Measuring Interest Rates

Measuring Interest Rates
Bond Interest Rate is more
formally called its Yield to Maturity
Yield to Maturity -- the interest rate
which equates the present value of
all future payments with the
current bond price
Present Value
Present Value – an equation that
converts future payments into
their current dollar equivalent
Example 1 – Find the present
value of payment received one
year from now.
Given P dollars today, with interest
rate i, how much will you have one
year from now (F)?
Answer to Example 1
F = Repayment of principal
+ Payment of Interest
F = (P) + (i)(P) = (P)(1 + i)
To obtain the present value of the
future payment, solve for P
P = F/(1 + i) -- Present value of payment
(F) received one year from now
Example 2 -- Present Value
of Fixed Payment (F)
Received n Years From Now
After One Year: F = P(1 + i)
Two Years:
F = [P(1 + i)](1 + i)
F = P(1 + i)2
Three Years:
F = P(1 + i)3
n Years:
F = P(1 + i)n
Obtaining The Present Value
To convert to current dollars,
solve previous equation for P
P = F/(1 + i)n
Present Value of Payment
Received n Years From Now
Example 3 -- Present Value
of Annual Stream of
Suppose one receives a payment
of A1 at the end of year 1, A2 at the
end of year 2, A3 at the end of year
3, …, and An at the end of year n.
What is the present value (current
dollar equivalent) of that series of
Answer to Example 3
Present Value = Sum or the
present values of each payment
P = A1/(1 + i)
+ A2/(1 + i)2
+ A3/(1 + i)3 + … + An/(1 + i)n
Present Value -Applications
Consider formula (for simplicity,
let A1 = A2 = A3 = … = An = A)
P = A/(1 + i)
+ A/(1 + i)2
+ A/(1 + i)3 + … + A/(1 + i)n
Given any 2 variables, we can
solve for the third.
Application #1 -Given A and i, Solve for P
Examples -- Multiyear Contracts,
Lottery Winnings
Example -- You win $100,000 for year 1
$125,000 for year 2 and $150,000 for
year 3, with i = 0.08.
P = $100,000/(1 + 0.08)
+ $125,000/(1 + 0.08)2
+ $150,000/(1 + 0.08)3
= $318,834.78
Application #2 -Given P and i, Solve for A
Computing Annual Loan Payments
P = Amount Borrowed
i = Interest rate on the loan
An Example
You take out a 5 year loan of
$20,000 to buy a car, at a loan rate
of 9% (0.09). What is your annual
Answer to Car Loan
$20,000 = A/(1 + 0.09)
+ A/(1 + 0.09)2
+ A/(1 + 0.09)3
+ A/(1 + 0.09)4
+ A/(1 + 0.09)5,
Solve for A
A = $5141.85
Monthly Loan Payments
Example -- Car Loan Problem
Same Present Value Formula -Minor Adjustments
i = 0.09/12 = 0.0075
(monthly interest rate)
n = 5 x 12 = 60 months
Monthly Loan Payment
$20,000 = A/(1.0075) + A/(1.0075)2
+ A/(1.0075)3
… + A/(1.0075)60
Solve for A (ugh!!)
A Compressed Formula for
Computing Loan Payments
 Consider again the present value
P = A/(1 + i)
+ A/(1 + i)2
+ A/(1 + i)3 + … + A/(1 + i)n.
 For loan payment, given P and i,
solve for A.
Solution for A
Based upon the solution to a geometric
series, one can show that the equation
solves as:
A = (i)(P)/[1 – 1/(1 + i)n].
Monthly loan payment:
A = (0.0075)($20,000)/[1 – 1/(1.0075)60]
A = $415.17
Application #3 -Given P and A, Solve for i
Example: Yield to Maturity
(interest rate) on Bonds
Apply present value equation to
determine bond interest rates
 Based upon the series of future
payments and the current bond
price (PB)
Yield to Maturity:
Long-Term Bonds
Information printed on the face
of the bond
-- Coupon rate (iC)
-- Face value (F)
Structure of Repayment:
Long-Term Bond
Series of Future Payments: Coupon
(interest) payment each year equal to C
= (iC)(F) along with the face value (F) (or
par value) at maturity.
These payments are fixed, no matter
what the bond sells for.
Long-Term Bonds: Bond
Price and Interest Rate
Bond price (PB) -- determined by
market conditions, constantly
PB < F -- the bond sells at a discount
PB > F -- the bond sells at a premium
PB = F -- the bond sells at par
Interest Rate (Yield to Maturity) -solution to the present value equation,
given future payments and bond price
A General Formula
Yield to Maturity: Long-Term Bond
PB = C/(1 + i) + C/(1 + i)2
+ C/(1 + i)3 + … + C/(1 + i)n
+ F/(1 + i)n
Solve for i (ugh!!)
An Example
Find the yield to maturity for a 20
year Corporate Bond, with a
coupon rate of 7% (0.07), a face
value of $1000, which sells for
Coupon payment: C =
(0.07)($1000) = $70 per year
Bond also pays $1000 at maturity
(year 20).
Solving the Problem
$975 = $70/(1 + i)
+ $70/(1 + i)2
+ $70/(1 + i)3 + …
+ $70/(1 + i)
+ $1000/(1 + i)
Solve for i (ugh!!)
The Yield to Maturity
and the Coupon Rate
One can show the following
If PB = F (coincidentally) then i = iC.
If PB < F, then i > iC.
If PB > F, then i < iC.
Important Property: Bonds
Bond Prices and Bond interest
rates are inversely related, by
In other words, PB  i
Key reason: future payments are
fixed, no matter what price the
bond sells for.
Special Cases: Yield to
Maturity, Long-Term Bonds
Consol (Perpetuity) -- Pays fixed
payment C each year, no maturity
PB = C/(1 + i) + C/(1 + i)2
+ C/(1 + i)3 + … , Solve for i
PB = C/i, which implies that i = C/PB.
Zero Coupon Bond -- No annual
payment, just face value (F) at
PB = F /(1 + i)n, Solve for i
 i = (F/PB)1/n - 1
Yield to Maturity -Money Market Bonds
Method of repayment -- Holder just
receives face value at maturity
Formula -- One year bond
PB = F /(1 + i), Solve for i
 i = (F - PB)/PB
Bonds With Maturities of
Less Than One Year
Simple Adjustment: Multiply the
formula for the 1 year one by an
annualizing factor.
i = [(F - PB)/PB][365/(# of days until
An Example
Suppose that a 90-day Treasury-Bill
has a face value of $100000 and 59
days until maturity. It sells on the
secondary market for $99800. Find the
Yield to Maturity (i).
i = [($100000 - $99800)/($99800)]
x [365/59] = 0.0124 = 1.24%
Other Measures of Yield or
Return on Financial Assets
Current Yield (iCUR),
Yield on a Discount Basis (iDB), or
Discount Yield
i = [(F - PB)/F][360/(# of days until
Rate of Return
Rate of Return (RET) -- Annual return
based upon financial asset’s current
value (bonds sold before maturity,
Formula for Rate of Return (bond)
RETt = [C + (PBt - PB,t-1)]/PB,t-1
Rate of Return: An Example
Suppose that a long-term bond has a
coupon rate of 5% and a face value of
$1000. It sold for $990 last year and
currently sells for $975. Find the Rate
of Return (RET).
C = (0.05)($1000) = $50
RET = [$50 + ($975 - $990)]/$990
= 0.0354 = 3.54%
Implications: Rate of Return
Investors can lose money
(RET < 0) holding bonds.
Formula also applies to stocks.
Bonds and stocks are substitutes,
existence of bond traders.
The possibility of unknown capital
gains or losses introduces
Another Inconvenience:
Market Risk
Market (Asset Price) Risk -- Uncertainty
due to bond prices (and interest rates)
changing, affecting rate of return
Market Risk  i
Factors affecting Market Risk
Interest rate volatility (σB), or degree
of interest rate fluctuation
Real Versus Nominal
Interest Rates
Nominal Interest Rate -- Observed,
unadjusted yield to maturity
Real Interest Rate -- Interest Rate
adjusted for inflation
Key issue -- Must align interest
rate and inflation measure so that
they cover the same time span.
The Ex-Post
Real Interest Rate
 Ex-Post Real Interest Rate (r)
r = iPAST - ,
iPAST = past interest rate
= actual measured
inflation rate (from
past period to now)
The Ex-Ante
Real Interest Rate
 Ex-Ante Real Interest Rate (re)
r e = i -  e,
i = current interest rate
e = expected inflation rate
(from now through the
maturity of the bond)
The most commonly used measure of
the real interest rate
The Fisher Effect
Fisher Effect -- The current nominal
interest rate is constantly 2%-4% above
the inflation rate expected over the life
of the bond.
Crude initial theory of interest rate
determination, shows important role of
expected inflation in affecting nominal
interest rates
Inflation-Indexed Bonds
Inflation-Indexed Bonds (I-Bonds) -- T-Bonds
or Savings Bonds that pay a base rate (e.g.
2%) plus an adjustable interest rate based
upon the existing rate of inflation (over a the
given period from the most recent past).
Seeks to approximate a constant real
interest rate, even though it’s actually
neither the ex-ante nor ex-post measure.

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