The market for bond and loans measuring interest rates and returns
Mishkin, Chap 4
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Chap 4 discusses:
I.
Measuring interest rate: step I - concept of present value
(PV);
II. Measuring interest rate: step II - classifying debt
instruments according to timing and nature of payments
and defining yield to maturity (YTM)
III. Calculating YTM for different kinds of bonds and loans,
relationship between YTM and other characteristics of an
asset
IV. Simpler approximations of YTM – current yield, yield on a
discount basis; concepts of nominal vs. real interest rates
V.
Interest rate vs. rate of return on bonds; Interest rate risk
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I.
Measuring interest rate : step I - Concept of Present value (PV):
•
What is “present value” and why do we need it?
Q: Suppose you are promised a payment of $100 a year from now. Will you be as
happy as if you were given $100 today? Why or why not?
Most people have “______________”. They value the future ________________,
that is they ________ the future.
present value or PV of $100 received a year later =
Say you are as happy receiving $1 today as receiving $1.05 a year later.
PV or value today of $100 received a year later =
General formula for PV =
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Interest rates on debt assets are
is one factor influencing interest rates. Others are
. The
Say, the annual interest rate on a Savings Account = 10%. $1 invested in SA today
becomes,
at the end of 1 year:
at the end of 2 years:
…
at the end of “n” years:
If $1 invested today becomes __________ “n” years later, the amount that we
need to invest today to get $1, “n” years later is ___________.
If interest rate = i, the amount we need to invest today to get $1, “n” years later
is ______________.
PV of a cash inflow of $x, “n” years later, if the interest rate today is i =
The PV is the amount you need to
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II . Measuring interest rate: step II - defining yield to maturity (YTM)
A debt asset pays unequal amounts at different points of time through out their
terms.
• a 10-year US savings bond pays:
• a 10-year US T-note pays:
• a consol pays:
All payments need to be considered when determining an interest rate.
Present Value of an asset =
Some numerical examples
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Classifying debt assets
4 major types depending on the timing and broad nature of their payments:
•
Simple loans:
•
Fixed payment loans
•
Coupon bonds
•
Discount bonds
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Major difference between loans and bonds:
“interest rate” on a loan or Yield To Maturity (YTM) is
Formula:
“interest rate” on a bond or Yield To Maturity (YTM) is
Formula
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III. Calculating YTM on different types of debt assets
1. A simple loan pay $110 in a year’s time at a cost of $100 today.
YTM =
A commercial borrower gets a loan of $1m today and pays back $2m, 5 years later.
The annual interest rate he pays (YTM) =
2. Suppose a car loan offers to pay $3000 every year for 5 years, at a cost of
$10,000 today.
The YTM or annual interest rate =
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3. A coupon bond costing $850 (market price), pays $100 yearly for 10 years,
$1000 on maturity (its face value).
Its YTM =
4. A T-bill maturing a year from now, has a face value of $1000 and a market
price of $950 today.
Its YTM =
The YTM on a US-Savings bond which matures in15 years from today, pays a
face value of $1000 and costs $800 today =
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Why do we choose to measure the interest rates on debt assets as their yield to
maturity?
Fundamental theorem (FT) of finance: “An asset is worth no more or no less (to
a rational investor) than its present value in a competitive financial market. In a
competitive market therefore, the market price of an asset = PV ”
You have $3000 to invest with two options: open a savings account at 10%
annually or lend to a (honest) friend who promises to pay $1100 the first year,
$1210 the second year, $1331 the third year. Which is better?
Suppose the interest rate on the SA is 15%. Your friend cannot change the
payments but is willing to accept a smaller or larger loan. What would you do?
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The PV of an asset is what it takes to ____________________________
__________________________________________________________
The PV of an asset is often described as the fundamental worth of the asset.
Suppose your friend has a history of default. What would you do if the rate on
SA is 10% and the size of the loan is negotiable? What is the fundamental worth
of the loan?
FT in essence: If the price of an asset (P) is determined by market forces of
demand and supply, it settles at it’s fundamental value. Why?
Hence YTM assuming P=PV, is the rate it takes __________________________
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Relationship between YTM and other characteristics of an asset:
C = annual coupon payments, FV = face value, P = market price of a bond;
coupon rate for a coupon bond, r = C/FV
LV = loan value or principal amount, FP = annual fixed payment on a loan
n = term to maturity
• Given C, FV and n, as P increases, the YTM of a coupon bond
• Given FV and n, as P increase, the YTM of a discount bond
• Given LV, and n, as FP increases the YTM for a fixed payment loan
• Given FP and n, as LV increases, the YTM for a fixed payment loan
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The following relationships can be proved:
1.
When P = FV, yield to maturity = _______________.
2.
When P < FV, yield to maturity __________________.
3.
When P > FV, yield to maturity ___________________.
Note: The other way implications are also true.
a consol or a perpetuity is a ____________________________________.
The YTM for a consol:
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IV.
Simpler approximations to YTM
the Current Yield on coupon bonds: xc =
Obsv. 1. For a consol _____________________________. In general the longer the
term to maturity of a bond __________________________________________.
2. If P = FV for a coupon bond _________________________________.
The closer P is to FV, ______________________________________________.
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the nominal interest rate is the interest rate calculated
the real interest rate is the
Relationship between the two: ir = ________________, where
The equation is known as _________________________.
Real interest rates are more relevant for decision making because
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V. Difference between YTM and the rate of return
Most buyers don’t hold on to their assets throughout their term to maturity, but
carry them for shorter periods.
Rate of return on a bond held for one period from date t to date t+1
=
where
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Calculating the one –period rate of return on a bond at date t
• If Pt + 1 is known at date t, use formula on previous slide
• Pt + 1 may not be known at date t; it has to be calculated using the present value
formula.
Pt + 1 = present value of the asset over its remaining term to maturity starting at
date t + 1, at the interest rate at date t + 1.
Example: An asset pays $1000 at the end of year 1, $1000 at the end of year 2, and
$1000 at the end of year 3. Market interest rate is 5% currently. At the end of year
1, it increases to 8% and remains at that level for the next two years.
P year 1 =
Pyear 2 =
If an investor buys the asset at the beginning of year 1 and sells it off at the
beginning of year 2, the one year rate of return is equal to
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Properties of the rate of return on a bond
1.
RET differs from xc if _________________________________________
2.
A rise in interest rates is accompanied by a ______ in bond prices for bonds
whose terms to maturity are _______ than the holding period
3.
The ______ the term to maturity, the _______ the percentage ______.
4.
Prices and rates of return for longer term bonds are _______ volatile
compared to prices and rates of return for shorter term bonds.
interest rate risk is
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