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Unformatted text preview: Chapter 4: Understanding Interest Rates After this chapter you should have an
understanding of the following. 1.What present value is and how to calculate
the present value of bonds. 2.Understand the deﬁnitions of interest rates,
yield to maturity, current yield, and rate of
capital gain. 3.Understand interest—rate risk.
4.Understand and apply the Fisher equation. 5.Understand the differences between nominal
and real interest rates. 6.Understand the difference between ex ante
and ex post interest rates. 27‘ 0' W ﬂ a")? "" .1: Is it possible for a market to have a negative rate
of return? V435 ML 664:4 quyfem ( {4% Max—m? ma Is it possible for very high interest rates on short
term bonds to indicate a lose credit market as
opposed to a tight credit market? V55 1 W “Wi— To answer the above questions, we have to know
how to evaluate the value of bonds. In particular, the value of the _fQ_L_1r general types of credit
market instruments in consideration for such things as inﬂation and income tax. l.Simple Loan a saga [m4 (ampwmdrby) g
2.Fixed Payment Loan ’Mov’flgjagﬂﬁ , pay/wads “f
3.Coupon Bond 4.Discount Bond hy‘ﬂwﬁ ymf‘a
grpA 555575 (30!! (V m if? How to calculate Present Value Basic idea is that money received in the future is
not as/yaluable as money received today. WM If you were to put $1,000 in the bank today and it
paid a simple interest rate of 10% to you at the
end of the year, you would have $1,100 at the end
of the year. This can be written as the following, Let i = the interest rate, PV = Present Value, @194 F“
= future cash ﬂow and I1: number of periods,
then your cash ﬂow one year from now is ﬂ/ _ q
or : PV*(1+i)n 
F” In this case 1,100 = 1000 a 1:" If we left the $1,000 in the bank for two years we
would have $1,100 after the ﬁrst and $1,210 after the second.
F \/
W = We: . ' Note that we can rearrange this equation to ﬁnd
the presednt value of future cash ﬂow.
i PV = @F/ (1+1)n ./
To calculate present value we will need to
discount future cash ﬂows by the discount factor,
1/ (l + opportunity cost)“, when the interest rate is
the opportunity cost than the discount factor
becomes l/(l + i)”, for each period n and add up
the PV of those cash ﬂows. More generally, N
PV = ;) {CF11 * [ 1/ (1+in)"]} For example: Suppose that you own a bond that
will give you a cash ﬂow of $1,000 for 3 years
and at the end of the four year give you acash
ﬂow of $1 ,100. If the interest rate is 10%What is the PV of this bond? W: The interest rate that equates
the present value of cash ﬂow payments received from a debt instrument with its value today. I The yield to maturity is what economist
generally considers being the market interest rate. Let’s calculate the Yield to Maturity for four
types of credit market instruments, Simple Loan,
Fixed Payment Loan, Coupon Bond, Discount Bond. Simple Loan
From our earlier example if the amount borrowed is $1,000 and the end of the year cash ﬂow to
repay the loan is $1 ,100, what is the yield to
maturity of this credit instrument? For simple loans, the simple interest rate equals Fixed Payment Loan (or fully amortized loan) is
a loan where principle and interested are paid by
a ﬁxed amount at predetermined intervals for the duration of the loan. LV = loan value (present value)
FP = ﬁxed yearly payment (Cash Flow)
11 = number of years until maturity  P
Lve= PP + FP + PP .+...+ F 1+i a+o2 a+o3 0+0" a MIL/i Md: will pay a ﬁxed “stated” interest
rate (also called the coupon rate) or coupon
payment every year until the maturity date. At the maturity date it will pay the face value or par
value of the bond. P = price of coupon bond C = yearly coupon payment 4%???" _’ f; i' " F 4}: ._ a: . F = face value of the bond
n = years to maturity date
C C A C C F P=———+_ ,,+ _ l+...+_ _+ I
Hi (l+f)“' (1+z')’ (1+i)” (1+i)” Let’s look at an example of different loan values
(their PV “‘price”) and yields to maturities. Table 1: Yields to Maturity on a 10%Coupon—
Rate Bond Maturing in Ten Years (Face Value = $1,000) 0 When the coupon bond is priced at its face
value, theyield to maturity equals [am mr‘ e. o The price of a coupon bond and the yield to
maturity are m wow/53C related a The yield to maturity is Maw than the
coupon rate when the bond price is below its facexalue _
We”? 352‘ Pmﬁﬁboﬂv MEI/M525 ““““““ H
——W 2:056 C'sfar}...
66/15 .27" ;ch ﬂeets 0% WWW? Consol or Mil: A bond with no maturity
date that does not repay principal but pays ﬁxed
coupon payments forever. (a. I. L,”
{d . W v :i P = C it /'
R = price of the consol
C : yearly interest payment if 2 yield to maturity of the consol can rewriteabove equationas this: x", = C/ I: For coupon bonds, this equation gives the current yield, an
easy to calculate approximation to the yield to maturity Discount Bond or zero—coupon bond does not
pay any cpupons (or yearly interest payments). It
just pays the face value of the bond at maturity.
Money is then made by the holder of the bond
only through the difference in the purchase price
and par value of the bond (ortheir selling price if
sold before maturity) For any one year discount bond: i = (F  P) / P
@LW Fahd/c1 [C] F = Face value of the discount bond P = current price of the discount bond The yield to maturity equals the increase in price over the year divided by the initial price. As with
a coupon bond, the yield to maturity is negatively
related to the current bond price. Let’s consider the difference between Interest
rates and rates of return. 10 Rate of Return:_ The payments to the owner plus
the change in value expressed as a fraction of the
purchase price. VH7 V! r“! ' r’ '" ’ ” " 0w RET = C/Pt + (Pu — Pt) / Pt RET = return from holding the bond from time t
to time t + 1 Pt = price of bond at time t Pu = price of the bond at time t + l C = coupon payment OR = current yield = i0 (Pu  Pt) /Pt = rate of capital gain = g o The return equals the yield to maturity only if the WIS the time to maturity
Link,1} 9.3}  M 0 A liiﬁeininterest rates is associated with a fall
in bond prices, resulting in a cap_i_t_al_loss if time
to maturity/is longer than the holding period a The more distant @QIMW, the greater
the sizeef the percentage prieeehepge
associated with an interest—rate_change 11 ' WW, the lower
the rate of return theloeeurs as a result of an
increase in the interest rate Mmsmurwmm 0 Even if a bond has a substantial initial interest
rate, its return ean be negative if interest rates
rise Table 2: OneYear Returns on DifferentMaturity
lO%—C0up0n—Rate Bonds When Interest Rates
Rise from 10% to 20% 101/ f; /, 05:3, [‘5 5 /, 2.00 ft? m??? [VACav 6 %& 5 Cpl/CM MW) 660% lit/“CWWJZocrﬂz) (x) = 2 55%., e ,[gemsIMk is the possible reduction in returns associated with changes in market interest
rates. 0 Prices and returns for longterm bonds are
We than those for shorterterm bonds 0 There is generally no interest—rate risk for any
bond whose time to maturity matches the holding period 13 Real and Nominal Interest Rates 0 Nominal interest rate makes no allowance
for inﬂation 0 Real interest rate is adjusted for changes in
price level so it more accurately reﬂects the w. m Whorrowing
A 0 IE? ante real interest rate is adjusted for M changes in the price level L. 0 Ex p’o‘st real interest rate is adjusted for actual
changes in the price level . k ‘ . 'I. .  l {. .,_ _
_I ,..[_. Fisher E uation: i 9—— ir + rte i = nominal interest rate
ir = real interest rate
are I expected inﬂation rate When the real interest rate is low, there are
greater incentives to borrow and fewer incentives " . (b252,) r 5‘5; 14 FIGURE 1: Real and Nominal Interest Rates
(ThreeMonth Treasury Bill), 195 3—2008 Interest Rate (%)
16 12 Nominal Rate Estimated Real Rate 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 ...
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 Fall '11
 Intro

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