Lecture 7: Measuring interest rate Mishkin chapter 4 – part A Page 67-78

Future value Deposit $1 in bank, annual interest rate i=0.1, how much you would get after 1 year? 2 years? … n years?

Present value – discounting $1 1.33 1 2 3 n 1.21 1.1 1*(1 + .1)n $1 received after n years has a present value of:

Present value - meaning A dollar paid to you one year from now is less valuable than a dollar paid to you today. Present value of $1 is the minimum number of dollars that you would have to give up today in return for receiving $1 in year n. Why? impatient forgone interest

Present value – discounting a future cash flow

Present value – discounting multiple future cash flows $100 $150 $200 Start date Maturity date

Example Your bank offers you a CD with 3% interest rate for five year investment. You wish to invest $1500 for five years. How much your investment will be worth then? Known current value and need to calculate future value.

Example You have been offered $40,000 to sell your printing business, payable in two years. Suppose the market interest rate is 8%. How much is the offer worth to you today? (In other words, what is the offer’s present value?)

Compare debt instruments Simple loan Fixed payment loan Coupon bond Discount bond Difference: repayment schemes. Calculate yield to maturity for each.

Yield to maturity (YTM) Yield to maturity (YTM) is the interest rate (i) that equates the present value of cash flow payments received from a debt instrument with its value today. the most accurate measure of interest rates.

Simple loan Payment scheme: pay the loan value (LV) together with an interest payment (I) on the maturity date. Time line: Loan value (LV) + Interest payment (I) borrower receives Lender receives maturity date

Simple loan - YTM LV = loan value I = Interest payment n = years to mature Example: borrow a simple loan of $100, interest rate is 0.1, or, need to pay interest of $10, mature in one year. What’s the yield to maturity?

Example – Cont’d

Fixed payment loan Payment scheme: example: home mortgage makes periodic fixed payments to the lender until a specified maturity date. These periodic fixed payments include both principal (loan value) and interest, so at maturity there is no lump-sum repayment of principal. example: home mortgage Loan value (LV) Fixed payments (FP) FP borrower receives Lender receives maturity date

Fixed payment loan - YTM

Example Consider a particular fixed-payment loan contract with a loan value $5000, annual fixed payments $660.72, and a maturity of 20 years. What is the yield to maturity for this loan contract? Answer: Yield to maturity for this fixed-payment loan contract is 0.12 or 12 percent.

Coupon bond Payment scheme: purchase price may not equal face value (1)pay a fixed amount of funds (the coupon payment) periodically; (2) pay the face value (or par value) of the bond on maturity date. purchase price may not equal face value Purchase price (P) Coupon payments (C) C + Face value (F) borrower receives Lender receives

Coupon bond - YTM

Example Consider a coupon bond whose purchase price is $94, face value is $100, whose coupon payment is $10 and maturity is 10 years. What’s yield to maturity i? Cash flow is: ( $10, $10, $10, $10, $10, $10, $10, $10, $10, [$10 + $100] ).

Bond price and YTM 1. When bond is at par (sold at face value), yield to maturity equals coupon rate. 2. Price and yield to maturity are negatively related.

Coupon rate and YTM Other things unchanged: P increase  i decrease. C increase  i increase. Coupon rate or coupon payment is positively related to yield to maturity.

Discount bond Payment scheme: on maturity date, pay the face value (F) Sold at a discount: price < face value Time line: Purchase price (P) Face Value (F) borrower receives Lender receives

Discount bond - YTM

Example A discount bond selling for $15,000 with a face value of $20,000 in one year has a yield to maturity of _____?

Recap Present value Yield to maturity Calculate yield to maturity for: definition Calculate yield to maturity for: Simple Loan Fixed Payment Loan Coupon Bond Discount Bond