Money and Banking: Chapter 4 (The meaning of Interest Rates)
Future Value of money equation
$100 * (1 + i)ⁿ i = interest rate n = number of periods (in this case years)
Question What is the present value of $250 to be paid in two years if the interest rate is 15%.
$189.04
$1000 face value bond with a coupon rate of 10% Held for one year, and then sold for $1,200 Yearly coupon payments of $100 Change in the bonds value = 200 (1200-1000) What's the coupon rate?
($100 + $200) / $1000 = 30%
Rate of capital gain
the change in the bond's price relative to the initial purchase price Pt = price of the bond at time t [P(t+1) - Pt / Pt]
Coupon Rate
The dollar amount of the yearly coupon payment expressed as a percentage of the face value of the bond Coupon payment/ Face Value Coupon payment = 100 Face value = 1000 Coupon Rate = 100/1000 = 0.1 or 10%
What is the real interest rate if the nominal interest rate is 8% and the expected inflation rate is 10% over the course of the year.
-2% r = i - π^e i = nominal interest rate = 0.08 π^e = expected inflation rate = 0.10 r = 0.08 - 0.1 = -0.02 = -2%
Rate of Return
How well a person does financially by holding a bond or any other security over a particular time the amount of each payment to the owner plus the change in the security's value
YTM on a fixed payment loan
Fixed payment = same cash flow payment for every period Loan value = $1000 Yearly payment = $126 time = 25 years PV = 126/(1+i)ⁿ $126/(1+i)²⁵ LV = loan value FP = Fixed Yearly Payments n = number of years to maturity LV = FP/(1+i)ⁿ (just use a financial calculator)
Four types of credit market instruments
1. Simple Loan = already discussed 2. fixed-payment loan (fully amortized loan) 3. coupon bond 4. discount bond
One-Year Returns on Different-Maturity 10%-Coupon-Rate Bonds When Interest Rates Rise from 10% to 20%
1. The only bonds whose returns will equal their initial yields to maturity are those whose times to maturity are the same as their holding periods 2. A rise in interest rates is associated with a fall in bond prices, resulting in capital losses on bonds whose terms to maturity are longer than their holding periods. 3. The more distant a bond's maturity date, the greater the size of the percentage price change associated with an interest rate change 4. The more distant a bond's maturity date, the lower the rate of return that occurs as a result of an increase in the interest rate 5. Even though a bond may have a substantial initial interest rate, its return can turn out to be negative if interest rates rise
How do interest rates affect our daily decisions/businesses/economy.
1. They influence us whether to consume or save, buy a house, purchase bonds, add funds to savings 2. businesses = invest in new equipment? Save money?
Types of discount bonds
1. U.S. T bills (treasury bills) 2. U.S. savings bonds 3. long-term zero-coupon bonds
Four pieces of info that describes a coupon bond
1. bond's face value 2. corporation or agency that issues the bond 3. maturity date of the bond 4. bond's coupon rate
Discount Bond
A bond that's bought at a price below its face value (at a discount) No interest is made. However, since you're buying this bond below the face value, you'll still be able to make money when the maturity date reaches. $1000 discount bond will be purchased for $900 (discount). $100 profit
Fixed payment loan (fully amortized loan)
A loan in which the borrower must repay a fixed amount of funds (making the same payment per period) for a set number of years. If you borrow $1,000, a fixed payment loan will require you to pay $126 every year for 25 years.
Present Value
Based on the commonsense notion that a dollar paid to you one year from now is less valuable than a dollar paid to you today. This is used to compare the value of one kind of debt instrument with another (to compare interest rates).
Why is a dollar now more valuable than a dollar in the future?
Because you can earn interest if you deposit the dollar right now. (Also inflation in the future)
True or False? Prices and returns for long-term bonds are less volatile than short-term bonds
False Long-term bonds are more volatile (due to inflation and more economic activity)
True or False? The yield to maturity is less than the coupon rate when the bond price is below face value and is less than the coupon rate when the bond price is above its face value.
False The yield to maturity is greater than the coupon rate when the bond price is below its face value and is less than the coupon rate when the bond price is above its face value.
True or False? When the coupon bond is priced at its future value, the yield to maturity equals the coupon rate.
False When the coupon bond is priced at its FACE value, the yield to maturity equals the coupon rate.
Example of yield to maturity (for a simple loan)
If Pete borrows $100 from his sister and next year she wants $110 back from him, what is the yield to maturity on this loan PV = CF/(1+i)ⁿ 100 = $110/(1+i) Solve for i i = 10% Wait a minute! The YTM is the same as the interest rate!
____ are among the most closely watched variables in the economy
Interest Rates
Simple interest rate
Interest payment / principle payment = simple interest rate With the Jane example $10/$100 = 0.10 or 10% (you can also use this formula to find interest payment or principal payment with an interest rate)
Nominal Interest Rate
Interest rate that doesn't account for inflation
Real interest Rate
Interest rate that is adjusted by subtracting expected changes in the price level (inflation)
Current bond prices and interest rates are _____ related
Negatively related When the interest rate rises, the price of the bond falls, and vice versa.
True or False? The return on a bond will equal the yield to maturity on that bond (coupon bonds)
No The return on a bond will not necessarily equal the yield to maturity on that bond.
Question Assume that you just hit the $20 million jackpot in the State Lottery, which promises you to pay $1 million every year for the next twenty years. Did you really win $20 million?
No, because today's dollars are worth more than dollars in the future. Thus, you'd have to take the interest rate for each $1 million dollar payment for every year. Year 0 = 1 million Year 1 = 1 million / (1+0.1) = 909,090 Year 2 = 909,090 / (1+0.1) = 826,446 Rinse and repeat. Thus, your earnings will depreciate over-time with interest.
Yield to Maturity for a Coupon bond
P = price of the coupon bond C = yearly coupon payment F = face value of the bond n = years to maturity date P = C ÷ (1+i)
Discounting the Future Value equation (present value equation)
PV = CF ÷ (1+i)ⁿ CF = future value of money (future cash flow)
What is the yield to maturity on a one-year, $1000 Treasury bill with a current price of $900?
PV = CF/(1 + i)ⁿ $900 = $1000/(1+i) 11.1%
Coupon bond
Pays the owner of the bond a fixed interest payment every year until the maturity date (when the face value is repaid) Coupon bond with $1000 face value, will pay you a coupon payment of $100 for ten years. Then repay you the face amount of $1000 at the maturity date.
Price of a perpetuity equation
Pc = C/ic Pc = price of the perpetuity C = yearly payment ic = yield to maturity of the perpetuity
For simple loans, the ___ ______ ____ equals to the yield of maturity
Simple Interest Rate
Simple loan
Simplest kind of debt instrument A lender provides the borrower with an amount of funds that must be repaid to the lender at the maturity date, along with an additional payment for interest
How do you find the future value of a principle payment (with simple interest rate of course)?
Take the principle amount In Jane's case, $100 Multiply the principal amount by (1 + simple interest rate) Simple interest rate = 10% $100 * (1 + 0.10) = $110 $110 is the future value for year 1 To find year 2, take the principle of year 2 and repeat the equation $110 * (1 + 0.10) = 121 Year 3 = 121 * (1 + 0.10) = 133
Interest
The additional payment received to the lender when the maturity date reaches. Thus it provides an incentive for the lender to hand-out money to the borrower.
Maturity date
The amount that the borrower must pay to the lender at the end of a loan's life
Principal
The initial amount of funds that the lender provides the borrower
Yield to Maturity
The interest rate that equates the present value of cash flow payments received from a debt instrument with its value today. present value of future cash flow payments
Cash Flows
The various streams of cash payments to the holder of a debt instrument (bonds) the movement of cash into or out of an account, business, or investment
True or False? The price of a coupon bond and the yield to maturity are negatively related. As the yield to maturity rises, the price of the bond falls.
True
True or False? When the real interest rate is low, there are greater incentives to borrower and fewer incentives to lend.
True
Simple loan example
With your friend Jane (borrower), you gave her a loan (which makes you the lender) of $100 for one year. At the maturity date, you would require her to repay the principal amount ($100) and an additional payment for interest ($10).
Table 1 Yields to Maturity on a 10%-Coupon-Rate Bond Maturing in Ten Years (Face Value = $1000) Price of a Bond ($) 1,200 1,100 1,000 900 800
YTM 1. 7.13% 2. 8.48% 3. 10% 4. 11.75% 5. 13.81%
What is the yield to maturity on a bond that has a price of $2,000 and pays $100 of interest annually, forever?
YTM = 5% C = yearly payment = $100 Pc = price of perpetuity = $2000 ic = $100/$2000 = 0.05 or 5%
What type of tool can you use to differentiate the most valuable credit market instrument.
Yield to Maturity
What does the timeline for interest (future value of money) represent?
You are just as happy with owning $100 right now than you are owning $110 in one year, $121 in two years, $133 in three years. etc.
Perpetuity
a bond with no maturity date and no repayment of principal fixed coupon payments for cash flow forever.
Current yield
frequently used as an approximation to describe interest rates on long-term bonds Used to calculate YTM on a Discount Bond
Find the price of a 10% coupon bond with a face value of $1000, a 12.25% yield to maturity, and eight years to maturity? (financial calculator)
n = years to maturity = 8 FV = face value of the bond (F) = 1000 i = annual interest rate = 12.25% PMT = yearly coupon payments (C) = 100 Push the PV button on a financial calculator $889.20
Real interest rate formula
r = i - π^e r = real interest rate i = nominal interest rate π^e = expected rate of inflation
real terms
real goods and services you can buy
Long term maturity =
similar to a perpetuity (20 years generally) Cash flows with 20 years has small present discount values.
