Loans

Outlines Loan balance: prospective method and retrospective method Amortization schedule Sinking fund

Outstanding Balance Loan are usually paid through annuities and payments are also called installments and they can be level or varying payments. Total loan amount : 𝐿 = 𝑃∙ 𝑎 𝑛 |

Example A loan of $5,000 to be repaid over 6 years with a 6-payment annuity-immediate at effective rate of interest of 6% per year. Calculate the installment amount .

Example A loan of $5,000 to be repaid over 6 years with a 6-payment annuity-immediate at effective rate of interest of 6% per year. Calculate the installment amount . 5000 = 𝑃∙ 1− 1.06 −6 0.06 𝑃=1016.813142

Amortization schedule Installment + Interest Principal

Amortization schedule Interest First ! Whatever left reduces the loan

Example Construct an amortization schedule of a loan of $5,000 to be repaid over 6 years with a 6-payment annuity-immediate at effective rate of interest of 6% per year. We have calculated the installment to be $𝟏𝟎𝟏𝟔.𝟖𝟏

Example- Continued Installment Interest Principal Outstanding balance 5,000 𝟏𝟎𝟏𝟔.𝟖𝟏

Example- Continued Installment Interest Principal Outstanding balance 5,000 𝟏𝟎𝟏𝟔.𝟖𝟏 300 716.81 4,283.19 256.99 759.82 3,523.37 211.40 805.41 2,717.96 163.08 853.73 1,864.23 111.85 904.96 959.26 57.55

Exercises 𝐺𝑖𝑣𝑒𝑛 : 𝑖 = 0.06 𝑂 𝐵 2 = 3,523.37 𝐼𝑛𝑠𝑡𝑎𝑙𝑙𝑚𝑒𝑛𝑡=1016.81 𝑂 𝐵 4 =? 𝐼 4 =? 𝑃 4 =?

Exercises 𝐺𝑖𝑣𝑒𝑛 : 𝑖 = 0.06 𝑂 𝐵 2 = 3,523.37 𝐼𝑛𝑠𝑡𝑎𝑙𝑙𝑚𝑒𝑛𝑡=1016.81 𝑂 𝐵 4 =1864.209332 𝐼 4 =𝑂 𝐵 4−1 ∙𝑖= 2717.9522 ∙ 0.06 =163.077132 𝑃 4 =1016.81−163.077132=853.732868

Important Formulas 𝑃 𝑡 =𝑅 𝑣 𝑛−𝑡+1 𝑃 𝑡+𝑘 = 𝑃 𝑡 1+𝑖 𝑘

Important Formulas - Example 𝑃 𝑡 =𝑅 𝑣 𝑛−𝑡+1 𝑃 𝑡+𝑘 = 𝑃 𝑡 1+𝑖 𝑘 𝐹𝑜𝑟 𝑎 𝑙𝑜𝑎𝑛 𝑤𝑖𝑡ℎ 𝑙𝑒𝑣𝑒𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 1018.52 𝑡ℎ𝑒 𝑝𝑟𝑖𝑐𝑖𝑝𝑎𝑙 𝑟𝑒𝑝𝑎𝑖𝑑 𝑖𝑛 𝑡ℎ𝑒 5𝑡ℎ 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 297.3 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑝𝑎𝑖𝑑 𝑖𝑛 𝑡ℎ𝑒 16𝑡ℎ 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑖𝑠 325.33 . 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒.

Important Formulas - Example 𝑃 𝑡 =𝑅 𝑣 𝑛−𝑡+1 𝑃 𝑡+𝑘 = 𝑃 𝑡 1+𝑖 𝑘 𝐹𝑜𝑟 𝑎 𝑙𝑜𝑎𝑛 𝑤𝑖𝑡ℎ 𝑙𝑒𝑣𝑒𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 1018.52 𝑡ℎ𝑒 𝑝𝑟𝑖𝑐𝑖𝑝𝑎𝑙 𝑟𝑒𝑝𝑎𝑖𝑑 𝑖𝑛 𝑡ℎ𝑒 5𝑡ℎ 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 297.3 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑝𝑎𝑖𝑑 𝑖𝑛 𝑡ℎ𝑒 16𝑡ℎ 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑖𝑠 325.33 . 𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒. 𝑃 16 = 𝑃 16 297.3 1+𝑖 11 =1018.52−325.33 𝑖=0.08

Annual Interest Payments Only + Repay loan As one lump sum payment Sinking Fund Annual Interest Payments Only + Repay loan As one lump sum payment Risky Risky Risky Risky

Sinking Fund Lender may ask the borrower to make deposits to a fund to accumulate the loan amount. (Sinking Fund)

Sinking Fund If you borrow 100,000 at 4% interest rate for 30 years .You will pay: 4,000 as interest every year + Deposit R into a sinking fund to accumulate the loan 100,000

Sinking Fund If you borrow 100,000 at 4% interest rate for 30 years .You will pay: 4,000 as interest every year + 𝑹 ∙𝑠 30| =100,000 𝑹=1783.009913

Sinking Fund

Sinking Fund 𝑖=0.065