Measuring Interest Rates
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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
Payments
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
payments?
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
payment?
Answer to Car Loan
Problem
$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
Computing
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
formula.
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
fluctuating.
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
$975.
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 + …
20
+ $70/(1 + i)
20
+ $1000/(1 + i)
Solve for i (ugh!!)
The Yield to Maturity
and the Coupon Rate
One can show the following
properties.
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
definition.
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
maturity
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.
Formula:
i = [(F - PB)/PB][365/(# of days until
maturity)]
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),
iCUR = C/PB
Yield on a Discount Basis (iDB), or
Discount Yield
i = [(F - PB)/F][360/(# of days until
maturity)]
Rate of Return
Rate of Return (RET) -- Annual return
based upon financial asset’s current
value (bonds sold before maturity,
stock)
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
uncertainty.
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
Maturity
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
Application:
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.