Transcript Understanding Interest Rates

Understanding the Concept of
Present Value
Interest Rates, Compounding,
and Present Value
• In economics, an interest rate is known as
the yield to maturity.
• Compounding is the process that gives us
the value of a sum invested over time at a
positive rate of interest.
• Present value is the process that tells us how
much an expected future payment is worth
today.
Compounding
• Assume you have $1 which you place in an
account paying 10% annually.
• How much will you have in one year, two
years, etc?
– An amount of $1 at 10% interest
– Year
1
2
3
•
$1.10
$1.21
• Formula: FV = PV(1 + i)
$1.33
n
$1(1 + i)n
Compounding over Time
• Extending the formula over 2 years
– FV = PV(1 + i) (1 + i) or FV = PV(1 + i)2
• 3 years
– FV = PV(1 + i) (1 + i) (1 + i) = PV(1 + i)3
• n years
– FV = PV(1 + i)n
Present Value
• Present value tells us how much an
expected future payment is worth today.
• Alternatively, it tells us how much we
should be willing to pay today to receive
some amount in the future.
– For example, if the present value of $1.10 at an
interest rate of 10% is $1, we should be willing
to spend $1 today to get $1.10 next year.
Present Value Formula
• The formula for present value can be found
by rearranging the compounding formula.
– FV = PV(1 + i)
• FV/(1 + i) = PV
solve for PV
Present Value over Time
• Extending the formula over 2 years
– FV = PV(1 + i)2
– PV = FV/(1 + i)2
• 3 years
– FV = PV(1 + i)3
– PV = FV/(1 + i)3
• n years
– FV = PV(1 + i)n
– PV = FV/(1 + i)n
Things to Notice
• An increase in the interest rate causes present
value to fall.
– Higher rates of interest mean smaller amounts
can grow to equal some fixed amount during a
specified period of time.
• A decrease in the interest rate causes present
value to rise.
– Lower rates of interest mean larger amounts are
needed to reach some fixed amount during a
specified period of time.
Example:
How much must I invest today to get $10,000 in five years if interest
rates are 10%?
PV = FV/(1 + i)n
PV = $10,000/(1 + .10)5 = $10,000/1.6105 = $6,209.2
How much must I invest today to get $10,000 in five years if interest
rates are 5%?
PV = FV/(1 + i)n
PV = $10,000/(1 + .05)5 = $10,000/1.2763 $7,835.15
More Things to Notice
• Present value is always less than future
value.
– (1 + i)n is positive so FV/(1 + i)n < FV
• In addition, PV4 < PV3 < PV2 < PV1
– (1 + i)1 < (1 + i)2
• The longer an amount has to grow to some fixed
future amount, the smaller the initial amount
needs to be.
Time Value of Money
• The longer the time to maturity, the less we
need to set aside today. This is the principal
lesson of present value. It is often referred
to as the “time value of money.”
Example:
If I want to receive $10,000 in 5 years, how much do I have to invest
now if interest rates are 10%?
$10,000 = PV(1 + .10)5
$10,000/1.5105 = $6209.25
If I want to receive $10,000 in 20 years, how much do I have to invest
now if interest rates are 10%?
$10,000 = PV(1 + .10)20
$10,000/6.7275= $1486.44
Understanding Interest Rates
Yield to Maturity
• Yield to maturity is the interest rate that
equates the present value of payments
received from a debt instrument with its
value today.
• Yield to maturity can be calculated using
the present value formula.
– PV = FV/(1 + i)
– i = (FV – PV)/PV
Simple Example:
•
•
•
•
•
PV = FV/(1 + i)
PV(1 + i) = FV
PV + PVi = FV
PVi = FV - PV
i = (FV – PV)/PV
– $1.00 = $1.10/(1 + i)
– $1.00 + $1.00i = $1.10
– i = ($1.10 - $1.00)/$1.00 = 0.10 = 10%
Relationship between Yield to
Maturity and Price
Yields to maturity on a 10% coupon rate bond with a face value
of $1000 maturing in 10 years
Price of Bond
1200
1100
1000
900
800
Yield to Maturity
7.13
8.48
10.00
11.75
13.81
Relationship between Yield to
Maturity and Price
• Three interesting facts:
– Price and yield are negatively related.
– When the bond is at par, yield equals coupon
rate.
– Yield is greater (less than) than the coupon rate
when the bond price is below (above) par value.
Current Yield
• In more complicated cases, yield to maturity
can be difficult to calculate. Tables are
available that can be used . And, of course,
calculators do a fine job.
• There are also simple formulas that can
approximate yield to maturity such as
current yield.
Current Yield
• Current yield is an approximation for yield
to maturity that is used to calculate the
interest rate on a bond quickly.
• Formula:
– Current yield = Coupon/Bond Price
• Current yield is a better approximation to yield to
maturity, the nearer price is to par and the longer is
the maturity of the security.
• A change in current yield always signals a change in
the same direction as yield to maturity.
Inverse Relationship
• We can use the current yield formula to see
clearly the inverse relationship between
interest rates and bond prices.
– Current yield = Coupon/Bond Price
• The coupon is a fixed payment, it does not change.
Therefore, if yields rise, bond prices must fall, and if
yields fall, bond prices must rise.
Intuition
• Assume you buy a $1,000 bond today with
a fixed coupon of $100. You are receiving a
10% return. Let a year pass, and you find
you want to sell you bond. You call your
broker and say, “Sell!” Your broker sighs
and tells you that bonds just like yours now
yield 12%. What price can you expect to
receive?
Example
• Use the current yield formula:
– 0.12 = $100/PB
– 0.12PB = $100
– PB = $100/.12 = $833.33
• You must reduce your price until $100
represents a 12% rate of return.
Yield on a Discount Basis
• The yield to maturity calculation for a
discount bond is similar to our example of
yield to maturity.
• A discount bond is one that is sold at a
discount from its face value.
• The yield or interest received is determined
by the difference between the price paid and
its face value.
Discount Bonds
• Yield formula:
– i = (Face - Price Paid)/Price Paid
• U.S. Treasury bills are sold on a discount
basis. The formula used to calculate the
yield is:
– i = (Face - Price Paid)/Face * (360/# Days to
Maturity)
Discount Bonds: Characteristics
• The formula used to calculate yield
understates yield to maturity. The longer the
maturity, the greater the understatement.
• A change in discount yield always signals a
change in the same direction as yield to
maturity.
Coupon Bond
• A bond is a debt instrument. A coupon bond
is a bond that pays its owner a fixed coupon
payment every year until maturity, at which
time a specified final amount (face value) is
repaid
– We expect to get:
• Coupon payments each year
• Principal at maturity.
• How much should we be willing to pay
today for a stream of income?
Coupon Bond
• PB = C/(1 + i) + C/(1 + i)2 + C/(1 + i)3 +
……..C/(1 + i)n + P/(1 + i)n
– where
•
•
•
•
•
C is a fixed coupon
i is the rate of interest
PB is the price or present value of the bond
P is the principal
n is years to maturity
Coupon Bond Example
• Let the coupon payment be $100, the rate of interest
10%, and the principal equal to $1000. If n is 4, how
much should we pay today for this bond?
– PB = 100/(1 + 0.10) + 100/(1 + 0.10)2 + 100/(1 + 0.10)3 +
100/(1 + 0.10)4 + 1000/(1 + 0.10)4
– PB = 100/(1.10) + 100/(1.21) + 100/(1.331) + 100/(1.4641)
+ 1000/(1.4641)
– PB = 90.9 + 82.64 + 75.13 + 68.3 + 683.01 =
– $1,000 (Note that you don’t pay $1,400).
Things to Notice
• When a coupon bond is priced at its face
value, the yield to maturity equals the
coupon rate.
More Things to Notice
• The yield to maturity and the coupon rate
do not have to be the same.
– If the bond price is less than the face value, the
yield to maturity is greater than the coupon rate.
• In this case, the difference between the bond price
and the face value adds to the total return.
– If the bond price is greater than the face value,
the yield to maturity is less than the coupon
rate.
• In this case, the difference subtracts from the total
return.
Even More Things to Notice
• The price of a coupon bond and the yield to
maturity are inversely related.
– An increase in the interest rate decreases the
bond price.
– A decrease in the interest rate increases the
bond price.
• This is the reason the market participants
are so interested in the actions of the
Federal Reserve.
Interest Rates and Returns
• For any security, the rate of return is defined
as the payments to the owner plus the
change in its value, expressed as a ratio to
its purchase price.
• The return to a bond depends on its stream
of coupon payments and the price the bond
receives when it is sold.
Interest Rates and Returns
• If the bond sells at a price in excess of its
original purchase price, the owner receives
a capital gain which increases his/her total
return.
• If the bond sells at a price below its original
purchase price, the owner suffers a capital
loss, which decreases his/her total return.
Return on a Bond
• The return on a bond may be expressed by
the formula:
– Ret = C/Pt + (Pt+1 - Pt)/Pt
where
• C = Coupon payment
• Pt = Price of the bond in time t
• Pt+1 = Price of the bond in time t + 1
Returns on Different Maturity
10% Coupon Rate Bonds
Term
30
20
10
5
2
1
Initial i
10%
10%
10%
10%
10%
10%
Initial P
1000
1000
1000
1000
1000
1000
New i
20%
20%
20%
20%
20%
20%
New P
503
516
597
741
917
1000
K Gain
-49.7
-48.4
-40.3
-25.9
-08.3
0
ROR
-39.7
-38.4
-30.3
-15.9
+ 1.7
+10.0
Things to Notice
• The only bond whose return is certain to
equal the initial yield is the one whose time
to maturity is the same as the holding
period.
• 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 the holding period.
More Things to Notice
• The longer the bond’s maturity, the greater
is the size of the price change associated
with an interest rate change.
• The longer a bond’s maturity, the lower is
the rate of return that occurs as a result of
the increase in the interest rate.
• Even though the bond had a good interest
rate, its return became negative when
interest rates rose.
Reinvestment Risk
• Reinvestment risk occurs
– when an investor holds a series of short bonds
over a long holding period and interest rates are
uncertain.
• If interest rates rise, the investor gains
• If interest rates fall, the investor loses
Reinvestment Risk: Example
• Assume a holding period of two years and
an investor who has decided to buy two one
year bonds sequentially.
– Year 1 bond:
• Face = $1000, initial interest rate = 10%
– At the end of the year, the investor has $1100.
– Year 2 bond:
• Face = $1100, interest rate = 20%
– At the end of year 2, the investor has $1320.
Reinvestment Risk: Example
• The investor’s two year return will be:
– ($1320 - $1000)/$1000 = 0.32 = 32% over two
years.
– In this case the investor has benefited by buying
two one year bonds.
– Conversely, if interest rates had fallen to 5%,
the investor would done less well.
Reinvestment Risk: Example
• Year 1:
– ($1000 x (1 + 0.10)) = $1100
• Year 2:
– ($1100 x (1 + 0.05)) = $1155
• Return = ($1155 - $1000)/$1000 = 15.5%
over two years.
• The investor now loses from a change in
interest rates.
Real and Nominal Interest Rates
• Nominal interest rate is the rate of interest
that makes no allowance for inflation.
• The real interest rate is the rate of interest
that is adjusted for expected changes in the
price level.
– It more accurately reflects the true cost of
borrowing and lending.
Fisher Equation
• The Fisher equation states that the nominal
interest rate equals the real interest rate plus
the expected rate of inflation
– in = i r + p
• Rearranging terms we find:
– ir = i n - p
Logic behind the Inflation
Premium
• Lenders want to be compensated for the loss
in buying power due to inflation.
• Buyers understand that they will be
repaying debt with dollars that buy less.
• The interest rate must reflect these facts.