Reading 28: The Term Structure and Interest Rate Dynamics Flashcards
Spot Rates
-Annualized market interest rates for a single payment to be received in the future.
Ex:
4% annual pay $1,000 face value bond given S1=5%, S2=6%, S3=7%
(40/1.05) + (40/1.06^2) + (1040/1.07^3) = $922.64
Forward Rate
-an interest rate (agreed to today) for a loan to be made at some future date.
Expected Return and bond yield
- will be equal if:
1) bond is held to maturity
2) all payments (coupon and principal) are made on time and in full
3) All coupons are reinvested at the original YTM
Forward pricing model
-values forward contracts based on arbitrage-free pricing
Ex:
Calc price two years from now for a $1, zero-coupon, three year bond given following spot rates: S2 = 4%, S5 = 6%
1/(1.04^2) = .9246 1/(1.06^5) = .7473
.7473/.9246 = .8082
$.8082 us the price agreed to today, to pay in two years, for a three-year bond that will pay $1 at maturity.
Forward Rate Model
-Investor would be indifferent between buying a five-year zero-coupon bond versus buying a two-year zero-coupon bond at maturity and reinvesting the principal for three additional years.
Ex:
(1. 06^5)/(1.04^2) = f(2,3) = 7.35%
* notice f(2,3) > s5 b/c the yield cure is upward sloping
Par Rate
- YTM of a bond trading at par.
- Par curve = collection of par rates w/ differing maturities
-Bootstrapping allows spot rates or zero-coupon rate to derived from par curve.
Bootstrapping
Maturity
1 = 1.00% 2 = 1.25% 3 = 1.50% S1 = 1.00%
100 = (1.25/1.01) + (101.25/(1+S2^2) S2 = 1.252% 100 = (1.50/1.01) + (1.50/1.01252^2) + (101.50/(1+S3^3) S3 = 1.51%
Forward Price Evolution
- if the future spot rates actually evolve as forecasted by the forward curve, the forward price will remain unchanged.
- a change in the forward price indicates that the future spot rate did not conform to the forward curve
- when spot rates turn out to be lower (higher) than the implied forward curve, the forward price will increase (decrease)
Rolling Down the Yield Curve
Ex:
Maturity = 5 Yield = 3% Price = 100 Maturity = 30 Yield = 5.5% Price = 63.67
***investment horizon of 5%
Instead of investing in 5-year bond and earning 3% , but no capital gains….the investor could purchase a 30-year bond for $63.67, hold it for five years, and sell it for $71.81, earning additional return beyond the 3% coupon
The Swap Rate Curve
- In a plain vanilla interest rate swap, one party makes payments based on a fix rate while the counter party makes payments based on a floating rate.
- Fixed Rate = swap rate
-Swap rates for various maturities = the swap rate curve
Why is swap rate curve preferred?
1) swap rates reflect credit risk of commercial banks rather than the credit risk of governments
2) Swap rate is not regulated by any govt, which makes swap rates in different countries more comparable. Govt. bond yield curves reflect sovereign risk to each country
3) Swap curve typically has yield quotes at many maturities, while the US govt bond yield curve has on-the-run issues trading at only a small number of maturities
Swap Spread
-refers to the amount by which the swap rate exceeds the yield of a government bond with the same amturity
swap spread = swap rate - treasury yield
-swap spreads are almost always positive, reflecting the lower credit risk of governments compared to the credit risk of surveyed banks that determine the swap rate.
I-spread
- Ispread for a credit-risky bond is the amount by which the yield on the risky bond exceeds the swap rate for the same maturity.
- missing swap rate can be estimated from the swap rate curve using linear interpolation
Ex: 6% Zinni, Inc bonds are currently yielding 2.35% and mature in 1.6 years
Tenor Swap Rate 0.5 1.00% 1 1.25% 1.5 1.35% 2 1.50%
1.6 falls in b/w 1.5 and 2 yr interval
Interpolated rate = rate for lower bound + (#of years for interpolated rate - #of years for lower bound rate)(higher bound rate - lower bound rate)/(# of years for upper bound - # of years for lower bound)
1.35 + (0.10*(1.50-1.35))/.5 = 1.38%
I-spread = 2.35% - 1.38% = 0.97%
***A bonds yield reflects time value as well as compensation for credit and liquidity risk, I-spread only reflects compensation for credit and liquidity risk,
Z-spread
-the spread that when added to each spot rate on default-free spot curve, makes the present value of a bond’s cash flows equal to the bond’s market price.
EX:
One year spot rate is 4%, two year spot rate is 5%. Market price of the bond with annual coupon payments of 8% is $104.12. Z-spread is:
$104.12 = ($8/(1.04 + Z)) + ($108/(1.05 + Z)^2)
z-spread = .008
***Assumes zero interest volatility…not appropriate to use to value bonds with embedded options.
TED Spread
- TED spread is the amount by which the interest rate on loans between banks (formally, three month-LIBOR) exceeds the interest rate on short-term U.S. govt debt
- Because t-bills are considered to be risk free while LIBOR reflects the risk of lending to commercial banks, TED spread is seen as an indication of the risk of interbank loans.
- A rising TED spread indicated that market participants believe banks are increasingly likely to default on loans and that risk-free Tbills are becoming more valuable in comparison.
