Lecture 4 - Interest Rates

Interest Rates
Ryan Herzog, Ph.D.
Associate Professor Economics

Key Questions
• What are interest rates?
• How do we calculate bond prices?
• What is the relationship between interest
rates and bond prices?
7/5/2020 GONZAGA UNIVERSITY 2

Learning Objectives
• Calculate the present value of future cash
flows and the yield to maturity on the four
types of credit market instruments.
• Recognize the distinctions among yield to
maturity, current yield, rate of return, and rate
of capital gain.
• Interpret the distinction between real and
nominal interest rates.

Interest Rates
• Commercial Paper:
– Short-term note (1-270 days), money market (follows federal
funds rate)
• Federal funds rate:
– Short-term overnight lending market. Between banks
• Corporate Bonds:
– Long-term note (different yields, follow treasury notes)
• Governments Bonds:
– Varying maturities (riskless)
• Conventional Mortgages:
– 30 year maturities (follow 10 year bond)

What are interest rates?
• Interest rates are the opportunity cost of
borrowing money.
• They help us determine the pricing of assets
(stocks, bonds, real)
• Many factors go into determining interest
rates (risk, time, amount)
• In this chapter we will use interest rates in
determining asset prices.

Valuing monetary payments
now and in the future
• We must learn how to calculate and compare
rates on different financial instruments.
• We need a set of tools:
– Future value
– Present value
• How and why is the promise to make a payment
on one date more or less valuable than the
promise to make it on a different date?

The Time Value of Money
• Future Value:
– is the value on some future date of an investment made today.
• Compound Interest:
– interest on the interest.
• Base Point:
– one-hundredth of a percentage point — 0.25% = 25 base points.
• Present Value:
– is the value today of a payment promised in the future.
• Rule of 72:
– to find the number of years it takes for an investment to double take 72
divided by the interest rate.
• Internal Rate of Return:
– is the interest rate the equates the present value of an investment with its
cost. .

Future Value and Compound
Interest
• Future value is the value on some future date
of an investment made today.
– $100 invested today at 5% interest gives $105 in a
year. So the future value of $100 today at 5%
interest is $105 one year from now.
– The $100 yields $5, which is why interest rates are
sometimes called a yield.
– This is the same as a simple loan of $100 for a
year at 5% interest.

Interest
• If the present value is $100 and the interest
rate is 5%, then the future value one year
from now is:
$100 + $100 0.05 = $105
• This also shows that the higher the interest
rate, the higher the future value.
• In general:
𝐹𝑉 = 𝑃𝑉 + 𝑃𝑉 𝑖 = 𝑃𝑉(1 + 𝑖)

Interest
• The higher the interest rate or the higher the
amount invested, the higher the future value.
• Most financial instruments are not this simple, so
what happens when time to repayment varies.
• When using one-year interest rates to compute
the value repaid more than one year from now,
we must consider compound interest.
– Compound interest is the interest on the interest.

Interest
• What if you leave your $100 in the bank for
two years at 5% yearly interest rate?
• The future value is:
$100 + $100 0.05 + $100 0.05 + $5 0.05 = $110.25
$100 1.05 1.05 = $100 1.05 2
• In general:
𝐹𝑉 𝑛 = 𝑃𝑉(1 + 𝑖) 𝑛

Interest
• Converting n from years to months is easy, but
converting the interest rate is harder.
– If the annual interest rate is 5%, what is the
monthly rate?
• Assume im is the one-month interest rate and
n is the number of months, then a deposit
made for one year will have a future value of
$1 𝑥 (1 + 𝑖 𝑚)12
.

Interest
• We know that in one year the future value of $1 is
$1.05 so we can solve for im:
(1 + 𝑖𝑚)12= 1.05
1 + 𝑖𝑚 = 1.05 (
1
12)
= 1.0041
• This is the monthly yield for a 5% annual interest rate.
– This is different than a 5% interest rate compound monthly

Measuring Interest Rates
• Present value: a dollar paid to you one year
from now is less valuable than a dollar paid to
you today.
– Why: a dollar deposited today can earn interest
and become $1×(1 + i) one year from today.

Present Value
• Financial instruments promise future cash
payments so we need to know how to value
those payments.
• Present value is the value today (in the present)
of a payment that is promised to be made in the
future.
• Or, present value is the amount that must be
invested today in order to realize a specific
amount on a given future date.

Present Value
• Solve the Future Value Formula for PV:
𝐹𝑉 = 𝑃𝑉(1 + 𝑖) 𝑛
P𝑉 =
𝐹𝑉
(1 + 𝑖) 𝑛
• This is just the future value calculation
inverted.

Present Value
• We can generalize the process as we did for
future value.
• Present Value of payment received n years in
the future:
P𝑉 =
𝐹𝑉
(1 + 𝑖) 𝑛

Present Value
• From the previous equation, we can see that
present value is higher:
– The higher future value of the payment, FVn.
– The shorter time period until payment, n.
– The lower the interest rate, i.
• Present value is the single most important
relationship in our study of financial
instruments.

Present Value
• Doubling the future value of the payment,
without changing the time of the payment or
the interest rate, doubles the present value.
This is true for any percentage.
• The sooner a payment is to be made, the
more it is worth.

Simple Present Value
• Cannot directly compare payments scheduled
in different points in the time line

Yield to Maturity
• Yield to maturity: the interest rate that
equates the present value of cash flow
payments received from a debt instrument
with its value today

Yield to Maturity on a Simple
Loan
• PV = amount borrowed = $100
• CF = cash flow in one year = $110
• n = number of years = 1
• For simple loans, the simple interest rate equals the
yield to maturity
1
$110
$100 =
(1 + )
(1 + ) $100 = $110
$110
(1 + ) =
$100
= 0.10 = 10%
i
i
i
i

Fixed-Payment Loan
• The same cash flow payment every period
throughout the life of the loan
LV = loan value
FP = fixed yearly payment
n = number of years until maturity
     n
iiii 







1
FP
...
1
FP
1
FP
1
FP
LV 32

Coupon Bond
• Using the same strategy used for the fixed-
payment loan:
P = price of coupon board
C = yearly coupon payment
F = face value of the bond
n = years to maturity date
       nn
iiiii 









1
F
1
C
...
1
C
1
C
1
C
P 32

Coupon Bond
• When the coupon bond is priced at its face value, the yield to
maturity equals the coupon rate.
• The price of a coupon bond and the yield to maturity are negatively
related.
• The yield to maturity is greater than the coupon rate when the
bond price is below its face value.
– Table 1 Yields to Maturity on a 10%-coupon-Rate Bond Maturing in Ten Years
(Face Value = $1,000)
Price of Bond ($) Yield to Maturity (%)
1,200 7.13
1,100 8.48
1,000 10.00
900 11.75
800 13.81

Coupon Bond
• Consol or perpetuity: a bond with no maturity date that
does not repay principal but pays fixed coupon payments
forever
Pc = to the price of the console
C = to the price of the console
ic yield to maturity of the console
• For coupon bonds, this equation gives the current yield, an
easy to calculate approximation to the yield to maturity

c
C
P
i
c
c
C
i
P

Discount Bond
• For any one year discount bond
F = Face value of the discount bond
P = current price of the discount bond
• The yield to maturity equals the increase in price
over the year divided by the initial price. As with
a coupon bond, the yield to maturity is negatively
related to the current bond price.


F P
P
i

Internal Rate of Return
• Imagine that you run a tennis racket company
and that you are considering purchasing a new
machine.
– Machine costs $1 million and can produce 3000
rackets per year.
– You sell the rackets for $50, generating $150,000 in
revenue per year.
– Assume the machine is only input, have certainty
about the revenue, no maintenance and a 10 year
lifespan.

• If you borrow $1 million, is the revenue
enough to make the payments?
• We need to compare the internal rate of
return to the cost of buying the machine.
• The interest rate that equates the present
value of an investment with its cost.

• Balance the cost of the machine against the
revenue.
– $1 million today versus $150,000 a year for ten
years.
• To find the internal rate of return, we take the
cost of the machine and equate it to the sum
of the present value of each of the yearly
revenues.
– Solve for i - the internal rate of return.

• Solving for i , i = 0.0814 or 8.14%
$1,000,000 =
$150,000
(1 + 𝑖)1 +
$150,000
(1 + 𝑖)2 +
$150,000
(1 + 𝑖)3 + … +
$150,000
(1 + 𝑖)10
• So long as your interest rate at which you
borrow the money is less than 8.14%, then
you should buy the machine.

Key Terms
• Fixed-Payment Loan: A loan where the borrows pays a fixed
amount every period (typically months/years).
• Bond: is a promise to make a series of payments on specific
future dates.
• Coupon Bond: Issuer is required to make annual payments,
called coupon payments. The annual interest the borrower
pays (ic) is the coupon rate. The date on which the
payments stop and the loan is repaid (n), is the maturity
date or term to maturity. The final payment is the principal,
face value, or par value of the bond.
• Discount Bond: A bond that makes one payment at the
time of maturity.

Calculating Return
• Rarely do people hold bonds until maturity, how
do we calculate the rate of return?
• After purchasing a bond, the bondholder can be
susceptible to interest rate risk, the risk that the
market price of a bond or other instrument will
decrease due to increase in the interest rate.
– If interest rates increase - bond prices fall - capital
losses.
– If interest rates decrease - bond prices rise - capital
gains.

Interest-Rate Risk
• Prices and returns for long-term bonds are
more volatile than those for shorter-term
bonds.
• There is no interest-rate risk for any bond
whose time to maturity matches the holding
period.

Interest-Rate Risk
• Nominal interest rate makes no allowance for
inflation.
• Real interest rate is adjusted for changes in
price level so it more accurately reflects the
cost of borrowing.
– Ex ante real interest rate is adjusted for expected
changes in the price level
– Ex post real interest rate is adjusted for actual
changes in the price level

Real and Nominal Interest
• Borrowers care about the resources required to
repay.
• Lenders care about the purchasing power of the
payments they received.
• Neither cares solely about the number of dollars,
they care about what the dollars buy.
– Nominal Interest Rates (i): The interest rate expressed
in current-dollar terms.
– Real Interest Rates (r): The inflation adjusted interest
rate.

Real and Nominal Interest
• The nominal interest rate you agree on (i) must be based on
expected inflation (πe ) over the term of the loan plus the real
interest rate you agree on (r).
𝑖 = 𝑟 + 𝜋 𝑒
This is called the Fisher Equation.
• The higher expected inflation, the higher the nominal interest rate.
• Financial markets quote nominal interest rates.
– When people use the term interest rate, they are referring to the
nominal rate.
• We cannot directly observe the real interest rate; we have to
estimate it.

Estimating the Real Interest Rate

Lecture 4 - Interest Rates

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