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Relationship between interest rates and maturity

(BKM - 15)

interest rates increase as maturity increases


Yield curve

(BKM - 15)

relationship between YTM and maturity


Uses for the yield curve (2)

(BKM - 15)

1. bond valuation
2. to gauge expectations for future interest rates against the market


Spot rates

(BKM - 15)

YTM on zero-coupon bonds (for the given duration)


General relationship between individual coupon values and total bond value and arbitrage opportunities if this relationship is violated (2)

(BKM - 15)

sum of individual coupon values should equal the total bond value

if it does not, arbitrage opportunity exists
1. bond stripping
2. bond reconstitution


Bond stripping

(BKM - 15)

if bond price < sum of individual coupon values investors can buy the bond then strip each coupon payment into stand-alone zero-coupon bonds and sell (resulting in an arbitrage opportunity)


Bond reconstitution

(BKM - 15)

if bond price > sum of individual coupon values investors can buy the individual zero-coupon bonds then re-assemble them into a coupon bond and sell (resulting in an arbitrage opportunity)


Pure yield curve

(BKM - 15)

yield curve for zero-coupon bonds


On-the-run yield curve

(BKM - 15)

yield curve for recently issued coupon bonds selling at or near par value


Short rates

(BKM - 15)

rate for a specific period length at different points in time


Expected future short rate formula

(BKM - 15)

(1 + r(n)) = ((1 + y(n))^n) / ((1 + y(n-1))^(n-1))

y(n) = YTM for n-period maturity
r(n) = short rate


Forward rate description

(BKM - 15)

"break-even" interest rate that forces identical returns b/w an n-period zero-coupon bond and an (n-1) period zero-coupon bond rolled over into a 1-yr bond in year n


Forward rate formula (3)

(BKM - 15)

(1 + f(n)) = ((1 + y(n))^n) / ((1 + y(n-1))^(n-1))

(1 + f(n)) = price of (n-1) yr zero-coupon bond / price of n-yr zero-coupon bond

forward rate = expected future short rate + liquidity premium


Reason that the forward rate does not necessarily equal the future short rate

(BKM - 15)

interest rate uncertainty


Liquidity premium definition & formula

(BKM - 15)

compensation for uncertainty in bond price due to changes in short rates demanded by investors

liquidity premium = forward rate - expected future short rate


Expected holding period return (HPR)

(BKM - 15)

expected HPR = change in price / original price


Liquidity premiums desired by short-and long-term investors

(BKM - 15)

short-term: positive liquidity premium
long-term: negative liquidity premium

(to invest in short-term bonds)


Reasons forward rates can be high (2)

(BKM - 15)

1. investors expect rising interest rates
2. large liquidity premium required for holding longer-term bonds


Theories of interest rate term structure (2)

(BKM - 15)

1. expectations hypothesis
2. liquidity preference theory


Expectations hypothesis assumptions (3)

(BKM - 15)

1. the forward rate = expected future short rate
2. liquidity premiums are 0
3. upward-sloping yield curve indicates expectation of increasing interest rates


Liquidity preference theory

(BKM - 15)

assumes short-term investors dominate the market, so in general, the forward rate > expected short rate resulting in a positive liquidity premium, on average


Implication of liquidity preference theory

(BKM - 15)

means that an upward-sloping yield curve does not mean there is an expectation of increasing interest rates

>> possible to have an upward curve w/declining expected future short rates if the liquidity premium is increasing more than expected future short rates are decreasing


Relationship between yield curve and forward rates

(BKM - 15)

upward-sloping yield curve must indicate an increase in forward rates (which consist of expected short rates and liquidity premiums)


Problems with constant liquidity premium assumption (2)

(BKM - 15)

1. difficult to obtain precise estimates of liquidity premiums
2. no reason to assume liquidity premiums are constant


Nominal interest rate

(BKM - 15)

nominal interest rate = real interest rate + inflation rate


Causes for changes in interest rates (3)

(BKM - 15)

1. increase in nominal interest rates can be driven by real rate increases or changes in inflation
2. rapidly expanding economy generally leads to an increase in rates
3. supply-side shocks can lead to an increase in rates (e.g. interruptions in oil supply)


Forward contracts

(BKM - 15)

arrangements that effectively lock in a future interest rate


Synthetic forward loan (investor = borrower)

(BKM - 15)

buy 1-yr zero-coupon bond for initial CF = - B(1)
sell (1 + f(2)) 2-yr zero-coupon bonds for CF = B(2) * (1 + f(2))

total initial CF = 0


When is it advantageous to lend and borrow a synthetic forward loan?

(BKM - 15)

lend: if believe interest rates will fall

borrow: if believe interest rates will rise