Chapter 4 Interest Rates Notes

 Chapter 4  Interest Rates Notes

Interest, the opportunity cost of money. Interest rate is the price of borrowing money.

If rates are low, businesses will be more likely to borrow money, expand production, and hire you. 

If rates are high, businesses will be less likely to expand or to hire you.

Interest = the payment it takes to induce a lender to part with their money for some period of time.

To measure interest rates, yield to maturity, the method preferred by economists for its accuracy. 

To compare the value of money today, called present value(PV and aka present discounted value or price), to the value of money tomorrow, called future value (represented here by the variable FV).

4.2 Present and Future Value

Money today is always worth more than money tomorrow. It means to forgo so many opportunities and the quality of life for a period of time. 

Paying interest on the interest every year, called annually compounding interest.

FV = PV(1 + i)n

FV = the future value (the value of your investment in the future) 

PV = the present value (the amount of your investment today)

(1 + i)n = the future value factor (aka the present value factor or discount factor in the equation below) 

i = interest rate 

n = number of terms (here, years; elsewhere days, months, quarters)

A higher interest rate means a higher opportunity cost for money. 

Question: 

$100 million payable in $5 million installments over 20 years! Did you really win $100 million? (Hint: Calculate the PV of the final payment with interest at 4 percent.)

FV = PV(1 + i)^n

PV = FV / (1+i)^n

The money payable next year and in subsequent years is not worth $5 million today. For example, the last payment, with interest rates at 4 percent compounded annually, has a PV of only 5,000,000/(1.04)^20 = $2,281,934.73. 

4.3 Compounding Periods

Interest does not always compound annually. Sometimes it compounds quarterly, monthly, daily, even continuously. The more frequent the compounding period, the more valuable the bond or other instrument, all else constant. 

$1,000 invested at 12 % for a year compounded annually = $1,000 × (1.12)^1 = $1,120.00. 

Compounded monthly

12%/12 months = 1% per month

$1,000 × (1.01)^12 = $1,126.83 

*Notice

i as the interest paid per period (.12 interest/12 months in a year = .01) and n as the number of periods (12 in a year; 12 × 1 = 12), rather than the number of years. 

Compounded quarterly

$1,000 × (1.03)^4 = $1,125.51. 

Compounded annually

$1,000,000 × (1.04) = $1,040,000

The same terms compounded quarterly

$1,000,000 × (1.01)4 = $1,040,604.01. 

4.4 Pricing Debt Instruments

4 major types of instruments

Discount coupon bonds, Simple loans, Fixed-payment loans, and Coupon bonds. 

A discount bond (zero-coupon bond) makes only one payment, its face value on its maturity or redemption date, so its price is easily calculated using the present value formula.

A simple loan is a loan where the borrower repays the principal and interest at the end of the loan. Use the future value formula to calculate the sum due upon maturity. 

A fixed-payment loan (fully amortized loan) is one in which the borrower periodically (weekly, bimonthly, monthly, quarterly, or annually) repays a portion of the principal along with the interest. Like auto loans and home mortgages, all payments are equal. There is no big balloon or principal payment at the end because the principal shrinks, slowly at first but more rapidly as the final payment grows nearer.

Web sites tools for calculation

http://ray.met.fsu.edu/cgibin/amortize

http://www.yona.com/loan/

http://realestate.yahoo.com/calculators/amortization.html. 

If you wanted to buy this mortgage (to purchase the right to receive the monthly repayments of $2,997.75) from the original lender, you’d simply sum the present value of each of the remaining monthly payments. 

A coupon bond makes one or more interest payments periodically (for example, monthly, quarterly, semiannually, annually, etc.) until its maturity or redemption date, when the final interest payment and all of the principal are paid. 

The sum of the present values of each future payment = the price. 

Exp. A $10,000 face or par value coupon bond that pays 5% interest annually until its face value is redeemed (its principal is repaid) in exactly five years:

PV1 = $500/(1.06) = $471.70 (This is the interest payment after the first year.)

The $500 is the coupon or interest payment = $10,000 x “coupon rate" = $10,000 × .05 = $500

PV2 = $500/(1.06)^2 = $445.00 

PV3 = $500/(1.06)^3 = $419.81

PV4 = $500/(1.06)^4 = $396.05 

PV5 = ($10,000+$500)/(1.06)^5 = $10,500/(1.06)^5 = $7,846.21

4.5 The Yield

Suppose you have $1,000 to invest for a year and two ways of investing it: a discount bond due in one year with a face value of $1,000 for $912 or a bank account at 6.35 percent compounded annually. Which should you take?

The bond

The yield = 9.65 % = (1000 − 912)/912 = .0965