Fixed Income

Question 1

What is the term structure of interest rates?

Answer 1

is how the interest rate evolves with time or what the interest rates are across different time maturities.

Question 2

What is a Spot rate in interest rate term structure?

Answer 2

A spot rate is the interest rate on a security that makes a single payment at a future point in time.

Question 3

What is the discount factor?

Answer 3

The price or the present value of a risk-free single unit payment after N periods is called the discount factor.

Question 4

How to calculate discount factor?

Answer 4

It’s the PV, so DF = 1 / (1 + Zn)^N where Zn = spot rate (YTM).

Question 5

What is the Discount Function?

Answer 5

It’s a combination of Discount factors (PV) for a range of maturities.

Question 6

What is a spot curve?

Answer 6

It shows the annualized return on an ‘option-free’ and ‘default-risk-free’ zero-coupon bond with a single payment at maturity, for various maturities.

Question 7

What is a Forward Rate?

Answer 7

An interest rate that is determined today for a loan that will be initiated in the future time period.
fa,b-a is how we write the forward rate. (small ‘f’ represents the rate)

Question 7

How is a forward contract price represented?

Answer 7

Fa,b-a represents the forward contract price(PV) for a zero-coupon bond with unit principal which starts in the future (at time a and ends at time b).

Question 7

What is the forward pricing model?

Answer 7

DFb = Fa,b-a * DFa to prevent any arbitrage opportunity.

Question 8

What is the forward rate model?

Answer 8

Instead of calculating Forward prices, we calculate the forward rates.
Simple maths (expand the formula for Forward price!).

Question 9

What are the implications of the Forward Rate model?

Answer 9

When the spot curve is upward sloping, the forward curve will lie above the spot curve & when the spot curve is downward sloping, the forward curve will lie below the spot curve.

Question 10

What is a par curve?

Answer 10

Basically a par curve represents the YTM (like spot rates) of a government bond which pays coupons over a range of maturities, priced at par. (like discount function, it’d have a par function).

Question 11

What is Bootstrapping?

Answer 11

A process which can be used to construct a spot curve from a par curve.

Question 12

What is YTM?

Answer 12

YTM is the single discount rate when applied to the bond’s promised cash flows, equates those cash flows to the bond’s market price.

Question 13

How is YTM related to spot rates?

Answer 13

YTM is the weighted average of spot rates used in the valuation of the bond.

Question 14

How is YTM calculated?

Answer 14

Using spot rates and finding the discount factor, then with the DF we can come up with YTM.

Question 15

Is YTM a good estimate of expected returns, yes or no? Why?

Answer 15

It’s not, it can be a poor estimate of expected return when -
Interest rates are volatile, so coupons can be reinvested in different rates.
Risk of default and also the bond can have embedded options(put, call or convert) in it.

Question 16

What is the strategy Riding the yield curve or Rolling down the Yield curve?

Answer 16

You expect the spot rates to remain static! So, they don’t change when you bought a bond, then the price of the bond will be higher if you buy a bond with longer maturity!
The returns will be higher if the gap between the spot curve and forward curve is higher and also more returns the longer the maturity.

Question 17

What is a swap rate?

Answer 17

is the interest rate for the fixed-rate leg of an interest rate swap. The floating rate can be a market reference rate like LIBOR.

Question 18

What is swap rate curve?

Answer 18

Swap curve or swap rate curve is the yield curve of swap rates.

Question 19

Why is the swap market highly liquid?

Answer 19

A) a swap doesn’t have multiple borrowers or lenders, only counterparties who exchange cash flows.
B) swaps provide one of the most efficient ways to hedge interest rate risk.

Question 20

When to use government spot rates for bond valuation and when to use swap rates? What factors affect the choice of which to use?

Answer 20

a major factor which contributes to which one to use is relative liquidity of both markets.
Organisations like wholesale banks frequently use swap curves for valuation, as they have many items on their balance sheet which they hedge with swaps.
Retail banks on the other hand have little exposure to the swap market and use government spot curve as a benchmark.
So it depends upon the organisation to decide which is the best curve to use for bond valuation.

Question 21

What is a swap price?

Answer 21

The swap price is the fixed rate of interest in the swap.

Question 22

How do we calculate a swap price?

Answer 22

We solve for a constant fixed rate that sets the PV of the fixed leg payments equal to the PV of the floating-leg payments over the life of the swap.

Question 23

What is the PV of a floater?*

Answer 23

If there is no change in credit risk, the PV of a floater at a reset point will be the par value (1 or 100).

Question 24

Are spot rates and swap rates mathematically linked?

Answer 24

YES! the swap pricing formula shows the link. The formula is [PV of all fixed leg payments over the period of the swap(total) + par value discounted back with spot rate] = 1 (Par value of a floater at a reset date)

Question 25

What is a swap spread?

Answer 25

The spread paid by the fixed-rate payer of an interest rate swap over the rate of the "on-the-run"(most recently issued) government security with the same maturity as the swap.

Question 26

Reasons for the swap market to be more popular than government spot market?

Answer 26

A) Swap market is unregulated, so swap rates are more comparable across different countries relative to government securities. (Cross border comparison) B) Swap market has more maturities with which to construct a yield curve than do government bond markets. C) Swap spread reflects the default risk of private entities at a rating of about A1/A+ (roughly the equivalent of commercial banks)

Question 27

What is the Z-spread?

Answer 27

A more accurate measure of credit and liquidity risk is called the Zero spread(Z-spread). It's the constant basis point spread that would need to be added to a government (or interest rate swap) spot curve so that the discounted cash flows of a bond are equal to its current market price.

Question 28

What is the TED Spread?

Answer 28

It's an indicator of perceived credit risk in the general economy. Difference between LIBOR and the yield on a T-Bill of matching maturity. An increase in the TED spread indicates a greater perceived credit and liquidity risk.

Question 29

What is LIBOR-OIS spread?

Answer 29

It's difference of LIBOR and OIS (overnight indexed swap) rate. OIS is a swap in which the floating rate is the geometric average of an overnight index rate (typically the rate for overnight unsecured lending between banks - the federal rate for US dollars). LIBOR-OIS spread is considered an indicator of the risk and liquidity of money market securities.

Question 30

What is SOFR?

Answer 30

Secured overnight financing rate (SOFR) is the overnight cash borrowing rate collateralized by US treasuries and will replace LIBOR in the future.

Question 31

What do the traditional theories of the term structure try to explain?

Answer 31

They try to explain the shape of the yield curve.

Question 32

What is the unbiased expectations theory?

Answer 32

It's pure expectations theory. Forward rate is an unbiased predictor of the future spot rate. Can be used to explain any shape of the yield curve. Does not consider risk. -> Investors are risk neutral.

Question 33

What is the local expectations theory?

Answer 33

More rigorous than the unbiased expectations theory. Expected return for every bond over short time periods is the risk-free rate. Bond pricing doesn't allow for traders to earn arbitrage profits. No risk premium for short time periods. However, accommodates risk premium for long time periods.

Question 34

What is the liquidity preference theory?

Answer 34

asserts that liquidity premiums exist to compensate investors for the added interest rate risk they face when lending long term and these premiums increase with maturity. expectation of unchanging short-term spot rates, liquidity preference theory predicts an upward-sloping yield curve. Forward rate provides an estimate of the expected spot rate that is biased upward by the amount of the liquidity premium. implies that the yield curve will typically be upward sloping.

Question 35

What is the segmented markets theory?

Answer 35

Assumes that the market is segmented where each maturity sector is considered a 'market segment'. Different credit categories such as investment grade vs high-yield can also be considered market segments. Thus, yields are solely a function of the supply and demand for bonds in a particular 'market segment'.

Question 36

What is the preferred habitat theory?

Answer 36

Similar to the segmented markets theory in proposing that many borrowers and lenders have strong preferences for particular maturities but it states yields at different maturities are determined independently of each other. if expected additional returns to be gained are large enough, then institutions will be willing to deviate from their preferred maturities or habitats. Agents and institutions will accept additional risk in return for additional expected returns.

Question 37

What is shaping risk?

Answer 37

The shape of the yield curve changes continually and yield curve shifts are rarely parallel. Shaping risk is defined as the sensitivity of a bond's price to the changing shape of the yield curve. Shaping risk also affects the value of many options, which is very important because many fixed-income instruments have embedded options.

Question 38

How are yield curve movements described?

Answer 38

by a combination of three independent movements: A) Level B) Steepness C) Curvature

Question 39

What is yield volatility?

Answer 39

The term structure of interest rate volatilities is a representation of the yield volatility of a zero-coupon bond for different maturities.

Question 40

How do we measure yield curve risk?

Answer 40

The volatility curve or volatility term structure measures yield curve risk.

Question 41

Why is it important to measure yield curve risk?

Answer 41

A) Most fixed-income instruments and derivatives have embedded options B) Fixed-income interest rate risk management.

Question 42

Imp points about yield curve volatility?

Answer 42

Short term rates are more volatile than long-term rates (Due to changes/effects of monetary policies)

Question 43

How do we manage yield curve risks?

Answer 43

We do that using a measure called Duration.

Question 44

What is Effective Duration vs Key Rate Duration?

Answer 44

Effective Duration measures the sensitivity of a bond's price to a small parallel shift in a benchmark yield curve. Key Rate Duration measures a bond's sensitivity to a small change in a benchmark yield curve at a specific maturity segment.

Question 45

How to calculate KeyRate Duration to measure yield curve risk?

Answer 45

KeyDurFULL = -KeyDur1*z1 - KeyDur5*z5 - KeyDur10*z10

Question 46

What is Bond risk premium?

Answer 46

Expected excess return of a default-free long-term bond (T-bill 10 years) less the expected excess return of an equivalent short-term bond (1 year T-bill) or the one-period risk-free rate. It is a forward-looking expectation and must be estimated.

Question 47

Which factors explain short term and intermediate term bond yield?

Answer 47

Inflation explains about 2/3rds and rest is explained by economic growth factors & monetary policy. For long term bonds, Monetary policy explains 2/3rds and remaining 1/3rd is explained by inflation.

Question 48

Which other factors influence the bond risk premium?

Answer 48

A) Fiscal Policy B) Maturity structure of Debt c) Investor Demand.