ❞❡ ❥✉r♦s ♠❡♥s❛❧ i✳
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ✶✳✷✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s✿
130.000 = 100.000(1 + 8i),
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❞♦ ✶♦ ❣r❛✉❀
130.000
100.000 = 1 + 8i 1,3 = 1 + 8i
1,3−1 = 8i
8i= 0,3
i= 0,3 8
i= 0,0375×100 ✱
♦❜t❡♠♦s❀
i= 3,65%a.m. ▲♦❣♦✱ ❛ t❛①❛ ❞❡ ❥✉r♦s é ❞❡ ✸✱✻✺✪ ❛✳♠✳
❊①❡♠♣❧♦ ✶✳✷✳✶✶ ❯♠ ❝❛♣✐t❛❧ ❞❡ ❘✩ ✼✳✺✵✵✱✵✵ ❢♦✐ ❛♣❧✐❝❛❞♦ ❛ ✉♠❛ t❛①❛ ❞❡ ❥✉r♦s s✐♠✲ ♣❧❡s ❞❡ ✷✵✪ ❛✳❛✳✱ ❡♠ q✉❛♥t♦ t❡♠♣♦ s❡rá r❡s❣❛t❛❞♦ ❘✩ ✾✳✵✵✵✱✵✵❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✿ ❖ ♠♦♥t❛♥t❡ é M = 9.000.
❖ ❝❛♣✐t❛❧ éC = 7.500.
❆ t❛①❛ ❞❡ ❥✉r♦s ♣❡r❝❡♥t✉❛❧ é i= 20%a.a. ◆❛ ❢♦r♠❛ ❞❡❝✐♠❛❧ ✜❝❛i= 20
100 = 0,2.
◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ♦ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ n✳
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ✶✳✷✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s✿
9.000 = 7.500(1 + 0,2n),
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❞♦ ✶♦ ❣r❛✉❀
9.000
7.500 = 1 + 0,2n 1,2 = 1 + 0,2n
0,2n= 1,2−1
0,2n = 0,2
n = 0,2 0,2 ✱
♦❜t❡♠♦s❀
n = 1ano.
▲♦❣♦✱ ❡♠ ✉♠ ❛♥♦ ♦ ✈❛❧♦r ❞❡ ❘✩ ✾✳✵✵✵✱✵✵ é r❡s❣❛t❛❞♦✳
✶✳✸ ❏✉r♦s ❈♦♠♣♦st♦s
❉❡✜♥✐çã♦ ✶✳✸✳✶ ❏✉r♦s ❈♦♠♣♦st♦s é q✉❛♥❞♦ ♦s ❥✉r♦s ✐♥❝✐❞❡♠ s♦❜r❡ ♦ ♠♦♥t❛♥t❡ ❞♦ ♣❡rí♦❞♦ ❛♥t❡r✐♦r✱ ♣❛ss❛♥❞♦ ♦ ♥♦✈♦ ♠♦♥t❛♥t❡ ❛ ♣r♦❞✉③✐r ❥✉r♦s ♥♦ ♣❡rí♦❞♦ s❡❣✉✐♥t❡✳
❊①❡♠♣❧♦ ✶✳✸✳✷ ❯♠ ❝❛♣✐t❛❧ ❞❡ ❘✩ ✶✵✳✵✵✵✱✵✵✱ ❢♦✐ ❛♣❧✐❝❛❞♦ ❞✉r❛♥t❡ três ♠❡s❡s ❛ ✉♠❛ t❛①❛ ❞❡ ✶✪ ❛✳♠✳✱ ❡♠ r❡❣✐♠❡ ❞❡ ❥✉r♦s ❝♦♠♣♦st♦s✳
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✿ ❖ ❝❛♣✐t❛❧ éC = 10.000.
❆ t❛①❛ ❞❡ ❥✉r♦s ♣❡r❝❡♥t✉❛❧ é i= 1%a.m.
1% = 1001 = 0,01.
❱✐♠♦s q✉❡✿ J =C×i✱ ❡♥tã♦✿
J1 = 10.000×0,01 = 100,00 ✱
❧♦❣♦❀
• ◆♦ ✶♦ ♠ês ♦ ♠♦♥t❛♥t❡ é✿
M1 =C+J = 10.000 + 100 = 10.100.
• ◆♦ ✷♦ ♠ês✱ ♦ ❥✉r♦s é ❝❛❧❝✉❧❛❞♦ ❡♠ ❝✐♠❛ ❞♦ ♠♦♥t❛♥t❡ ❛♥t❡r✐♦r✿
M2 =M1+M1×i= 10.100 + 10.100×0,01 = 10.100 + 101 = 10.201.
• ◆♦ ✸♦ ♠ês✱ ♦ ❥✉r♦s é ❝❛❧❝✉❧❛❞♦ ❡♠ ❝✐♠❛ ❞♦ ♠♦♥t❛♥t❡ ❛♥t❡r✐♦r✿
M3 =M2+M2×i= 10.201 + 10.201×0,01 = 10.201 + 102,01 = 10.303,01.
❯s❛♥❞♦ ❡st❡ ❛r❣✉♠❡♥t♦ ❡♠ ❝♦♠❜✐♥❛çã♦ ❝♦♠ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥❞✉çã♦ ✜♥✐t❛✱ ❝♦♥✲ ❝❧✉✐♠♦s q✉❡✿
• ▼♦♥t❛♥t❡ ❛♣ós ♦ ✶♦ ♣❡rí♦❞♦✿
M1 =C+Ci=C(1 +i).
• ▼♦♥t❛♥t❡ ❛♣ós ♦ ✷♦ ♣❡rí♦❞♦✿
M2 =M1+M1×i=M1(1 +i) =C(1 +i)(1 +i) = C(1 +i)2.
• ▼♦♥t❛♥t❡ ❛♣ós ♦ ✸♦ ♣❡rí♦❞♦✿
M3 =M2+M2×i=M2(1 +i) =C(1 +i)2(1 +i) = C(1 +i)3.
• ▼♦♥t❛♥t❡ ❛♣ós n ♣❡rí♦❞♦s✿
M =C(1 +i)n .
P♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♣❛r❛ ❛❝❤❛r ♦ ♠♦♥t❛♥t❡ ❡♠ ❥✉r♦s ❝♦♠♣♦st♦s✱ ✉t✐❧✐③❛♠♦s ❛ ❋ór♠✉❧❛✿
Mn =C(1 +i)n ✭✶✳✸✮
❖❜s❡r✈❛çã♦ ✹ ❊♠ ❏✉r♦s ❈♦♠♣♦st♦s t❛♠❜é♠ ✉t✐❧✐③❛♠♦s ✈ár✐♦s ❝♦♥té✉❞♦s ❞♦ ❊♥✲ s✐♥♦ ❇ás✐❝♦✱ ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ♦ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦✱ ✉t✐❧✐③❛♠♦s ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛✱ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ t❛①❛ ❞❡ ❥✉r♦s✱ ✉t✐❧✐③❛♠♦s r❛❞✐❝✐❛çã♦ ❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❝r❡s❝✐♠❡♥t♦ ❞♦ ❝❛♣✐t❛❧ ✐♥✐❝✐❛❧ ❛♦ ✜♥❛❧ ❞❡ ❝❛❞❛ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦✱ é ✉♠❛ Pr♦❣r❡ssã♦ ●❡♦♠étr✐❝❛ ✭P✳●✮ ❞❡ r❛③ã♦ ✭✶ ✰ ✐✮✳
✶✳✸✳✶ ❯t✐❧✐③❛♥❞♦ ❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛ ❍P ✶✷❈
❆ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛ ❍P ✶✷❈ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ♣❡❧♦ ♠❛t❡♠át✐❝♦ ♣♦❧♦♥ês ❏❛♥ ▲✉❦❛s✐❡✇✐❝③✳ ➱ ❝❛r❛❝t❡r✐③❛❞❛ ♣♦r tr❛❜❛❧❤❛r ❝♦♠ ❧ó❣✐❝❛ ❘P◆ ✭❞♦ ✐♥❣❧ês ❘❡✈❡rs❡ P♦❧✐s❤ ◆♦t❛t✐♦♥ ✱ ♦✉ ♥♦t❛çã♦ ♣♦❧♦♥❡s❛ r❡✈❡rs❛✮✱ ♣❡r♠✐t✐♥❞♦ ✉♠❛ ❡♥tr❛❞❛ ♠❛✐s rá♣✐❞❛ ❞❡ ❞❛❞♦s ❡ ❛ ❡①❡❝✉çã♦ ♠❛✐s ❡✜❝✐❡♥t❡ ❞♦s ❝á❧❝✉❧♦s✳ ❊ss❡ ♠ét♦❞♦ s❡ ❛❞❡q✉♦✉ ❜❡♠ ❛♦ ✉s♦ ♥❛ ❝❛❧❝✉❧❛❞♦r❛✱ ✉♠❛ ✈❡③ q✉❡ ❞✐s♣❡♥s❛ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♣❛rê♥t❡s❡s✳ P♦ss✉✐ ♠❛✐s ❞❡ ✶✷✵ ❢✉♥çõ❡s ❡s♣❡❝í✜❝❛s✱ q✉❡ ♣❡r♠✐t❡♠ tr❛❜❛❧❤❛r ❝♦♠ ✷✵ ❞✐❢❡r❡♥t❡s ✢✉①♦s ❞❡ ❝❛✐①❛✱ ❥✉r♦s ❝♦♠♣♦st♦s✱ ❛♠♦rt✐③❛çã♦ ✭q✉❡ ✈❛♠♦s ❡st✉❞❛r ♥♦ ❈❛♣ít✉❧♦ ✸✮✳
✶✳✸✳✷ ❆♣r❡s❡♥t❛♥❞♦ ❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛ ❍P ✶✷❈
❋✐❣✉r❛ ✶✳✶✿ ❈❛❧❝✉❧❛❞♦r❛ ❍P ✶✷❈
❋✐❣✉r❛ ✶✳✷✿ ❋✉♥çõ❡s ❋✐♥❛♥❝❡✐r❛s ❞❛ ❈❛❧❝✉❧❛❞♦r❛ ❍P ✶✷❈
❋✐❣✉r❛ ✶✳✸✿ ■❞❡♥t✐✜❝❛çã♦ ❞❛s ❋✉♥çõ❡s ❋✐♥❛♥❝❡✐r❛s ♥❛ ❈❛❧❝✉❧❛❞♦r❛ ❍P ✶✷❈
P❛r❛ ❛♣❛❣❛r ❛s ❢✉♥çõ❡s ✜♥❛♥❝❡✐r❛s ❛❝✐♦♥❛♠♦s ❛s t❡❝❧❛s✿
❋✐❣✉r❛ ✶✳✹✿ ❈♦♠♦ ▲✐♠♣❛r ♦ ❱✐s♦r ❞❛ ❈❛❧❝✉❧❛❞♦r❛
❖❜s❡r✈❛çã♦ ✺ ❖ ✢✉①♦ ❞❡ ❝❛✐①❛ é ✉♠ ♦❜❥❡t♦ ♠❛t❡♠át✐❝♦✱ q✉❡ ♣♦❞❡ s❡r ✉s❛❞♦ ❡♠ tr❛♥s❛çõ❡s ✜♥❛♥❝❡✐r❛s✱ ♠♦str❛♥❞♦ ❣r❛✜❝❛♠❡♥t❡ ❡♠ ✉♠❛ ❧✐♥❤❛ ❤♦r✐③♦♥t❛❧ ♦ t❡♠♣♦✳ ❆s ❡♥tr❛❞❛s ♦✉ r❡❝❡❜✐♠❡♥t♦s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r s❡t❛s ✈❡rt✐❝❛✐s ♣❛r❛ ❜❛✐①♦ ❡ ❛s s❛í❞❛s ♦✉ ♣❛❣❛♠❡♥t♦s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r s❡t❛s ♣❛r❛ ❝✐♠❛✳
❋✐❣✉r❛ ✶✳✺✿ ❋❧✉①♦ ❞❡ ❈❛✐①❛ ♥♦ P♦♥t♦ ❞❡ ❱✐st❛ ❞♦ ❊♠♣r❡st❛❞♦r
❋✐❣✉r❛ ✶✳✻✿ ❋❧✉①♦ ❞❡ ❈❛✐①❛ ♥♦ P♦♥t♦ ❞❡ ❱✐st❛ ❞♦ ❚♦♠❛❞♦r ❞♦ ❊♠♣rést✐♠♦
❈♦♠♦ ❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ❍P ✶✷❈ tr❛❜❛❧❤❛ ❝♦♠ ❛ ✐❞❡✐❛ ❞♦ ✢✉①♦ ❞❡ ❝❛✐①❛✱ s❡♠♣r❡ q✉❛♥❞♦ ✐♥s❡r✐♠♦s ♦ ✈❛❧♦r ♣r❡s❡♥t❡ ♦✉ ♦ ✈❛❧♦r ❢✉t✉r♦ ♥❛ ❝❛❧❝✉❧❛❞♦r❛ t❡♠ q✉❡ ♠✉❞❛r ♦ s✐♥❛❧✱ ♣♦✐s ♥♦ ✢✉①♦ ❞❡ ❝❛✐①❛ q✉❛♥❞♦ ❛ s❡t❛ ❞♦ ✈❛❧♦r ♣r❡s❡♥t❡ ❡stá ♣❛r❛ ❜❛✐①♦✱ ❛ ❞♦ ✈❛❧♦r ❢✉t✉r♦ ❡stá ♣❛r❛ ❝✐♠❛ ❡ ✈✐❝❡ ✈❡rs❛✱ ❡ss❛s s❡t❛s ❢✉♥❝✐♦♥❛♠ ❝♦♠♦ ♦ s✐♥❛❧ ♥❡❣❛t✐✈♦ ♦✉ ♣♦s✐t✐✈♦✳ ❙❡ ♦ s✐♥❛❧ ♥ã♦ ❢♦r ♠✉❞❛❞♦✱ ♦ r❡s✉❧t❛❞♦ s❛✐rá ♥♦ ✈✐s♦r ❞❛ ❝❛❧❝✉❧❛❞♦r❛ ♥❡❣❛t✐✈♦✳
❊①❡♠♣❧♦ ✶✳✸✳✸ ❯♠ ❝❛♣✐t❛❧ ❞❡ ❘✩ ✺✳✵✵✵✱✵✵ ❢♦✐ ❛♣❧✐❝❛❞♦ ❛ ❥✉r♦s ❝♦♠♣♦st♦s ❞✉r❛♥t❡ ✻ ♠❡s❡s✱ à t❛①❛ ❞❡ ✷✪ ❛✳♠✳✳ ◗✉❛❧ ♦ ♠♦♥t❛♥t❡❄
❖ ❝❛♣✐t❛❧ éC = 5.000.
❖ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ é n= 6 ♠❡s❡s✳
❆ t❛①❛ ❞❡ ❥✉r♦s ♣❡r❝❡♥t✉❛❧ i= 2% a.m. ◆❛ ❢♦r♠❛ ❞❡❝✐♠❛❧i= 1002 = 0,02. ❖ ♠♦♥t❛♥t❡ é M✳
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ✶✳✸✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s✿
M = 5.000(1 + 0,02)6 ✱
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧❀
M = 5.000(1,02)6 ✱
♦❜t❡♠♦s❀
M = 5.630,81.
❱❛♠♦s ✐♥s❡r✐r ♣r✐♠❡✐r♦ ♦s ❞❛❞♦s ✐♥❢♦r♠❛❞♦s ♥♦ ♣r♦❜❧❡♠❛ ❡ s❡♠♣r❡ ❝♦❧♦❝❛♠♦s ♣♦r ú❧t✐♠♦ ♦ ❞❛❞♦ ❛ s❡r ❡♥❝♦♥tr❛❞♦✳
◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛✿
❋✐❣✉r❛ ✶✳✼✿ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✶
❖ ♠♦♥t❛♥t❡ ❢♦✐ ❞❡ ❘✩ ✺✳✻✸✵✱✽✶✳
❊①❡♠♣❧♦ ✶✳✸✳✹ ◗✉❛❧ ♦ ❝❛♣✐t❛❧✱ q✉❡ ❛♣❧✐❝❛❞♦ ❛ ❥✉r♦s ❝♦♠♣♦st♦s à t❛①❛ ❞❡ ✸✪ ❛✳♠✳✱ ♣r♦❞✉③ ✉♠ ♠♦♥t❛♥t❡ ❞❡ ❘✩ ✸✵✳✵✵✵✱✵✵ ❛♣ós ✉♠ ❛♥♦❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✿ ◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ♦ ❝❛♣✐t❛❧C✳
❆ t❛①❛ ❞❡ ❥✉r♦s ♣❡r❝❡♥t✉❛❧ i= 3% a.m. ◆❛ ❢♦r♠❛ ❞❡❝✐♠❛❧i= 3
100 = 0,03✳
❖ ♠♦♥t❛♥t❡ é M = 30.000✳
❖ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ n = ✶ ❛♥♦ ❂ ✶✷ ♠❡s❡s✱ ♣♦✐s ❛ t❛①❛ ❞❡ ❥✉r♦s ❡stá ❛♦ ♠ês✱
tr❛♥s❢♦r♠❛♠♦s ✶ ❛♥♦ ❡♠ ♠❡s❡s✱ ♦❜t❡♥❞♦ ✶✷ ♠❡s❡s✳
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ✶✳✸✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s❀
30.000 = C(1 + 0,03)12 ✱
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧❀
30.000 =C(1,03)12 30.000 =C×1,4258
❆ss✐♠✱
C = 30.000 1,4258 ✱
♦❜t❡♠♦s❀
C = 21.041,40. ◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛✿
❋✐❣✉r❛ ✶✳✽✿ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✷
❖ ❝❛♣✐t❛❧ é ❞❡ ❘✩ ✷✶✳✵✹✶✱✹✵✳
❊①❡♠♣❧♦ ✶✳✸✳✺ ❯♠ ❝❛♣✐t❛❧ ❞❡ ❘✩ ✶✵✳✵✵✵✱✵✵ ❢♦✐ ❛♣❧✐❝❛❞♦ ❛ ❥✉r♦s ❝♦♠♣♦st♦s ❞✉r❛♥t❡ q✉❛tr♦ ♠❡s❡s✱ ♣r♦❞✉③✐♥❞♦ ✉♠ ♠♦♥t❛♥t❡ ❞❡ ❘✩ ✶✷✳✵✵✵✱✵✵✳ ◗✉❛❧ ❛ t❛①❛ ♠❡♥s❛❧ ❞❡ ❥✉r♦s❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✿ ❖ ❝❛♣✐t❛❧ éC = 10.000.
❖ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ é n= 4. ❖ ♠♦♥t❛♥t❡ é M = 12.000.
◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ❛ t❛①❛ ❞❡ ❥✉r♦si.
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ✶✳✸✱ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s✿
12.000 = 10.000(1 +i)4 ✱
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧❀
12.000
10.000 = (1 +i)
4
1,2 = (1 +i)4
4
p
1,2 = p4
(1 +i)4
1,0466 = 1 +i
i= 1,0466−1
i= 0,0466×100
i= 4,66%a.m. ◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛✿
❋✐❣✉r❛ ✶✳✾✿ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✸
❆ t❛①❛ ❞❡ ❥✉r♦s é ✹✱✻✻✪a.m.
❊①❡♠♣❧♦ ✶✳✸✳✻ ❉✉r❛♥t❡ q✉❛♥t♦ t❡♠♣♦ ✉♠ ❝❛♣✐t❛❧ ❞❡ ❘✩ ✸✳✺✵✵✱✵✵ ❞❡✈❡ s❡r ❛♣❧✐❝❛❞♦ ❛ ❥✉r♦s ❝♦♠♣♦st♦s à t❛①❛ ❞❡ ✶✷✪ ❛✳❛✳ ♣❛r❛ r❡s✉❧t❛r ❡♠ ✉♠ ♠♦♥t❛♥t❡ ❞❡ ❘✩ ✼✳✵✵✵✱✵✵❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✿
◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ♦ ♣ér✐♦❞♦ ❞❡ t❡♠♣♦ n✳ ❖ ❝❛♣✐t❛❧ éC = 3.500.
❆ t❛①❛ ❞❡ ❥✉r♦s ♣❡r❝❡♥t✉❛❧ i= 12% a.a. ◆❛ ❢♦r♠❛ ❞❡❝✐♠❛❧i= 12
100 = 0,12.
❖ ♠♦♥t❛♥t❡ é M = 7.000,00✳
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ✶✳✸✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s✿
7.000 = 3.500(1 + 0,12)n ✱ r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧❀
7.000
3.500 = (1,12)
n
2 = (1,12)n
log2 =log(1,12)n ✱ ✉t✐❧✐③❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞♦ ❧♦❣❛rít♠♦ ❞❛ ♣♦t❡♥❝✐❛❀
log2 =n×log(1,12)
log2
log1,12 =n ✱
♦❜t❡♠♦s✿
n = 6,17anos. ❉✉r❛♥t❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✻✱✶✼ ❛♥♦s✳
◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛✿
❋✐❣✉r❛ ✶✳✶✵✿ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✹
❖❜s❡r✈❛çã♦ ✻ ◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛ ❍P ✶✷❈✱ ❛ ✈❛❧♦r ❞❡ ♥ é ❛rr❡❞♦♥❞❛❞♦ ♣❛r❛ ♠❛✐s q✉❛♥❞♦ ✉t✐❧✐③❛♠♦s ❛s ❢✉♥çõ❡s ✜♥❛♥❝❡✐r❛s✳ ❘❡❝♦♠❡♥❞❛✲s❡ ✉s❛r ❛ ❢ór♠✉❧❛✱ q✉❛♥❞♦ ♥ã♦ s❡ s❛❜❡ s❡ ♥ é ✐♥❢❡r✐♦r ♦✉ ♥ã♦✳
✶✳✹ ❚❛①❛s ❊q✉✐✈❛❧❡♥t❡s
❉❡✜♥✐çã♦ ✶✳✹✳✶ ❚❛①❛s ❊q✉✐✈❛❧❡♥t❡s sã♦ t❛①❛s q✉❡ q✉❛♥❞♦ ❛♣❧✐❝❛❞❛s ❛♦ ♠❡s♠♦ ❝❛✲ ♣✐t❛❧✱ ♥✉♠ ♠❡s♠♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✱ ♣r♦❞✉③❡♠ ♠♦♥t❛♥t❡s ✐❣✉❛✐s✳
P❛r❛ ❝♦♥✈❡rt❡r ❛ t❛①❛ ❞❡ ❥✉r♦s✿
• ❊♠ ❏✉r♦s ❈♦♠♣♦st♦s✱ ❝♦♠♦ ♥ã♦ é ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r✱ é ✉♠ ♣♦✉❝♦ ♠❛✐s ❝♦♠✲
♣❧❡①♦✳ P❡r❝❡❜❛♠ q✉❡ t❡♠♦s q✉❡ ❛❝❤❛r ✉♠❛ t❛①❛ ❡♠ ♦✉tr❛ ✉♥✐❞❛❞❡✱ ♠❛s ♦ ♠♦♥t❛♥t❡ ♥♦ ♠❡s♠♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ t❡♠ q✉❡ s❡r ✐❣✉❛❧✱ ❧♦❣♦✿
C(1 +i1)n1
=C(1 +i2)n2
,
❛ss✐♠✱
(1 +i1)n1 = (1 +i2)n2,
❞♦♥❞❡ ♦❜t❡♠♦s✱
n1p
(1 +i1)n1 = n1p(1 +i
2)n2,
♦✉ s❡❥❛✱
(1 +i1) = (1 +i2)
n2 n1,
P♦rt❛♥t♦✱
i1 = (1 +i2)
n2 n1 −1.
P❛r❛ ❞❡t❡r♠✐♥❛r ❛ t❛①❛ ❛♥✉❛❧✱ ❝♦♥❤❡❝✐❞❛ ❛ t❛①❛ ♠❡♥s❛❧✳
ia = (1 +im)12−1.
❊①❡♠♣❧♦ ✶✳✹✳✷ ❉❡t❡r♠✐♥❛r ❛ t❛①❛ ❛♥✉❛❧ ❡q✉✐✈❛❧❡♥t❡ ❛ ✷✪ ❛✳♠✳✿
ia= (1 +im)12 ✲ ✶ = (1,02)12 ✲ ✶= 1,2682
−1 = 0,2682 ♦✉ ✷✻✱✽✷✪✳
❊①❡♠♣❧♦ ✶✳✹✳✸ ❉❡t❡r♠✐♥❛r ❛ t❛①❛ ♠❡♥s❛❧ ❡q✉✐✈❛❧❡♥t❡ ❛ ✻✵✱✶✵✸✪ ❛✳❛✳✿
im = (1 +ia)121 ✲ ✶ = (1,60103) 1
12 ✲ ✶ = 1,04✲ ✶ ♦✉ ✹✪ ❛♦ ♠ês✳
❈♦♠♦ ♥♦ ❞✐❛✲❛✲❞✐❛ ♦s ♣❡rí♦❞♦s ❛ q✉❡ s❡ r❡❢❡r❡♠ às t❛①❛s q✉❡ t❡♠♦s ❡ ❛s t❛①❛s q✉❡ q✉❡r❡♠♦s sã♦ ♦s ♠❛✐s ✈❛r✐❛❞♦s✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r ✉♠❛ ❢ór♠✉❧❛ ❣❡♥ér✐❝❛✱ q✉❡ ♣♦ss❛ s❡r ✉t✐❧✐③❛❞❛ ♣❛r❛ q✉❛❧q✉❡r ❝❛s♦✱ ♦✉ s❡❥❛✿
iq= (1 +it)
q
t −1. ✭✶✳✹✮
❖♥❞❡✿ iq =t❛①❛ ♣❛r❛ ♦ ♣r❛③♦ q✉❡ ❡✉ q✉❡r♦❀ it = t❛①❛ ♣❛r❛ ♦ ♣r❛③♦ q✉❡ ❡✉ t❡♥❤♦❀
q =♣r❛③♦ q✉❡ ❡✉ q✉❡r♦ ❡♠ ❞✐❛s❀ t= ♣r❛③♦ q✉❡ ❡✉ t❡♥❤♦ ❡♠ ❞✐❛s✳
❊①❡♠♣❧♦ ✶✳✹✳✹ ◗✉❛❧ ❛ t❛①❛ ♠❡♥s❛❧ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ t❛①❛ ❞❡ ❥✉r♦s ❞❡ ✶✺✪ ❛✳❛✳❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✿ ❆ t❛①❛ q✉❡ ❡✉ t❡♥❤♦it= 15% = 0,15. ❖ ♣r❛③♦ q✉❡ ❡✉ t❡♥❤♦ t= 1 ❛♥♦ =✸✻✵ ❞✐❛s✳
❖ ♣r❛③♦ q✉❡ ❡✉ q✉❡r♦q = 30 ❞✐❛s✳
❆ t❛①❛ q✉❡ ❡✉ q✉❡r♦ iq.
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛✶✳✹✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s✿
iq = (1 + 0,15)3060 −1 ✱
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧❀
iq = (1,15)121 −1
iq = 1,0117−1
iq = 0,0117×100
iq = 1,17%a.m. ◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛✿
❋✐❣✉r❛ ✶✳✶✶✿ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✺
❊①❡♠♣❧♦ ✶✳✹✳✺ ◗✉❛❧ ❛ t❛①❛ ❛♥✉❛❧ ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ t❛①❛ ❞❡ ❥✉r♦s ❞❡ ✹✪ ❛✳♠✳❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✿ ❆ t❛①❛ q✉❡ ❡✉ q✉❡r♦ it = 4% = 0,04. ❖ ♣r❛③♦ q✉❡ ❡✉ t❡♥❤♦ t= 1 ♠ês = 30 ❞✐❛s✳
❖ ♣r❛③♦ q✉❡ ❡✉ q✉❡r♦q = 360 ❞✐❛s✳
❆ t❛①❛ q✉❡ ❡✉ q✉❡r♦ iq.
❯t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ✶✳✹✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s✿
iq = (1 + 0,04)
360 30
−1 ✱
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦ ❡①♣♦♥❡♥❝✐❛❧❀
iq = (1,04)12
−1
iq = 1,601−1
iq = 0,601×100
iq = 60,1%a.m. ◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛✿
❋✐❣✉r❛ ✶✳✶✷✿ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✻
▲♦❣♦ ❛ t❛①❛ ♠❡♥s❛❧ é ✻✵✱✶✪a.a.
❈❛♣ít✉❧♦ ✷
❙ér✐❡s ❞❡ P❛❣❛♠❡♥t♦s
❉❡✜♥✐çã♦ ✷✳✵✳✻ ❙ér✐❡s ❞❡ ♣❛❣❛♠❡♥t♦s✱ s❡❣✉♥❞♦ ❙❖❇❘■◆❍❖ ✭✶✾✾✼✱ ♣✳✻✻✮✱ sã♦ ✈á✲ r✐♦s ♣❛❣❛♠❡♥t♦s ♦✉ r❡❝❡❜✐♠❡♥t♦s✱ ❝♦♥s❡q✉❡♥t❡s✱ ♣❛r❛ ✉♠ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦ ❞❡t❡r♠✐✲ ♥❛❞♦✳
➱ ♠✉✐t♦ ✉t✐❧✐③❛❞♦ ❡♠ ♦♣❡r❛çõ❡s ✜♥❛♥❝❡✐r❛s✱ ❝♦♠♦ ❡♠♣rést✐♠♦s ❡ ✜♥❛♥❝✐❛♠❡♥t♦s ❞❡ ❞✐❢❡r❡♥t❡s t✐♣♦s✳ ❆ sér✐❡ ❞❡ ♣❛❣❛♠❡♥t♦s ♣♦❞❡ s❡r✿
• ❆♥t❡❝✐♣❛❞❛✿ q✉❛♥❞♦ ♦ ♣❛❣❛♠❡♥t♦ ♦✉ ♦ r❡❝❡❜✐♠❡♥t♦ é ❢❡✐t♦ ♥♦ ❛t♦ ❞❛ ❝♦♠♣r❛✱
❝♦♠♦ ❡♥tr❛❞❛❀
• P♦st❡❝✐♣❛❞❛✿ q✉❛♥❞♦ ♦ ♣❛❣❛♠❡♥t♦ ♦✉ ♦ r❡❝❡❜✐♠❡♥t♦ é ❢❡✐t♦ ✉♠ ♣❡rí♦❞♦ ❞❡
t❡♠♣♦s ❛♣ós ❛ ❝♦♠♣r❛✳
✷✳✶ ❱❛❧♦r Pr❡s❡♥t❡ ♦✉ ❋❛t♦r ❞❡ ❱❛❧♦r ❆t✉❛❧
❉❡✜♥✐çã♦ ✷✳✶✳✶ ➱ ♦ s♦♠❛tór✐♦ ❞♦s ♣❛❣❛♠❡♥t♦s ♦✉ r❡❝❡❜✐♠❡♥t♦s ❞❛s ♣❛r❝❡❧❛s ♥✉♠ ♣❡rí♦❞♦ ❞❡ t❡♠♣♦✳ ❈❤❛♠❛♠♦s ❞❡ P▼❚ ♦s ✈ár✐♦s ♣❛❣❛♠❡♥t♦s ♦✉ r❡❝❡❜✐♠❡♥t♦s✳
✷✳✶✳✶ ❙ér✐❡ ❞❡ P❛❣❛♠❡♥t♦s P♦st❡❝✐♣❛❞❛
❱❛♠♦s ✈❡r✐✜❝❛r ♦ ✢✉①♦ ❞❡ ❝❛✐①❛ ❞❡ ✉♠❛ sér✐❡ ❞❡ ♣❛❣❛♠❡♥t♦s ♣♦st❡❝✐♣❛❞❛✳
❋✐❣✉r❛ ✷✳✶✿ ❋❧✉①♦ ❞❡ ❈❛✐①❛ ♣❛r❛ ❙ér✐❡s ❞❡ P❛❣✳ P♦st❡❝✐♣❛❞❛
P V = P M T (1 +i) +
P M T
(1 +i)2 +
P M T
(1 +i)3 +· · ·+
P M T
(1 +i)n. ❖✉ s❡❥❛✱
P V =P M T
1 (1 +i)+
1 (1 +i)2 +
1
(1 +i)3 +· · ·+
1 (1 +i)n
. ✭✷✳✶✮
❊♥tr❡ ❝♦❧❝❤❡t❡s✱ t❡♠♦s ✉♠❛ Pr♦❣r❡ssã♦ ●❡♦♠étr✐❝❛ ✭P✳●✮✱ ❞❡ r❛③ã♦ é 1
(1+i)✱ ♣r✐✲
♠❡✐r♦ t❡r♠♦ ❞❛ P✳● é 1
(1+i)✳ ❯t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s
❞❡ ✉♠❛ P✳● ✜♥✐t❛✱♣❛r❛ ♦s ❞❛❞♦s✿ a1 = (1+1i), q= (1+1i), Sn=P V. ❯s❛♥❞♦ ❛ ❡①♣r❡ssã♦✱
Sn =a1
qn−1
q−1 ✭❋ór♠✉❧❛ ❞❛ s♦♠❛ ❞❛ P● ✜♥✐t❛✳✮ ✭✷✳✷✮
❙✉❜st✐t✉í♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s❀
P V =P M T
1 (1+i)
h
1 (1+i)n −1
i
(1+1i)−1
P V =P M T
1 (1+i)
h
1−(1+i)n
(1+in)
i
1−(1+i)
(1+i)
.
❖✉ s❡❥❛✱
P V = P M T
(1 +i)n
−1
(1 +i)ni
. ✭✷✳✸✮
❖ q✉❡ ❡stá ❡♥tr❡ ❝♦❧❝❤❡t❡s é ❝❤❛♠❛❞♦ ❞❡ ❢❛t♦r ❞❡ ✈❛❧♦r ❛t✉❛❧✳ ◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛✿
❖ ❢❛t♦r ❞❡ ✈❛❧♦r ❛t✉❛❧ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞♦ ❞✐r❡t❛♠❡♥t❡✱ ✉t✐❧✐③❛♥❞♦ ❛s t❡❝❧❛s✿
❋✐❣✉r❛ ✷✳✷✿ ❇♦tõ❡s ❯s❛❞♦s ❡♠ ❙ér✐❡s ❞❡ P❛❣❛♠❡♥t♦s
❱❡❥❛♠♦s ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s✳
❊①❡♠♣❧♦ ✷✳✶✳✷ ❯♠ ❝♦♠♣✉t❛❞♦r é ✈❡♥❞✐❞♦ ❡♠ ✶✵ ♣r❡st❛çõ❡s ♠❡♥s❛✐s ❞❡ ❘✩ ✶✺✵✱✵✵✱ ✈❡♥❝❡♥❞♦ ❛ ♣r✐♠❡✐r❛ ♣r❡st❛çã♦ ✉♠ ♠ês ❛♣ós ❛ ❝♦♠♣r❛✳ ❙❡ ❛ t❛①❛ ❞❡ ❥✉r♦s é ❞❡ ✹✪ ❛✳♠✳✱ q✉❛❧ ♦ ✈❛❧♦r ❞♦ ❝♦♠♣✉t❛❞♦r à ✈✐st❛❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✿
n= 10 ♣r❡st❛çõ❡s ♠❡♥s❛✐s✳
❖ ✈❛❧♦r ❞❛s ♣r❡st❛çõ❡s éP M T = 150,00. ❆ t❛①❛ ❞❡ ❥✉r♦s é i= 4% = 0,04. a.m.
◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ♦ ✈❛❧♦r ❞♦ ❝♦♠♣✉t❛❞♦rP V.
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ✷✳✸✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s❀
P V = 150
(1 + 0,04)10−1
(1 + 0,04)100,04
,
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦✿
P V = 150
(1,04)10
−1
(1,04)100,04
P V = 150
0,48024 0,0592
P V = 150×8,11086 ✱
♦❜t❡♠♦s✿
P V = 1.216,63. ◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛✿
❋✐❣✉r❛ ✷✳✸✿ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✼
❖ ❝♦♠♣✉t❛❞♦r ❝✉st❛ à ✈✐st❛ ❘✩ ✶✳✷✶✻✱✻✸✳
❊①❡♠♣❧♦ ✷✳✶✳✸ ❯♠ ❛✉t♦♠ó✈❡❧ é ✈❡♥❞✐❞♦ à ✈✐st❛ ♣♦r ❘✩ ✸✺✳✵✵✵✱✵✵✱ ♠❛✐s ♣♦❞❡ s❡r ✈❡♥❞✐❞♦ ❡♠ ✹✽ ♣r❡st❛çõ❡s ✐❣✉❛✐s ❡ ❝♦♥s❡❝✉t✐✈❛s✱ ✈❡♥❝❡♥❞♦ ❛ ♣r✐♠❡✐r❛ ♣r❡st❛çã♦ ✉♠ ♠ês ❛♣ós ❛ ❝♦♠♣r❛✳ ❙❛❜❡♥❞♦ q✉❡ ❛ t❛①❛ ❞❡ ❥✉r♦s ❞♦ ✜♥❛♥❝✐❛♠❡♥t♦ é ❞❡ ✸✪ ❛✳♠✳✱ q✉❛❧ ♦ ✈❛❧♦r ❞❡ ❝❛❞❛ ♣r❡st❛çã♦❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✳
❖ ✈❛❧♦r ❞♦ ❛✉t♦♠ó✈❡❧ à ✈✐st❛ éP V = 35.000,00. n= 48 ♣r❡st❛çõ❡s ♠❡♥s❛✐s✳
❆ t❛①❛ ❞❡ ❥✉r♦s é i= 3% a.m.= 0,03. ◗✉❛❧ ♦ ✈❛❧♦r ❞❛s ♣r❡st❛çõ❡sP M T.
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ✷✳✸✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s ♦❜t❡♠♦s❀
35.000 = P M T
(1 + 0,03)48
−1
(1 + 0,03)480,03
,
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦✿
35.000 =P M T
1,0348
−1
1,03480,03
35.000 =P M T
3,13225 0,123968
35.000 =P M T ×25,2667 ✱
❧♦❣♦✱
P M T = 35.000 25,2667,
♦❜t❡♠♦s✿
P M T = 1.385,22. ◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛✿
❋✐❣✉r❛ ✷✳✹✿ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✽
❆s ♣r❡st❛çõ❡s sã♦ ❞❡ ❘✩ ✶✳✸✽✺✱✷✷✳
✷✳✶✳✷ ❙ér✐❡ ❞❡ P❛❣❛♠❡♥t♦s ❆♥t❡❝✐♣❛❞❛
❱❛♠♦s ✈❡r✐✜❝❛r ♦ ✢✉①♦ ❞❡ ❝❛✐①❛ ❞❡ ✉♠❛ sér✐❡ ❞❡ ♣❛❣❛♠❡♥t♦s ❛♥t❡❝✐♣❛❞❛✱ q✉❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ♣r❡st❛çã♦ é ❞❛❞❛ ❝♦♠♦ ❡♥tr❛❞❛✱ ♥♦ ❛t♦ ❞❛ ❝♦♠♣r❛✳
❋✐❣✉r❛ ✷✳✺✿ ❋❧✉①♦ ❞❡ ❈❛✐①❛ ♣❛r❛ ❙ér✐❡s ❞❡ P❛❣✳ ❆♥t❡❝✐♣❛❞❛
P V =❱❛❧♦r Pr❡s❡♥t❡❀
P M T =❙ã♦ ❛s ♣r❡st❛çõ❡s ✐❣✉❛✐s ❡ ❝♦♥s❡❝✉t✐✈❛s❀
P V =P M T + P M T (1 +i) +
P M T
(1 +i)2 +. . .+
P M T
(1 +i)n, ♦✉ s❡❥❛✱
P V =P M T
1 + 1 (1 +i) +
1
(1 +i)2 +. . .+
1 (1 +i)n
. ✭✷✳✹✮
❊♥tr❡ ❝♦❧❝❤❡t❡s✱ t❡♠♦s ✉♠❛ ♣r♦❣r❡ssã♦ ❣❡♦♠étr✐❝❛ ✭P✳●✮✱ ❞❡ r❛③ã♦ 1
(1+i)✱ ♣r✐♠❡✐r♦
t❡r♠♦ ❞❛ P✳● 1
(1+i)✱ ✉t✐❧✐③❛♥❞♦ ❛ ❢ór♠✉❧❛ ❞❛ s♦♠❛ ❞♦s n ♣r✐♠❡✐r♦s t❡r♠♦s ❞❡ ✉♠❛
P✳● ✜♥✐t❛ ✷✳✷✱ ❝♦♠ ♦s ❞❛❞♦s a1 = (1+1i), q= (1+1i), Sn=P V, ♦❜t❡♠♦s✿ P V =P M T
" 1
(1+i)n −1
1 (1+i) −1
#
P V =P M T
h
1−(1+i)n
(1+i)n
i
1−1−i
(1+i)
P V =P M T
( 1
(1 +i)n −1)
(1 +i)
−i
P V =P M T
1−(1 +i)n
(1 +i)n
(1 +i)
−i
,
❛ss✐♠✱
P V = P M T
(1 +i)n
−1
(1 +i)ni
(1 +i). ✭✷✳✺✮
❈♦♠♦ ❛ sér✐❡ ❞❡ ♣❛❣❛♠❡♥t♦s é ❛♥t❡❝✐♣❛❞❛ t❡♠♦s q✉❡ ❛t✐✈❛r ❛ ❢✉♥çã♦ ❜❡❣✐♥ ♥❛ ❝❛❧❝✉❧❛❞♦r❛✱ ♣❛r❛ ✐♥❞✐❝❛r q✉❡ ♦ ♣r✐♠❡✐r♦ ♣❛❣❛♠❡♥t♦ ♦✉ r❡❝❡❜✐♠❡♥t♦ é ❢❡✐t♦ ♥♦ ❛t♦ ❞❛ ❝♦♠♣r❛✱ ♦✉ ❝♦♠♦ ❡♥tr❛❞❛✳
◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛✿
❋✐❣✉r❛ ✷✳✻✿ ❆t✐✈❛r ❋✉♥çã♦ ❇❡❣✐♥
❊①❡♠♣❧♦ ✷✳✶✳✹ ❯♠❛ t❡❧❡✈✐sã♦ é ✈❡♥❞✐❞❛ ❡♠ ✶✵ ♣r❡st❛çõ❡s ♠❡♥s❛✐s ❞❡ ❘✩ ✶✺✵✱✵✵✱ ✈❡♥❝❡♥❞♦ ❛ ♣r✐♠❡✐r❛ ♣r❡st❛çã♦ ♥♦ ❛t♦ ❞❛ ❝♦♠♣r❛✳ ❙❡ ❛ t❛①❛ ❞❡ ❥✉r♦s é ❞❡ ✹✪ ❛✳♠✳✱ q✉❛❧ ♦ ✈❛❧♦r ❞❛ t❡❧❡✈✐sã♦ à ✈✐st❛❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✳ n= 10 ♣r❡st❛çõ❡s ♠❡♥s❛✐s✳
❖ ✈❛❧♦r ❞❛s ♣r❡st❛çõ❡s P▼❚= 150,00. ❆ t❛①❛ ❞❡ ❥✉r♦si= 4% a.m.= 0,04. ◗r❡r❡♠♦s ❡♥❝♦♥tr❛r ♦ ✈❛❧♦r ♣r❡s❡♥t❡P V.
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ✷✳✺✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s❀
P V = 150
(1 + 0,04)10
−1
(1 + 0,04)100,04
(1 + 0,04),
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦✿
P V = 150
(1,04)10
−1
(1,04)100,04
(1,04)
P V = 150
0,48024 0,0592
(1,04)
P V = 150(8,11086)(1,04), ♦❜t❡♠♦s✿
P V = 1.265,30.
◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛✿
❋✐❣✉r❛ ✷✳✼✿ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✾
❆ t❡❧❡✈✐sã♦ ❝✉st❛ à ✈✐st❛ ❘✩ ✶✳✷✻✺✱✸✵✳
❊①❡♠♣❧♦ ✷✳✶✳✺ ❯♠ ❛✉t♦♠ó✈❡❧ é ✈❡♥❞✐❞♦ à ✈✐st❛ ♣♦r ❘✩ ✸✺✳✵✵✵✱✵✵✱ ♠❛✐s ♣♦❞❡ s❡r ✈❡♥❞✐❞♦ ❡♠ ✹✽ ♣r❡st❛çõ❡s ✐❣✉❛✐s ❡ ❝♦♥s❡❝✉t✐✈❛s✱ s❡♥❞♦ ❛ ♣r✐♠❡✐r❛ ♣r❡st❛çã♦ ❝♦♠♦ ❡♥tr❛❞❛✳ ❙❛❜❡♥❞♦ q✉❡ ❛ t❛①❛ ❞❡ ❥✉r♦s ❞♦ ✜♥❛♥❝✐❛♠❡♥t♦ é ❞❡ ✸✪ ❛✳♠✳✱ q✉❛❧ ♦ ✈❛❧♦r ❞❡ ❝❛❞❛ ♣r❡st❛çã♦❄
■❞❡♥t✐✜❝❛♥❞♦ ♦s ❞❛❞♦s ❞♦ ♣r♦❜❧❡♠❛✳ ❖ ✈❛❧♦r ♣r❡s❡♥t❡ éP V = 35.000,00. n= 48 ♣r❡st❛çõ❡s ♠❡♥s❛✐s✳
❆ t❛①❛ ❞❡ ❥✉r♦s é i= 3%a.m= 0,03
◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r ♦ ✈❛❧♦r ❞❛s ♣r❡st❛çõ❡sP M T.
❯t✐❧✐③❛♥❞♦ ❛ ❋ór♠✉❧❛ ✷✳✺✱ ❡ s✉❜st✐t✉✐♥❞♦ ♦s ❞❛❞♦s✱ ♦❜t❡♠♦s✿
35.000 = P M T
(1 + 0,03)48
−1
(1 + 0,03)480,03
(1 + 0,03),
r❡s♦❧✈❡♥❞♦ ❛ ❡q✉❛çã♦✿
35.000 =P M T
(1,03)48
−1
(1,03)480,03
(0,03)
35.000 =P M T
3,13225 0,123968
(1,03) 35.000 =P M T(25,2667)(1,03)
35.000 = 26,0247×P M T.
❧♦❣♦✱
P M T = 35.000 26,0247
P M T = 1.344,88. ◆❛ ❝❛❧❝✉❧❛❞♦r❛ ✜♥❛♥❝❡✐r❛✿
❋✐❣✉r❛ ✷✳✽✿ ◆❛ ❈❛❧❝✉❧❛❞♦r❛ ❋✐♥❛♥❝❡✐r❛ ✶✵
❆s ♣r❡st❛çõ❡s sã♦ ❞❡ ❘✩ ✶✳✸✹✹✱✽✽✳
✷✳✷ ❙ér✐❡s ❞❡ P❛❣❛♠❡♥t♦s ❉✐❢❡r✐❞❛s
❉❡✜♥✐çã♦ ✷✳✷✳✶ ➱ ✉♠❛ sér✐❡ ❞❡ ♣❛❣❛♠❡♥t♦✱ q✉❡ ♦ ♣r✐♠❡✐r♦ ♣❛❣❛♠❡♥t♦ só ❛❝♦♥✲ t❡❝❡ ❞❡♣♦✐s ❞❡ ✉♠ ♣❡rí♦❞♦ ♠ ❞❡ t❡♠♣♦✱ ❛ q✉❡ s❡ r❡❢❡r❡ à t❛①❛ ❞❡ ❥✉r♦s ❝♦♠♣♦st♦s ❝♦♥s✐❞❡r❛❞❛✱ ❝♦♠ ♠>2.
