Samenvatting - Insurance: financial and actuarial methods (2113TEWFIN)

FORMULARIUM


CHAPTER 1: INTEREST AND DISCOUNT

1. Simple interest

• Amount of interest over a capital K outstanding during a period t (in years)
𝐼 =𝐾∗𝑖∗𝑡

• When term is expressed in days
𝑛
𝐼 =𝐾∗𝑖∗
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1.1 Effective interest rate of a loan

𝑖
• 𝑖∗ =
1−𝑖∗𝑡


• If period is expressed in days:
365𝑖
𝑖∗ =
365 − 𝑖 ∗ 𝑛



1.2 Future value of a capital

• 𝐾𝑡 = 𝐾0 ∗ (1 + 𝑖𝑡)



1.3 Discount

• Discount : 𝐸 = 𝐾 − 𝐾′

• Commercial discount: 𝐸ℎ = 𝐾 ∗ 𝑑 ∗ 𝑡

• Present value: 𝐾 ′ = 𝐾 − 𝐸ℎ = 𝐾(1 − 𝑑 ∗ 𝑡)




1.4 Effective simple interest rate of a commercial discount
𝑑
• 𝑖∗= 1−𝑑∗𝑡




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, 2. Compound interest

2.1 Equitations for compound interest

• Future value : 𝐾𝑛 = 𝐾0 ∗ (1 + 𝑖) 𝑛 = 𝐾0 ∗ 𝑢𝑛

• Total value of interest generated during n years: 𝐼 = 𝐾0 ∗ (𝑢𝑛 − 1)

• Present value : 𝐾0 = 𝐾𝑛 ∗ 𝑣 𝑛



2.2 The effective discount
• 𝐾0 = 𝐾 ∗ (1 − 𝑑)𝑛

• 𝐸 = 𝐾 − 𝐾0 = 𝐾 ∗ (1 − (1 − 𝑑)𝑛 )



2.3 Effective interest rate of a discount at effective discount rate d
𝑑
• 𝑖∗ = 1−𝑑




2.4 Nominal versus effective annual interest rate

• Future value of a capital K whose interest is capitalized at the end of every 1/k years:
𝑖(𝑘) 𝑘𝑛
𝐾𝑛 = 𝐾 ∗ (1 + )
𝑘


𝑖(𝑘) 𝑘
• Effective annual interest rate: 𝑖 = [(1 + 𝑘
) − 1]


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• Nominal annual interest rate: 𝑖(𝑘) = 𝑘 ∗ [(1 + 𝑖)𝑘 − 1]



2.5 calculation of continuous interest and continuous discounting
• The force of interest : 𝛿 = 𝑖(∞) = ln(1 + 𝑖) = ln (𝑢)

• Future value: 𝐾𝑡 = 𝐾0 ∗ 𝑒 𝛿∗𝑡

• Present value : 𝐾0 = 𝐾𝑡 ∗ 𝑒 −𝛿∗𝑡

𝑖 𝑖
• Proportionality factor: 𝑔(𝑘) = 𝑔(∞) =
𝑖(𝑘) 𝛿




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