Time Value of Money Loan

Loan A debt evidenced by a note between the borrower and lender. It specifies the original loan amount (principle) and the amount and date of repayments. The borrower initially receives an amount of money from the lender The borrower is obligated to pay back the principle and interests to the lender later

Loan Types How does the borrower repay the lender? Pure Discount Loan: A single lump sum paid at some time in the future Interest-only Loan: Interest paid each period and the entire principle repaid at some point in the future Amortized Loan: Part of the loan amount repaid over time Pure discount loans are very common when the loan term is short: Treasury Bill T-bill Most corporate bonds have the general form of an interest-only loan, and we leave this to bond chapter. The process of paying off a load by making regular principle reduction is called amortizing the loan

Pure Discount Loan If a Treasury Bill (T-bill) promises to repay $10,000 in 12 months and the market interest rate is 7%, how much will the bill sell for in the market? 𝑃𝑉= $10,000 1+7% =$9,345.79

Questions and Problems 57 This question illustrates what is known as discount interest. Imagine you are discussing a loan with a somewhat unscrupulous lender. You want to borrow $15,000 for one year. The interest rate is 14 percent. You and the lender agree that the interest on the loan will be .14 × $15,000 = $2,100. So, the lender deducts this interest amount from the loan up front and gives you $12,900. In this case, we say that the discount is $2,100. What is the interest rate on the loan? $15,000 = $12,900(1 + r) r = ($15,000 / $12,900) – 1 r = .1628

Amortized Loan: Equal Payments Consider a 4-year loan with equally annual payments. The interest rate is 8% and the principal amount is $5,000. What is the annual payment? 𝑝𝑚𝑡= 𝑃𝑉∙𝑟 1− 1 1+𝑟 𝑇 = 5,000×8% 1− 1 1+8% 4 = 400 1−0.735029 =1509.60 Each payment covers the interest expense plus reduces principal

Amortized Loan: Equal Principle Payments Consider a 4-year loan with equally annual principle payments. The interest rate is 8% and the principal amount is $5,000. What is the annual payment in each year? Principle payment: 5000 4 =1250 Each payment covers the interest expense plus reduces principal

"Balloon" or "Bullet" Loans Balloon loans are loans that are amortized over a relatively long schedule, but at some point during the life of the loan, the remaining principal of the loan is repaid Example in Excel

Annual Percentage Rate (APR) The annual rate quoted by law APR = Periodic compounding rate * # of compounding periods per year Periodic compounding rate is the effective rate compounded over the matched time period Suppose you can earn 1% per month on $1 invested today, what is the APR? 1%×12 =12%

Effective Annual Rate (EAR) The interest rate used for annual compounding Used to compare investments with different compounding periods

EAR and APR 𝐸𝐴𝑅= (1+ 𝐴𝑃𝑅 𝑚 ) 𝑚 −1 𝐴𝑃𝑅=𝑚 (1+𝐸𝐴𝑅) 1 𝑚 −1 𝐸𝐴𝑅= (1+ 𝐴𝑃𝑅 𝑚 ) 𝑚 −1 𝐴𝑃𝑅=𝑚 (1+𝐸𝐴𝑅) 1 𝑚 −1 𝑚= # of compounding periods per year Because you earn more interest on interest.

Example of EAR Suppose the bank quoted interest rate is 12% per year. How much are you effectively earning per year if you deposit $1 in the bank? In other words, what is the EAR? 𝐸𝐴𝑅= 1+ 12% 12 12 −1 =12.68%

Example of APR Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a quarterly basis. What APR must they pay? 𝐴𝑃𝑅=4× 1+12% 1 4 −1 =11.49%