1.6 Fixed Income Securities Flashcards by Jakub Trybull

How are agends linked in fixed income markets?

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What are fixed income securities?

Securities with a claim to a “fixed” set of cashflows at “fixed” dates. A form of debt financing.

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What are other forms of debt financing?

Bank loans.

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Differences betweeen fixed income securities and bank loans

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Types of fixed income security based on maturity

Bills (up to 1 year at issue)

Notes (1-10 years)

Bonds (10+ years)

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Types of fixed income security based on cashflow structure

Fixed rate or “straight” bonds - single equal “coupon” payment.

Floating rate or “floaters” - variable rate bonds tied to a reference interest rate.

Zero-coupon or discount bonds - no coupon payments, so interest is from the discount.

Perpetuities or “consols” - fixed coupon payment, but no repayment of principal.

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What is meant by option structure?

Bonds may have “options” built in, calls, puts, convertibles, exchanges.

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Types of fixed income security based on issuer type

Sovereign issuers (countries) and non-sovereign (everyone else).

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Types of fixed income security based on cashflow

Maturity (when do we get the money back?)

Principal (how much did we invest?)

Coupon (regular cashflow payments)

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What do fixed income securities differ by (time value)?

Interest rates (not a single value)

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How do fixed income securities vary based on risks?

Inflation

Credit (rating)

Timing (callability = bond issuer can decide to repay early)

Liquidity (if there is no trading in the bond, there will be a discount)

Currency risks (rouble bonds returns in US$ can look pretty ropey)

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How is the information on the time value of money given?

Spot interest rates

Prices of discount bonds (e.g., zero-coupon bonds)

Prices of coupon bonds

Forward interest rates

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What are zero-coupon bonds?

A zero-coupon (or discount) bond with maturity date t is a bond which pays a lump sum (the par or face value) at t and no periodic interest (coupons) at all.

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What is the formula for current price and interest rate for zero-coupon bonds?

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What is meant by par value?

The Par Value is the face value (FV) on the issuance of securities like bonds or stocks, as established on the issuer’s security certificate.

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Par values: bonds

The par value of a bond is the amount owed to the bondholders at maturity by the issuer.

Bonds are issued at or near their par (face) value, most often $1,000, unlike common stock.

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Par values: common stock

The par value of common stock is the stated value per share, ie. the minimum share price that future shares can be issued to the public.

Common stock is issued much higher than the par value, which is most often well below $0.01.

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Why are zero-coupon bonds useful?

The calculation of the interest rate is simple.

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Where do the returns arise for zero-coupon bonds?

The difference between the par value (received at t=1) and the bond price (paid in t=0).

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Example: what is the spot rate corresponding to the 5-year zero?

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What are spot rates?

A spot rate r_t is the current annualised interest rate for a maturity date t.

It is the “average” interest rate between now and t, and can vary depending on the maturity t.

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How are spot rates calculated?

Calculated using zero coupon (discount bonds) which mature at t. Thus, r₅ is the spot rate for 5 years.

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What is a coupon bond?

It pays a stream of regular coupons, as determined by a coupon rate C, as well as the notional amount of the principal N at maturity.

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What is the equation for the price of a coupon bond?

Where C is the coupon rate, N is the principal at maturity, B_t is current price

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How do we need to alter calculations it bonds pay coupons more/less often than once per year?

The appropriate infra-annual spot rates (e.g., for 0.25, 0.5, 0.75, 1, ... years) needs to be used instead.

Example: what should the price of the coupon be? What if it actually trades at a higher or lower price?

idk this lol

How can we calculate spot rates if we have non-zero coupon bonds?

Calculate spot rates using the cashflows of coupon bonds through bootstrapping.

How can we compare bonds?

Different maturities, different coupons so we can use Yield-to-Maturity.

What is Yield-to-Maturity?

YTM tells us what the single fixed interest rate is that explains the price of the bond, its coupons and maturity.

Equation for price of bond involving YTM

If bond’s coupon rate = YTM, what can we say about it?

The bond is selling at **par**, i.e. its price matches its face value (e.g., P=100).

If bond’s coupon rate \< YTM, what can we say about it?

Then the bond is selling at a **discount** (e.g., P\<100).

If bond’s coupon rate \> YTM, what can we say about it?

Then the bond is selling at a **premium** (e.g., P\>100).

Important consideration about YTM

YTM is an **approximation**. Actual cashflows are discounted by the spot rates NOT the YTM to give us the price of the bond.

How are forward rates different from spot rates?

Spot interest rates: rates for a transaction between today, *t₀*, and a future date, *t*. Forward interest rates: rates for a transaction between two future dates, e.g., *t₁* and *t₂*.

Forward rates and uncertainty

Future spot rates are uncertain and can be different from current corresponding forward rates

What are the elements behind a forward transaction on future borrowing?

Terms are agreed today, *t₀* Loan is received on future date, *t₁* Repayment of the loan occurs on date *t₂*

Equation for a forward interest rate

Diagram for the relationship between spot and forward rates

Example: what rate does the bank quote?

Define *term structure of interest rates*

The range of spot rates for different maturities.

What two hypotheses can we use to explain the term structure?

Expectations hypothesis Liquidity hypothesis

What is the expectations hypothesis? [term structure of interest rates]

What is the liquidity hypothesis? [term structure of interest rates]

Implication of the expectations hypothesis

Slope of term structure reflects the market’s expectations of future short-term interest rates.

Implications of the liquidity preference hypothesis

* Long bonds = higher returns than short terms (on average). * Forward rates on average “over-predict” future short-term rates. * Term structure reflects: * expectations of future interest rates. * risk premium demanded by investors in long bonds.

Why do investors fear a rising yield curve?

It reduces the value of their investment.

What is the Macaulay duration equation?

Implications of Macaulay duration equation

Higher duration = high the price change of the bond given a change in market interest rates (ceteris paribus). * Higher coupon rate → the lower the duration * Longer maturity → the higher the duration * Higher yield → the lower the duration

Diagram for value and duration

What is the modified duration equation?

Derivation for the modified duration equation

Example: 4-year T-note with face value $100 and 7% coupon, selling at $103.50, yielding 6%. Note: T-notes → coupons paid semi-annually. What is the coupons and yield rates using 6-month intervals?

Coupon rate is 3.5% and the yield is 3%. (semi-annually so they are halved)

Example: 4-year T-note with face value $100 and 7% coupon, selling at $103.50, yielding 6%. Note: T-notes → coupons paid semi-annually. Table for cash flow, present value and *t* × *PV*

Example: 4-year T-note with face value $100 and 7% coupon, selling at $103.50, yielding 6%. Note: T-notes → coupons paid semi-annually. What is the duration in 1/2 year units?

Example: 4-year T-note with face value $100 and 7% coupon, selling at $103.50, yielding 6%. Duration in 1/2 year units is 7.13 Note: T-notes → coupons paid semi-annually. What is the modified duration?

Example: 4-year T-note with face value $100 and 7% coupon, selling at $103.50, yielding 6%. Note: T-notes → coupons paid semi-annually. How does a semi-annual yield rise of 0.1% affect the bond price?

Moves the bond price by 0.692%

Define *credit risk*

The risk of default, where the issuer default fails to pay the cashflow that were promised.

Risk and bonds

Bonds carry the risk of failing to pay off as promised. Generally, sovereign bonds in the country’s own currency do not have default risk.

Moody’s and S&P credit rating table

How can we break down a yield?

* Promised YTM * Expected YTM * Default premium * Bond risk premium

Promised and expected yield

Promised YTM: the yield if default does not occur. Expected YTM: the probability-weighted average of all possible yields.

Default premium and bond risk premium

Default premium: the difference between promised yield and expected yield. Bond risk premium: the difference between the expected yield on a risky bond and the yield on a risk-free bond of similar maturity and coupon rate.

Example: calculate the promised YTM and expected YTM

Example: calculate default premium and risk premium Promised YTM = 12% Expected YTM = 9%