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∗
360
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
1
, 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]
1
• 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
2
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∗
360
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
1
, 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]
1
• 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
2
