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Interest Theory - Quiz 2 (Chapt. 4)

  1. The force of interest is the instantaneous rate of change in the accumulated value per unit of accumulated value. We denote it by 'r'. 

    More precisely, r=
    = d/dt*AVₜ / AVₜ
  2. AVₜ= PV₀(1+iᵐ/m)ᵐᵗ

    d/dt*AVᵗ=

    r=
    d/dt*AVᵗ= [PV₀(1+iᵐ/m)ᵐᵗ*ln(1+iᵐ/m)*d/dt(mt)] / AVₜ

    • r= [PV₀(1+iᵐ/m)ᵐᵗ *ln(1+iᵐ/m)*m] / PV₀(1+iᵐ/m)ᵐᵗ
    • r= mln(1+iᵐ/m)
    • r= ln(1+iᵐ/m)

    • Alternate Form
    • eʳ= (1+iᵐ/m)ᵐ
  3. Force of int. is 8%. We deposit $20 now. What's the AV after 5 years?
    r=8%, t=5, PVo=20

    • AV₅= PV₀ʳᵗ
    • = 20⁰.⁰⁸*⁵
    • = 29.8365
  4. Force of int. is 8%. Calculate the equiv:
    1) Annual eff. int rate
    2) Nominal annual int rate convertible quarterly 
    3) Nominal annual discount rate convertible monthly 
    4) Annual eff. discount rate
    • 1) eʳ= 1+i
    • e⁰.⁰⁸= 1+i
    • i= 0.0833

    • 2) Want i⁴
    • eʳ = (1+iᵐ/m)ᵐ
    • e⁰.⁰⁸ = (1+i⁴/4)⁴
    • e⁰.⁰² = 1+i⁴/4
    • 4[e⁰.⁰²-1] = i⁴
    • i⁴= 0.0808

    • 3) Want d¹²
    • e ʳ= (1-(dᵖ/p))⁻ᵖ
    • e⁰.⁰⁸ = (1-(d¹²/12))⁻¹²
    • e⁻⁰.⁰⁸/¹² = (1-(d¹²/12))
    • 12[1 - e⁻⁰.⁰⁸/¹²]= d¹²
    • d¹²= 0.07973

    • 4) Want d
    • eʳ = (1-d)⁻¹
    • e⁰.⁰⁸ = (1-d)⁻¹
    • e⁻⁰.⁰⁸ = 1-d
    • d= 0.07688
  5. $100 is lent for 5 years. It is repaid in a single payment at the end of 5 years. Calculate amount of payment if int. rate is:
    1) Annual eff. int rate of 7%
    2) i²= 7%, compounded semiannually
    3) convertible monthly & int¹²= 7%
    4) the force of int is 7%
    • 1) AV₅=100(1+0.07)⁵
    • = 140.26

    • 2) AV₅=100(1+(0.07/2))²*⁵
    • = 141.06

    • 3) AV₅= 100(1+(0.07/12)¹²*⁵
    • = 141.76

    • 4) AV₅= 100e⁰.⁰⁷*⁵
    • = 141.91
  6. Nominal annual int. rate is 10%. Calculate the equiv. force of int. if we assume a compounding frequency of:
    1) Once a yr
    2) Quarterly 
    3) Monthly
    4) 24 times per year
    5) 100 times per year
    • 1) AVₜ=PV₀(eʳᵗ) --> eʳ=1+i --> r=ln(1+i)
    • r= ln(1+0.10)
    • = 0.0953

    • 2) i⁴= 10%
    • eʳ= (1+iᵐ/m)ᵐ
    • r= ln[(1+iᵐ/m)ᵐ]
    • = 4ln(1+(0.10/4))
    • = 4ln(1.025)
    • = 0.0987

    • 3) r= 12ln(1+(0.10/12))
    • = 0.09959

    • 4) r= 24ln(1+(0.10/24))
    • = 0.09979

    • 5) r= 100ln(1+(0.10/100))
    • = 0.09995
Author
GoBroncos
ID
365743
Card Set
Interest Theory - Quiz 2 (Chapt. 4)
Description
Updated

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