Transcript For bond with a given P, Coupon and Face Value, YTM is

Jeffrey H. Nilsen
Bonds
 A firm needs to borrow – why would it prefer to take a
bank loan over issuing a bond (or vice versa) ?
 Do banks issue bonds ? Why ?
 We won’t use bond pricing in this class but we need to
have intuition for how financial instruments’ values
change when interest rates change
Questions
 What is the fundamental value of house (as asset) ?
 What is a bond’s yield to maturity (YTM) ?
 How does YTM differ from holding period yield ?
 Given a bond P, coupon and face value, what does the
YTM do?
 What is the YTM on a discount bond ?
 For a $100 face value with P of 95$ ?
Valuing An Asset
 The fundamental value of an asset today is the present
value of its future cash flows
 Your mortgage loan is asset to lender with
PV(expected future interest & principle payments you
make)
 Your house is an asset to you with value =
PV(consumption services + expected sales price)
Yield to Maturity
 Can measure YTM for all bonds, including zero coupon bonds
 For bond with a given P, Coupon and Face Value, YTM is the
rate equating bond’s value today (its P) with PV(cash flows until
maturity)
 if hold bond to maturity, then indeed YTM = return
 If sell prior to maturity, then (holding) return either greater or
smaller than YTM (changing interest rates affect value of bond)
Coupon Bond Questions
 What is the YTM on a coupon bond ?
 If the coupon rate = discount rate, what is Price of
Bond ?
 If you sell a bond prior to maturity, under what
conditions will you receive the YTM?
Coupon Bonds
(e.g. Treasuries, Corporate Bonds)
 Coupon bond (annuity): pays regular fixed coupons and
repays F (face value) at maturity
 Coupon rate = C/F
P0 
C
C
F2


2
1  i  1  i  1  i 2
 YTM equates P0 to PV(C) + PV(F)
 Fact: bond with coupon rate = discount rate has P0 = F
 E.g. 10%, 2-period bond with F = 1000, C = 100 & P0 = 1000
Bond Facts
 Coupon payments are fixed when you buy the bond
 While you hold the bond, market interest rates change
(changing the discount rate) and the value of the
bond
 If you sell the bond before it matures, you may NOT
earn the “yield to maturity”
 Your “holding period return” is the return you receive for the time
you have held bond
Coupon Bond Sold prior to
maturity questions
 You sell a 2 year, 10% $1000 face value bond after
receiving the 1st coupon, with 1 year remaining to
maturity. Interest rates have risen to 20%, what is the P
you sell it for?
Interest Rate vs.
Holding (or Actual) Return
Assume you must sell a 2-year bond you bought last year
(year 0). It has 1 year remaining until maturity
Holding Return = capital gain + CF earned holding bond
= (P sold – P bought) + (bond coupon)
rHold 
P1  P0   C
P0
Earned 1 coupon,
another remains
But what is Pt+1 (the price you sell it at) ??
Holding Return Example
(assume mkt rates rise to 20% in year 1)
You buy 2Youryr bond 100at P0 100
= 1000
1000, C = 100 & F = 1000, so YTM 10%
Bond
At yr 0
0
1
2
Sell in YR 1: Your 1 year bond has only one $100 coupon left
Your bond must compete with new 1
year bonds paying 20% coupon.
P1NEW 
200 1000

 $1000
1.2
1.2
To find your bond’s P1, discount
remaining CF at new market rate
P1YOUR 
100 1000

 $916
1.2 1.2
The buyer pays lower P1 vs. new bond. Your bond will give her
same return as new bond’s
Holding Return Example
If rates rise after you’ve bought a bond
you’ll take a capital loss.
ret 
P1  P0   C  916  1000  100  1.7%
P0
Your holding return is less than the YTM
1000
Term Structure of Interest Rates
(TS)
 Relation between interest rates of bonds differing only by term to
maturity
 Bonds identical in default risk, liquidity, and tax payments
 FT term structure TODAY
http://video.ft.com/4734867703001/US-recession-risk/editorschoice
Why Is Term Structure (TS) Interesting?
 It’s crystal ball: rise in slope of TS predicts higher economic
activity 4 quarters in future
 But how ?
From C. Harvey (1995)
Expectations Theory
 Explains how TS predicts future economic activity
 “pure” ET assumes short term and long term bonds are
perfect substitutes (i.e. saver chooses short or long
bond based only on return offered)
Expectations Theory
Interest rate on long bond equals avg. expected short rate over life of long bond
 Saver with $1 has 2-year horizon, selects between:
 Rollover short: Buy 1 yr bond, rollover at estimated t+1 1 year bond
interest rate
e
e
t
t+1
it
iet+1
 Long: Buy 2-yr bond:
t
t+1
1  it 1  it 1   1  it  it 1  it  ite1
1  i 1  i   1  2  i
2
t
2
t
2
t
 it2  it2
i2t
 Saver will switch to whichever bond offers higher return => yields
will be equal
it  ite1
2
 it 
2
ET => today’s long rate fixed by
expected future short rates
 Assume current 1-yr bond rate 6%
 Assume saver expects next year’s 1-yr bond
rate to be 8%
 So 2-yr bond rate must be
it2 
it  ite1
2
(6% + 8%)/2
= 7% per annum in order for saver to buy it
BUT: future short term rates can’t
Be observed on today’s market !!
Use ET to Infer
Future Short Term Rates
i2, 0
i0  E0 i1

2
 For 2 periods, today is time 0
 E0 i1 not observable
 If observe i2,0 > i0 pure ET predicts E0 i1 > i0 => market expects short
rate to rise in future
 But TS usually up-sloping (how can it be ?? How can rates usually be
expected to rise ??)
Segmented Markets Theory
 Theory assumes short & long term instruments
are not substitutes at all => separate markets so
short and long rates set independently
 Can explain consistently higher LT rates
 But can’t explain why all rates move together
Preferred Habitat
(Liquidity Premium) Theory
 Savers lend long only if receive term premium => TS
normally up-sloping
10
10, 0
i
i0  E0 i1  ...  E0 i10

 10,0
10
(10 year bond)
 where θ is premium savers are paid for lending long
 If LT < ST rates => inverted TS indicates steep fall in
expected short-term rates swamping θ
 But θ fluctuates over time; Shiller (1990): “...little
agreement on how term premium is affected by M policy!"
How ET predicts Future GDP
(via M Policy):
 (assume TS initially up-sloping)
 Fed tightens M policy: current short rates rise.
Y falls in short run
 M policy neutral in long run => future short
rates unaffected
 ET => short rate rise > long rate rise (TS
becomes more flat)
 Summarizing: contracting M
 (iLong – iShort) falls (slope falls)
 GDP slows
 => TS & GDP positively related
maturity