chapter 4 understanding interest rates. learning objectives calculate the present value of future...
TRANSCRIPT
Chapter 4
Understanding Interest Rates
Learning Objectives
• Calculate the present value of future cash flows and the yield to maturity on 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.
Present Value: Discounting the Future
• Present Value (PV) - the value today of a payment promised in the future
• A dollar paid to you one year from now is less valuable than a dollar paid to you today.
Discounting the Future (PV) First need to Understand Future Value (FV)
• Let i = 0.10 (10 percent interest rate)
• In one year: $100 +(.10 x $100) = $100 x (1+.10) = $110
• In two years: $110+(.10 x $110) = $110 x (1+.10) = $121
OR $100 x (1+.10)2 = $121
Future value
• In three years: $121+(.10 x $121)= $121x(1+.10) = $133
• OR $100 x (1+.10)3 = $133
• In n years: FV = $100 x (1+ .10)n
Future Value
• Future value in n years of an investment of PV today at interest rate i
(i is measured as a decimal, 10% = .10)
FVn = PV x (1+i)n
• Calculate one plus the interest rate (measured as a
decimal), raised to the nth power, multiply by the
amount invested (present value PV).
Future Value
Note:
When computing Future Value, both n and i must be measured in same time units—if i is annual, then n must be in years.
The future value of $100 in 18 months at 5% annual interest rate is:
FV = 100 *(1+.05)1.5
Future Value• The future value of $100 in one month at a 5%
annual interest rate is:
FV = $100 *(1+.05)1/12 = $100.4074
• (1+.05)1/12 converts the annual interest rates to a monthly rate.
• (1+.05)1/12 = 1.004074, which converted to percentage is 0.4074% or 0.41%(rounded)
Note: 0.05/12 = .004167
Basis Point
• Faction of a percentage point is called basis point.
• A basis point is one-one hundredth of a percentage point
• One basis point (bp) = 0.01 percent.
• On the previous slide: 0.41% is 41 basis points.
Discounting the Future PV Reverses the FV Calculation
n
PV = today's (present) value
CF = future cash flow (payment)
= the interest rate
CFPV =
(1 + )
i
i
FVn = PV x (1+i)n
PV = FVn /(1+i)nIn General:
Present Value
Present ValueExample 1:• Present Value of $100 received in 5 years discounted at
an interest rate of 8%. PV = $100 / (1.08)5 = $68.05
Example2:• PV of $20,000 received 20 years from now discounted at
8% is: PV = $20,000 / (1+0.08)20 = $20,000/ 4.6609 = $4,291 • Discounted at 9%: PV = $20,000 / (1+0.09)20 = $20,000/5.6044 = $3,568
Present Value
Example3:• PV of $20,000 received 19 years from now discounted at
8% is: PV = $20,000 / (1+0.08)19 = $20,000/ 4.3157 = $4,634
In general, present value is higher:
1. The higher the future value of the payment (CF).
2. The shorter the time period until payment (n).
3. The lower the interest rate. (i)
ni
CFPV
)1(
Mishkin Discusses Four Types of Credit Market Instruments
• Simple Loan
• Fixed Payment Loan (I will just mention this)
• Coupon Bond
- Special case: consol bond
• Discount Bond
We will focus on Bonds and look at three types of bonds• Coupon Bonds: which make periodic interest
payments and repay the principal at maturity. • U.S. Treasury Bonds and most corporate bonds are
coupon bonds.
• Discount or Zero-coupon bonds: which promise a single future payment, such as a U.S. Treasury Bill.
• Consols: which make periodic interest payments forever, never repaying the principal that was borrowed. (There aren’t many examples of these.)
Yield to Maturity -YTM
• The interest rate that equates the present value (PV) of cash flow payments (CF) received from a debt instrument with its value today
ni
CFPV
)1(
Given values for PV, CF and n, solve for i.
Yield to Maturity - One Year Simple Loan
1
PV = amount borrowed = $100
CF = cash flow in one year = $110
= number of years = 1
$110$100 =
(1 + )
(1 + ) $100 = $110
$110(1 + ) =
$100 = 0.10 = 10%
For simple loans, the simple interest rate equ
n
i
i
i
i
als the
yield to maturity
Fixed Payment Loan - YTM
2 3
The same cash flow payment every period throughout
the life of the loan
LV = loan value
FP = fixed yearly payment
= number of years until maturity
FP FP FP FPLV = . . . +
1 + (1 + ) (1 + ) (1 + )n
n
i i i i
Bonds: Our Objectives
• Bond price is a present value calculation.
• YTM is the interest rate.
• Supply and Demand determine the price of bonds.
- We will also discuss loanable funds and
money supply/money demand.
• Explain why bonds are risky.
Coupon Bond Price
A simple contract-
Price of a Coupon Bond
Present Value of Coupon Bond (PB) =
Present Value of yearly coupon payments (C)
+ Present Value of the Face Value (FV),
where: i = interest rate and n = time to maturity
nnB i
FV
i
C
i
C
i
C
i
CP
)1()1(......
)1()1()1( 321
Present Value of Coupon Payments
Present Value of Principal Payment
Example: Price of a n-year Coupon Bond
nnB iiiiP
)1(
1000$
)1(
100$......
)1(
100$
)1(
100$21
Coupon Payment (C) =$100, Face value (FV) = $1,000, and n = time to maturity
Definition: Coupon Payment / Face Value = Coupon Rate
Given values for i and n, we can determine the bond price PB
Price of a 10-year Coupon Bond
101021 )1.1(
1000$
)1.1(
100$......
)1.1(
100$
)1.1(
100$1000$
BP
Excel Example
If n = 10, i = 0.10, C = $100 and FV = $1000.
What’s the Coupon Rate?
Price of a 10-year Coupon Bond
101021 )12.1(
1000$
)12.1(
100$......
)12.1(
100$
)12.1(
100$887$
BP
If n = 10, i = 0.12, C = $100 and FV = $1000.
What’s the Coupon Rate?
YTM - Coupon Bond
101021 )1(
1000$
)1(
100$......
)1(
100$
)1(
100$950$
iiiiPB
What’s the Coupon Rate?
What’s the YTM?
YTM = .1085 or 10.85%
Suppose n = 10, PB = $950, C = $100 and FV = $1000.
2
.PFV
nPFV
CYTMApprox
C = the coupon paymentFV = Face Value P = Price n = years to maturity
Previous example:
n = 10, PB = $950, C = $100 and FV = $1000.
Using the Approximation Formula
Paid Price
PaymentCoupon Yearly YieldCurrent
For our 10-year, $1000 FV bond with a $100 coupon payment selling at $950:
Coupon rate = 10% YTM = 10.85% App. YTM = 10.77%Current Yield = 10.52%
Current Yield
Two Characteristics of Current Yield
1. Is a better approximation of yield to maturity, nearer price is to par (face value) and longer is maturity of bond
2. Change in current yield always signals change in same direction as yield to maturity
• 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
Consol – Special Case Coupon Bond
• Infinite maturity
• No face value.
• Fixed coupon payment of C forever.
• Pconsol = C/(1+i) + C/(1+i)2 + C/(1+i)3 + … + C/(1+i)t
• As t goes to infinity this collapses to:
PConsol = C / i => i = C / P
$2,000 = $100/.05
Discount or Zero Coupon Bond
• Definition: A discount bond is sold at some price P, and pays a larger amount (FV) after t years. There is no periodic interest payment.
Let P = price of the bond, i= interest rate, n = years to maturity, and FV = Face Value (the value at maturity):
ni
nFVP
)1(
Zero Coupon Bonds - PriceExamples: Assume i=4%
Price of a One-Year Treasury Bill with FV = $1,000:
Price of a Six-Month Treasury Bill with FV = $1,000:
53.961$)04.01(
1000
BP
58.980$)04.01(
10002/1
BP
Price of a 20-Year zero coupon bond at 8% and FV = $20,000:
96.290,4$)08.01(
20000$20
BP
YTM - Zero Coupon Bonds
ni
FVP
)1(
11)1()/1()/1(
nnn
P
FVi
P
FVi
P
FVi
Zero Coupon Bonds - YTM
• For a discount bond with FV = $15,000 and P = $4,200, and n = 20, the interest rate (or yield to maturity) would be:
i = 1.0657 -1=> i =6.57%
1)/1(
n
P
FVi
1200,4
000,15)20/1(
i
Note: This is the formula for compound annual rate of growth
Zero Coupon Bonds - YTM
1491,6
000,10)7/1(
i
i = 1.06368 – 1 = .06368 or 6.368%
• For a discount bond with FV = $10,000 and P = $6,491, and n = 7, the interest rate (or yield to maturity) would be:
From a Coupon Bond to Zero Coupon Bonds (called Strips)
nnB i
FV
i
C
i
C
i
C
i
CP
)1()1(......
)1()1()1( 321
Create n+1 discount bonds
Yield on a Discount Basis
SKIP
Distinction Between Interest Rates and Return
The payments to the owner plus the change in value
expressed as a fraction of the purchase price
RET = C
Pt
+ P
t1 - P
t
Pt
RET = return from holding the bond from time t to time t + 1
Pt = price of bond at time t
Pt1
= price of the bond at time t + 1
C = coupon payment
C
Pt
= current yield = ic
P
t1 - P
t
Pt
= rate of capital gain = g
Holding Period Return
• the return from holding a bond and selling it before maturity.
• the holding period return can differ from the yield to maturity.
Holding Period Return Example:• You purchase a 10-year coupon bond, with a 10%
coupon rate, at face value ($1000) and sell one year later.
1 1( 1)
( )t t
t
tt
tt
tt
C P PR
C
P
P PET
PP
Current Yield Capital Gain
Holding Period Return
If the interest rate one year later is the same at 10%:
One year holding period return =
or 10.0%
10.1000$
100$
1000$
1000$1000$
1000$
100$
Holding Period Return
If the interest rate one year later is lower, say at 8%:
One year holding period return =
or 22.5%
Where did the $1125 come from?
225.1000$
225$
1000$
1000$1125$
1000$
100$
Holding Period Returns
If the interest rate in one year is higher at 12%:
One year holding period return =
or -.70%
007.1000$
7$
1000$
1000$893$
1000$
100$
Where did the $893 come from. You need to know how to calculate the $1125 and $893
Key Conclusions From Table 2
• The return equals the yield to maturity (YTM) only if the holding period equals the time to maturity
• A rise in interest rates is associated with a fall in bond prices, resulting in a capital loss if the holding period is less than the time to maturity
• The more distant the maturity, the greater the percentage price change associated with an interest-rate change
Key Conclusions From Table 2
• The more distant a bond’s maturity, the lower the rate of return that occurs as a result of an increase in the interest rate
• Even if a bond has a substantial initial interest rate, its return can be negative if interest rates rise
Interest-Rate Risk
• Change in bond price due to change in interest rate
• Prices and returns for long-term bonds are more volatile than those for shorter-term bonds
• There is no interest-rate risk for a bond whose time to maturity matches the holding period
Reinvestment (interest rate) Risk• If investor’s holding period exceeds the term to maturity
proceeds from sale of bond are reinvested at new interest rate
the investor is exposed to reinvestment risk
• The investor benefits from rising interest rates, and suffers from falling interest rates
Real and Nominal Interest Rates• Nominal interest rate (i) makes no allowance for
inflation
• Real interest rate (r) 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 (πe)
• Ex post real interest rate is adjusted for actual changes in the price level (π)
Real and Nominal Interest Rates
Fisher Equation:
i = r + πe
From this we get -
rex ante = i - πe
rex post = i - π
Mankiw
Inflation and Nominal Interest Rates
Inflation and Nominal Interest Rates
Real and Nominal Interest Rates
Real Interest Rate:
Interest rate that is adjusted for expected changes in the price level
r = i -πe
if i = 5% and πe = 3% then:
r = 5% - 3% = 2%
if i = 8% and πe = 10% then
r = 8% - 10% = -2%
Figure 1 Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953–2014
Sources: Nominal rates from Federal Reserve Bank of St. Louis FRED database: http://research.stlouisfed.org/fred2/. The real rate isconstructed using the procedure outlined in Frederic S. Mishkin, “The Real Interest Rate: An Empirical Investigation,” Carnegie-Rochester Conference Series on Public Policy 15 (1981): 151–200. This procedure involves estimating expected inflation as a function of past interest rates, inflation, and time trends, and then subtracting the expected inflation measure from the nominal interest rate.
A measure in inflationary expectations
i = r + πe
πe = i - r
http://www.bloomberg.com/markets/rates-bonds/government-bonds/us/
http://research.stlouisfed.org/fred2/