Library/CFA (Chartered Financial Analyst)/JuiceNotes 2024 Fixed Income/The Term Structure of Interest Rates: Spot, Par, and Forward Curves

The Term Structure of Interest Rates: Spot, Par, and Forward Curves

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Maturity Structure and Yield Curve Shapes5 min
The maturity structure of interest rates relates the yields of bonds with identical credit risk to their time to maturity. The graphical depiction is the yield curve. A 'normal' yield curve is upward sloping, implying longer-term bonds have higher yields. A 'flat' curve indicates similar yields across maturities, while an 'inverted' curve shows short-term yields exceeding long-term yields. Spot rates are used to construct the spot curve, often derived from on-the-run government bonds. Interpolation is necessary to fill gaps in maturity data.

Key Points

  • Yield curve relates yield to maturity for similar credit quality bonds.
  • Normal curve: Upward sloping (Long-term rate > Short-term rate).
  • Inverted curve: Downward sloping (Short-term rate > Long-term rate).
  • Spot rates are zero-coupon rates used for specific cash flow discounting.
Bond Pricing with Spot Rates5 min
Pricing a bond using spot rates involves treating each cash flow as a standalone zero-coupon bond. Instead of a single Yield to Maturity (YTM), each coupon and principal payment is discounted by the specific spot rate for its time horizon (e.g., Year 1 cash flow at the 1-year spot rate, Year 2 cash flow at the 2-year spot rate). The sum of these present values equals the bond's no-arbitrage price. This method is more precise than using a single aggregate yield.

Key Points

  • Bonds are priced as a package of zero-coupon instruments.
  • Arbitrage-free price is the sum of PVs of cash flows using spot rates.
  • Spot rates remove the reinvestment assumption inherent in YTM.
Par Rates and Bootstrapping6 min
The par rate is the coupon rate required for a bond to trade at 100 percent of its face value. The par curve is constructed from these rates. Par rates are essential for term structure analysis as they normalize bonds to a common price point. Par rates are calculated using discount factors derived from the spot curve. The formula is $Par Rate = (1 - Z_n) / \Sigma(Z_i)$, where $Z$ represents the discount factor for each period.

Key Points

  • Par rate makes PV of cash flows equal to Par Value.
  • Par curve is a sequence of yields-to-maturity for par bonds.
  • Par rates are derived from spot rates via discount factors.
Forward Rates and Curve Relationships6 min
Forward rates are interest rates applicable to a future time interval. For example, '2y1y' refers to a 1-year rate starting 2 years from today. Forward rates are implied by the gap between spot rates of different maturities. The relationship between spot, par, and forward curves is strictly defined by the curve's slope: in an upward-sloping market, the forward curve sits above the spot curve, which sits above the par curve. In an inverted market, the order is reversed.

Key Points

  • Forward rates are implied future rates derived from current spot rates.
  • Notation 'AyBy': A = start time from today, B = tenor.
  • Upward Curve: Forward > Spot > Par.
  • Inverted Curve: Par > Spot > Forward.

Questions

Question 1

What does an upward-sloping yield curve indicate regarding the relationship between short-term and long-term yields?

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Question 2

Which curve represents the yields of bonds trading at 100 percent of their face value?

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Question 3

In the context of forward rates, what does the notation '3y1y' typically represent?

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Question 4

Given the following spot rates: Year 1 = 4 percent, Year 2 = 5 percent. What is the value of a 2-year annual pay bond with a 6 percent coupon and a Face Value of 100?

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Question 5

If the spot curve is downward sloping (inverted), which of the following orderings is correct?

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Question 6

Calculate the 1-year forward rate, 1 year from now (1y1y), given the following spot rates: 1-year spot = 5 percent, 2-year spot = 6 percent.

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Question 7

What is the primary characteristic of a 'flat' yield curve regarding the relationship between Spot, Par, and Forward curves?

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Question 8

Spot rates are also commonly referred to as:

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Question 9

Calculate the price of a 3-year, 5% annual coupon bond (Face Value 100) using the following spot rates: Year 1 = 2%, Year 2 = 3%, Year 3 = 4%.

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Question 10

Which rate is considered a 'breakeven reinvestment rate'?

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Question 11

Given 1-year spot rate = 3% and 2-year spot rate = 4%. What is the implied 1-year forward rate 1 year from now?

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Question 12

If the yield curve is upward sloping, the forward curve will be:

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Question 13

The par rate is essentially a:

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Question 14

Which technique is used to derive zero-coupon rates from the prices of coupon-paying bonds?

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Question 15

Given: 1-year spot = 4%, 2-year spot = 5%, 3-year spot = 6%. Calculate the 3-year Par Rate.

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Question 16

Using the '2y1y' forward rate notation, how long is the forward period (tenor)?

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Question 17

Given 2-year spot rate = 8% and 5-year spot rate = 15%. What is the implied '2y3y' forward rate?

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Question 18

If the spot curve is inverted (downward sloping), which yield is the highest?

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Question 19

Why are par rates derived for hypothetical government bonds used for term structure analysis?

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Question 20

Calculate the price of a Zero-Coupon bond with 5 years maturity and Face Value 100, if the 5-year spot rate is 10%.

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Question 21

What is the relationship between the Spot Curve and the Par Curve when the yield curve is flat?

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Question 22

The 1y1y rate is 8% and the current 1-year spot rate is 4%. What is the approximate 2-year spot rate?

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Question 23

Which curve is typically constructed using on-the-run government bonds?

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Question 24

Calculate the value of a bond with annual coupons of 10%, 2 years maturity, Face Value 1000, using Spot rates: 1yr = 5%, 2yr = 6%.

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Question 25

If 1-year spot = 5% and 1y1y forward = 5%, what is the 2-year spot rate?

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Question 26

What is the discount factor for year 2 if the 2-year spot rate is 6%?

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Question 27

Which curve is typically 'below' the spot curve in an upward-sloping market?

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Question 28

Calculate the 2-year spot rate given: 1-year spot = 3%, 2-year par rate = 4%.

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Question 29

A '3y1y' forward rate refers to:

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Question 30

In a 'normal' market conditions scenario, the yield curve is:

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Question 31

Calculate the 1-year discount factor given a 1-year spot rate of 5%.

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Question 32

Given 3-year spot rate = 7% and 4-year spot rate = 8%. What is the implied 1-year forward rate 3 years from now (3y1y)?

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Question 33

Interpolation is used in yield curve construction to:

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Question 34

If the spot curve is flat at 5%, what is the 1-year Par Rate?

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Question 35

Which bond pricing method provides an arbitrage-free value?

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Question 36

What is the 3-year Spot Rate if the 1-year rate is 2%, 2-year rate is 3%, and a 3-year annual par bond (coupon=4%) is priced at 100?

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Question 37

If a yield curve is downward sloping, what is likely true about future interest rate expectations?

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Question 38

Calculate the implied 2-year forward rate 1 year from now (1y2y) given: 1-year spot = 3%, 3-year spot = 5%.

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Question 39

The term 'Maturity Effect' generally implies that for the same change in yield:

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Question 40

In the notation '0y1y', what does this rate essentially represent?

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Question 41

Which of the following describes the relationship between Spot (S) and Forward (F) rates when the yield curve is upward sloping?

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Question 42

Calculate the 2-year Par Rate given: 1-year spot = 5%, 2-year spot = 6%.

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Question 43

Which rate is used to discount cash flows when valuing a bond using forward rates?

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Question 44

Given 1y1y = 8%, and 1-year spot = 4%. The 2-year spot rate is approximately:

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Question 45

In a 'humped' yield curve scenario, intermediate-term yields are:

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Question 46

Given: 2-year spot = 6%, 2y1y = 8%. What is the 3-year spot rate?

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Question 47

Why do on-the-run government bonds typically trade at lower yields than off-the-run bonds?

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Question 48

If the forward curve is below the spot curve, the spot curve must be:

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Question 49

Calculate the 1-year spot rate if a 1-year par bond has a coupon of 5%.

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Question 50

Using spot rates 3% (1yr) and 4% (2yr), value a 2-year Zero Coupon Bond (FV 100).

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