Time-Varying Rates of Return
and the Yield Curve
EC 1745
Borja Larrain
Page 1 of 1
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9/29/2008
Today:
1. Time-Varying Rates of Return.
2. Bonds and the Yield Curve.
3. Sensitivity to Changes in Interest Rates (Duration).
Readings: Chapter 5 Welch
Assignment 2 due in one week (10/06).
Goals of this class:
• Relax one simple assumption—that the interest rate
is constant in time—and see what happens.
• Important: all formulas apply, but with more painful
notation.
• Understand bonds and their IRRs (or YTM).
• Understand what the yield curve represents and what
happens when it moves.
1
Time-Varying Rates of Return
• Let’s continue to assume perfect capital markets and
no uncertainty.
• However, what if the interest rate applicable to the
second year flow is different from the interest rate
applicable to the third year flow?
• Basically, what I’m saying is what if:
1 + r0,t 6= (1 + r)t
• For example, if there are 3 periods and 3 different
rates:
³
1 + r0,3 = 1 + r0,1
´³
1 + r1,2
´³
1 + r2,3
´
• Which implies that the NPV of a project is now:
NP V
C1
C2
C3
´+³
´+³
´
= C0 + ³
1 + r0,1
1 + r0,2
1 + r0,3
C1
C2
´+³
´³
´
= C0 + ³
1 + r0,1
1 + r0,1 1 + r1,2
C3
´³
´³
´
+³
1 + r0,1 1 + r1,2 1 + r2,3
• More formally,
⎡
⎤
⎢
⎥
⎢
⎥
∞
X⎢
⎥
Ct
C
t
⎢
⎥
=
NP V =
⎢ t
⎥
Y
1
+
r
⎢
⎥
0,t
t=1
t=1 ⎣
(1 + rj ) ⎦
j=1
∞
X
where rj is an abbreviation for rj−1,j .
1.1
Annualized Rates of Return
• How can we compare rates of return over different
horizons? For example, what is bigger: 100% over 15 years
or 1% over 3 months?
• Analogous to ask: what is faster: 55 mph or 1,500
miles per day?
• Need to express all in the same units to make comparisons.
• It is standard to annualize interest rates to compare
them. For example, if the 1st year interest rate is 5% and the
2nd year interest rate is 3%, what is the annualized interest
rate?
q
(1 + 5%) (1 + 3%) − 1 = 3.99% ≈ 4%
• An annualized rate of return is like an "average" rate.
(For those of us who are nerds: it is a geometric
average.)
• More formally, if we are working with a multi-year
project:
q
1 + r0,t − 1 =
r³
´
³
´
1 + r0,1 ... 1 + rt−1,t − 1
= rannual
1.2
Inflation: Real and Nominal Rates
• One source of change in rates of return is inflation.
• So far we have assumed that $1 today buys you the
same stuff that $1 tomorrow. But what if there is
inflation? 40 years ago our parents were able to buy one
ticket to the movies for $1, nowadays you can buy only 1/10
of a ticket for the same money.
• A real cash flow is adjusted for inflation. A real dollar
always has the same purchasing power. For example,
Manhattan was bought for $24 in 1626. Just accounting for
the purchasing power of money, how much would that amount
buy you today? (Almost $3 billion with 5% inflation per year).
• Most contracts, at least in the U.S., are established
in nominal terms (not-inflation adjusted or just plain
dollars).
• Intuitively, inflation subtracts from the return of a
project. Inflation destroys purchasing power of dollars in the future.
• Example:
— C0 = 40, C1 = 60 implies a return of 50%.
— But what if $60 tomorrow buy you the same stuff
that $50 buy you today (think of the movie ticket
example). Then the real return is 50/40 − 1 =
25%.
— In this example, the loss in purchasing power of
$1, or the inflation rate, is 60/50 − 1 = 20%.
• More formally, if π is the inflation rate:
1 + rnominal = (1 + rreal) (1 + π)
1 + 50% = (1 + 25%)(1 + 20%)
• Or,
1 + rreal =
1 + rnominal
1+π
• Where is the minus sign we were expecting? (I said
inflation "subtracts" from your return). Well, for
small rates, the above formula can be approximated
quite accurately by:
rreal ≈ rnominal − π
• Real cash flows (with constant purchasing power)
are obtained multiplying cash flows by 1/ (1 + π) .
Example: if inflation is 5%, $110 next year are equivalent to
110/1.05 = $104.77 today
• Inflation changes in time because of business cycles
and the Fed’s actions. So, real returns fluctuate as
the purchasing power of the cash flows promised by
a project change.
1.3
Inflation and NPV
• Important: to get the right NPV you can either discount nominal dollars with nominal rates, or real dollars with real rates, but never mix.
• Example (try these at home without looking at the
answers):
— A project will return $110 in nominal cash next
year. The cost of capital is 10%. What is the
PV? A: 110/1.10 = 100.
— A project will return $110 in nominal cash next
year. The inflation rate is 4%. What is the purchasing power of the future $110 in today’s real
dollars? A: 110/1.04 = 105.77.
— The inflation rate is 4%. The cost of capital
is 10%. What is the real cost of capital? A:
(1.10/1.04) − 1 = 5.77%.
— What is the project’s real dollar value discounted
by the real cost of capital? A: 105.77/1.0577 =
100. Both ways of computing the NPV give the
same answer.
2
Bonds and the Yield Curve
2.1
The U.S. Treasuries Market
• This is basically a market where the U.S. government
sells annuity-like instruments of different maturities.
• Names: Bills (0-0.99y), Notes (1y-10y), Bonds (10y30y). I’ll usually just say bonds.
• This market is one of the most important markets in
the world:
— The outstanding amount in 2008 was > $9.5 trillion.
— Annual trading is $80-$100 trillion (Turnover =
10 times!)
— Foreigners are active participants in this market.
• This market is close to our definition of "perfect":
— Very low transactions costs.
— Few differences of opinion.
— Deep market, lots of buyers and sellers.
• Moreover, no uncertainty about payments (When is
the U.S. government going to default? Armageddon?).
2.2
Basic Concepts
• Bonds pay a "principal" or "face value" at the end.
• Most bonds pay a "coupon" at regular time intervals
(e.g., semi-annually), an x% of the principal. This is
the part that looks like an annuity.
• The PV of a bond with maturity T , coupon rate c,
principal F (for "face" value), and at the interest
rate of r is:
PV
T
X
F
cF
=
+
t
(1
+
r)
(1 + r)T
t=1
"
#
cF
1
F
=
1−
+
r
(1 + r)T
(1 + r)T
|
• Be careful:
{z
"annuity"
}
| {z }
PV of principal
— the coupon rate is NOT the interest rate applicable to this bond.
— the coupon rate is NOT the IRR of the bond.
• The IRR of a bond is the rate that equates the PV
of future payments with today’s price observed in
the market. In the case of bonds the IRR is called
"Yield to Maturity" (YTM). If P is the price of a
bond, then the YTM (y) is defined by the following
formula:
0 = −P +
"
#
1
F
cF
1−
+
y
(1 + y)T
(1 + y)T
• In some sense the YTM is the inverse of the price
of a bond: expensive bonds have low yields, cheap
bonds have high yields (all else equal).
• The YTM is the annualized return you get from buying the bond today and holding until maturity (hence
the name YTM).
• Probably the simplest financial instrument in the world
is a "zero-coupon" Treasury bond. These pay only
principal and no interim payments or coupons.
µ
¶
F 1/T
−1
y0-coupon =
P
2.3
The Yield Curve
• The "yield curve" or "term structure of interest rates"
is the graph that relates the YTM with the maturity
of different bonds, i.e., a graph of y and T.
• We know, from the previous class, that the YTM or
IRR is an annualized rate of return. In other words,
it is a per-period rate so we are not comparing apples
and oranges when we look at the YTM of two bonds
of different maturities. However, keep in mind that
these are per-period rates of investments of different
horizons (or maturities).
• The yield curve can have different shapes, although
the most common is upward sloping.
• Folk wisdom (aka Wall Street wisdom) says that a
downward-sloping yield curve precedes a recession.
US TREASURY YIELD CURVE
My opinion: maybe, but remember 2 things: (1)
preceding or predicting is not the same as causing,
and (2) if you actually look at the data this is not
always true.
2.4
Spot and Forward Rates
• A spot interest rate is the prevailing interest rate for
an investment starting today.
• A forward interest rate is the interest rate that will
be applicable to an investment that starts in some
future period.
• Does the yield curve tell us something about forward
rates? Yes, we’ll see how.
• Some more notation before we start: rt̄ is the annualized rate of r0,t, which, remember, was the holding
period interest rate. These two are related by:
(1 + rt̄)t ≡ 1 + r0,t
1 + rt̄ ≡
³
1 + r0,t
´1/t
• An annualized return is like an average. A holdingperiod return is like a sum.
• Example:
(1+r5̄)5 = 1+r0,5 = (1+r0,1)(1+r1,2)...(1+r4,5)
• The interest rate from, say, t = 1 to t = 2 is called
a 1-year forward rate.
• In the world of perfect certainty in which we are
working, the forward rate is the future spot rate. We
all know what’s going to happen, although it hasn’t
happened yet.
• Practice:
Spot/Forward
r0,1 = 5%
r1,2 = 10%
r2,3 = 15%
Holding
r0,1 =
r0,2 =
r0,3 =
Annualized
r1̄ =
r2̄ =
r3̄ =
• Compute these with the formula above. But before
let’s get the intuition:
— The three holding rates are roughly 5%, 15%,
and 30%...it’s like a sum.
— The three annualized rates are roughly 5%, 7.5%,
and 10%...it’s like an average.
• Conversely, if I give you r1̄, r2̄, and r3̄ you can back
out the spot/forward rates and the holding-period
returns.
• The yield curve is a set of annualized returns (r1̄,r2̄, r3̄,
etc.). So if I give you the complete yield curve you
can figure out the holding-period returns and forward
(future spot) rates.
• For econ nerds like myself: since the annualized rate
is the average, it can help to think of the forward
rate as the "marginal". If the average is going up,
then the forward has to be higher than the average
and vice versa if the average is going down.
• So, an upward-sloping yield curve implies higher future spot interest rates? This is what this simple
theory says. In practice it is more complicated, but
this intuition is not all bad.
• If the yield curve is upward-sloping, should we buy
longer-term bonds because they offer higher returns?
Not necessarily. All bonds are fairly priced.
• Example: the 1-yeer bond earns an annualized return
of 5%, the 2-year bond 10%, and the 3-year bond
15%. Is the 3-year bond a better investment if you
plan to sell next year?
— The cost of the 3-year bond today (per $1,000
of face value): 1, 000/1.153 = 657.52.
— The forward rates (future spot rates): r1,2 =
1.102/1.05−1 = 15.24%; r2,3 = 1.153/1.102−
1 = 25.69%.
— The 3-year bond will cost next year: 1, 000/(1.1524·
1.2569) = 690.45.
— What return does the 3-year bond give me from
today until t = 1: 690.45/657.52 − 1 = 5%.
— You earn the same return by buying long and
short bonds in this world with no uncertainty.
3
Sensitivity to Changes in Interest Rates
• What happens with the value of a bond when interest
rates go up? Does it depend on the maturity of the
bond?
• Imagine a set of zero-coupon bonds with face value
$100 and of different maturities.
• A 30-year bond discounted at 8% has a price today
of 100/1.0830 = $9.94.
• If the interest rate suddenly increases by 10 bp, the
price falls to 100/1.081030 = $9.67
• The holding period return is $9.67/$9.94 − 1 =
−2.74%. You lost $2.74 per $100 invested in the
blink of an eye!
• For a 1-year bond the same calculation gives: p1/p0−
1 = $92.50/$92.59 − 1 = −0.09%. You lost only
9 cents per $100.
• Bottom line: the interest rate sensitivity of longermaturity bonds is higher than the sensitivity of shortermaturity bonds.
• In this sense, longer bonds are "riskier" than shorter
bonds. But we haven’t discussed risk premium yet...next
few classes.
Read the appendix of Ch 5 about
Duration (It will be in the assignment too)