F I N A N C I A L

M A R K E T S

Structure, Par+cipants, Instruments Interest Rates and Valua+on of Bonds

DEBT MARKETS:

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FM: Objec+ves A?er successfully comple+ng this topic, you will be able to:

§ Apply basic pricing models to evaluate stocks and bonds § Describe the theoreIcal determinants of the level and term structure of interest rates

§ Explain the concept of “yield” and its rela+on to “interest rate” § Determine the price of coupon and discount bonds § Compute the dura+on and convexity of a bond § Differen+ate between Macaulay and modified dura+on § Understand the rela+onship between dura+on and convexity and bond price vola+lity

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FM: Bond. J. Bond. § Fixed income § Debt instrument § Main instrument

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FM: Bond Market Structure and Par+cipants § DomesIc market (local & foreign) § InternaIonal (eurobonds)

Issuers § Central Governments

(Treasury & Agencies)

§ Interna+onal En++es § Local Governments § Corpora+ons

Investors § Ins+tu+onal § Individuals Intermediaries RaIng Agencies Central Banks

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FM: Bond Characteris+cs

§ § § § §

Issuer Maturity Principal / Nominal Value Coupon / Interest Rate Provisions (issuer / holder)

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FM: Interest Rates – Defini+on and Structure § INTEREST RATES: § among the most closely watched variables in the economy § important for individuals, companies, ins+tu+ons, governments § defined as a measure of the price paid by a borrower (debtor) to a lender (creditor) for the use of resources during some Ime interval. § How many? § in theory: one short-­‐term, riskless, real interest rate that would prevail in the economy if the price levels remain constant and could provide an anchor. § in real world: a structure of interest rates that depend on a myriad of factors.

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FM: Nominal versus Real Interest Rates § Real Interest Rate: measures the amount of the commodity next period that can be exchanged for one unit of the commodity now. § Nominal Interest Rate: measures the amount of money to be repaid next period per unit borrowed now. § Fisher’s Law:

1 + inomin al = (1 + ireal ) ⋅ (1 + %Δp) ireal ≈ inomin al − %Δp

ireal

1 + inominal = −1 1 + %Δp

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FM: Features of Debt Instruments (1) § Base Interest Rate: § also referred to as the benchmark interest rate. § determined as the yield offered on a comparable maturity for an on-­‐the-­‐run Treasury security (most recently auc+oned issues for each maturity). § historically, the interest rates on Treasury bills (T-­‐bills, discount securi+es, maturity < 1 year) and Treasury Coupon SecuriIes (maturity > 2 yrs) served as the only benchmark given the full faith and credit of U.S. Government. § usually characterized as the riskless rate. § is the minimum interest rate that investors demand for inves+ng in a non-­‐Treasury security. § Interest rate (ri) = Base interest rate (rf) + Risk premium (πi)

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FM: Features of Debt Instruments (2) § Risk Premium: § is defined as the spread that reflects the addi+onal risks the investor faces by acquiring a security that is not issued by the government. § is affected by several factors: § the type of issuer § the issuer’s perceived creditworthiness § the term or maturity of the instrument § provisions that grant either the issuer or the investor the op+on to do something § the taxability of the interest received by investors § the expected liquidity of the issue

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FM: Features of Debt Instruments (3) § Term to maturity / Maturity: § number of years during which the borrower promises to meet the condi+ons of the debt. § Principal: § sum transferred from the lender to the borrower. § bullet maturity – a debt contract where the en+re principal is repaid by the borrower at the maturity date. § balloon payment – the remaining principal is repaid at the maturity date a?er various amounts of the principal were paid over the life of the debt contract. § for a bond the amount paid at maturity is called par value, maturity value, or face value.

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FM: Features of Debt Instruments (4) § Coupon: § the periodic interest payment made to owners during the life of the contract. § determined by mul+plying the coupon rate by the unpaid outstanding principal. § typically paid every 6 months (quarterly and yearly also used). § Discount (zero-­‐coupon) debt instruments: § bought at a price below its face value (at a discount) and repaid at the face value at the maturity date. § Price Quote: § for most debt contracts, prices are quoted as percentage of par value (89⅞ or 103½ or 95¾).

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FM: Types of Bonds

§ Government / Municipal / Corporate § Short term / medium / long / perpetui+es § Investment grade / junk bonds § Plain vanilla / discount § Bullet / amor+zing

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FM: Value(s) of a Bond § Nominal (Par) Value § Issue / Purchase Price § Redemp+on Value § Market Price § Intrinsic Value

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FM: Valua+on of Debt Instruments § FUNDAMENTAL PRINCIPLE OF VALUATION: § The value of any financial asset is the present value of the cash flow(s) expected. § Used for all financial assets (including bonds and other debt instruments).

§ PROCESS OF VALUATION: Œ Es+mate the cash flow(s)  Determine the appropriate interest rate for discoun+ng the cash flows (minimum interest rate plus risk premium) Ž Calculate the value of financial asset as the PV[E(CF)]

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FM: Discounted Cash-­‐Flow Method for Bonds !! = %!" !# $%& # " $
&=" '

( )

!! =

##$%" $

%"+ (&' $ $ $ +/ !! %)*+,-).+-,**+& = + +(((+ ' %"+ (& %"+ (& %"+ (&' $ ! " # / !! = & ("' )+ ' ( " %"+ (& $ %"+ (&' $ $ $ +/ !! %)*+,-).+-,**+& = + +(((+ %"+ (" & %"+ (" &&%"+ (' & %"+ (" &&%"+ (' &&…&%"+ (' &

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%"+ (& %"+ (&

+

##$%' $
'

+(((+

##$%' $

FM: Pricing Bonds and other Debt Instruments QUANT TIME: Pricing Debt Instruments

EXAMPLE 1: I. Determine the price of a 3-­‐year debt instrument today if the debtor is required to make yearly payments of 1000 LTL, 1200 LTL, and 1500 LTL, and also to repay the principal of 10.000 LTL at the maturity date. We assume that the 1-­‐year rates for the next 3 years are: (a) all equal to 7% / (b) 5%, 6%, and 10%. II. Compare the calculated price with that of a plain vanilla bond with the same maturity, a face value of 10,000 LTL and a coupon rate of 13%. Answer: _______ Hint: Use a financial calculator -­‐ hrp://www.epx.com.br/ctb/hp12c.php

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FM: Yield to Maturity § Represents a measure that can be used to compare instruments of different cash flows and different maturi+es. § Is defined as the interest rate that makes the present value of the cash flow equal to the market value (price) of the instrument. § For a coupon bond with a coupon payment C and a maturity value M, yield to maturity is the interest rate y that sa+sfies equa+on:

§ Interest Rates (and maturity) are customarily quoted per year, but typically all coupon bonds pay interest twice a year. "2T & § Solu+on y is a semiannual yield to # 1" (1+ y sem ) ( 1 maturity and can be converted to P0 = Csem ! % + 2T ( M M % y sem % ( (1+ y sem ) annual either by compounding, or

$ ' by doubling – bond equivalent yield.

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C C C +M P0 = + + ...+ 2 (1+ y) (1+ y) (1+ y)T

P0 C T 1 1 = !" + M M t=1 (1+ y)t (1+ y)T

FM: Yield to Maturity QUANT TIME: CalculaFng yield to maturity

EXAMPLE 2: Calculate the annual yield to maturity (both compounding and bond equivalent) for a 7%, 2-­‐year bond with a maturity value of 10.000 EUR, that is making semiannual coupon payments and is selling for: (a) 74.26% / (b) 100%.

Answer: _______ Hint: Use a financial calculator -­‐ hrp://www.epx.com.br/ctb/hp12c.php

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FM: Rela+onship between Price and Yield § P/M is called the par value relaIon and is usually expressed as %: § P/M = 1 – bond sells at “par” § P/M > 1 – bond sells at a “premium” § P/M < 1 – bond sells at a “discount” § If P/M = 1, the solu+on of the equa+on is y = C/M C #1" (1+ y)"T & 1 1= ! % + ( T y M
$ ' (1+ y)

§ If P/M < 1, the coupon rate is less than the yield to maturity

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FM: Current Yield C yc = P
§ Is a berer approxima+on to yield to maturity, nearer price is to par and longer is maturity of bond. § Change in current yield always signals change in same direc+on as yield to maturity. § For a consol, or perpetuity (special case of a coupon bond with no maturity date and no repayment of principal that makes coupon payments of C forever), yield to maturity is equal to the current yield.

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FM: Current Yield and Yield to Maturity EXAMPLE 3: (a) Calculate the current yield for a 7%, 2-­‐year bond with a maturity value of 10.000 EUR, that is making semiannual coupon payments and is selling for 100%. (b) Determine the yield to maturity on a bond that has a price of 1.000 GBP and pays 100 GBP annually forever.

QUANT TIME: CalculaFng yields

Answer: _______ Hint: Use a financial calculator -­‐ hrp://www.epx.com.br/ctb/hp12c.php

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FM: Yield to Maturity for Discount Bonds § One-­‐year discount (zero-­‐coupon) pays both the principal and the interest at the maturity date. P= M M "P !y= 1+ y P

QUANT TIME: CalculaFng yield to maturity

§ EXAMPLE 4: What is the yield to maturity of a “T-­‐bill” with the maturity of 1 year that is selling at a discount of 96 ⅝? § Answer: _______

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FM: Rela+onship between Rates and Returns § The only bond whose return = yield to maturity

is the one with maturity = holding period. § For bonds with maturity > holding period, when i ↑, P ↓ and capital loss. § Longer is the maturity, greater is the price change associated with an interest rate change. § Longer is the maturity, more return changes with change in interest rate. § Bonds with high ini+al interest rate can have nega+ve return if i ↑.

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FM: Reinvestment and Interest Rate Risk § Reinvestment Risk: § is the risk that future interest rates at which the coupon can be reinvested will be less than the yield to maturity at the +me the bond is purchased (gain from i ↑, lose when i ↓). § occurs if an investor’s holding period is longer than the term to maturity of the bond (the investor hold series of short bonds over long holding period). § Interest Rate (Price) Risk: § is the risk that a bond will have to be sold before its maturity date for less than its purchase price, resul+ng in a return that is less than the yield to maturity. § doesn’t occur when the maturity equals the holding period. § long-­‐term bonds have higher price risk (prices are more vola+le).

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FM: Reinvestment Risk and Rate of Return QUANT TIME: Reinvestment risk and rates of return

EXAMPLE 5: An investor has a holding period of 2 years and decides to purchase a £10.000 one-­‐year bond (with an ini+al interest rate of 8%) at face value and then purchase another one at the end of the first year. What is the annual rate of return if the interest rate on one-­‐year bonds becomes: (a) 15%; (b) 6.5%?

Answer: _______

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FM: Bond Price Vola+lity § A fundamental characteris+c of an op+on-­‐ free bond (not callable, putable, or conver+ble) is that its price changes in the opposite direcIon from the change in yield. § In general, absolute and percentage price change are asymmetric for equal changes in yield -­‐ increases when the yield declines are greater than decreases when the yield increases. § The primary determinants of price vola+lity: § Coupon (rate) – for a given term to maturity and ini+al market yield, percentage price vola+lity is greater the lower the coupon. § Term to maturity – for a given coupon rate and ini+al yield, the longer the term to maturity, the greater the price vola+lity, in terms of percentage price change.

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FM: Measuring Price Vola+lity -­‐ Dura+on § Price sensiIvity of a security to changes in interest rates can be measured by changing rates with a small number of basis points and calculate how the price or value of the security will change. !y = change in the yield of the security (in decimal form) V0 = initial value or price of a security (per 100 par value) V_ = the estimated value of a security per 100 of par value if the yield is decreased by !y V+ = the estimated value of a security per 100 of par value if the yield is increased by !y

§ DuraIon is defined as the approximate percentage price change for a 100 basis points change in yield.

V! !V+ Duration = 2 "V0 " #y

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FM: Dura+on EXAMPLE 6: I. Calculate the dura+on of a 20-­‐year 9% coupon bond trading to yield 9% it its face value (equal to maturity value) is 100 CHF. II. Explain the impact on dura+on of a change of: (a) the coupon rate to 5%; (b) the maturity to 5 years. Note: Suppose the yield changes 50 basis points. Answer: _______ Hint: Use a calculator – hrp://www.investopedia.com

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QUANT TIME: CalculaFng duraFon

FM: Features of Dura+on § Dura+on of coupon bond is less than its maturity. § Dura+on of a zero-­‐coupon bond is equal to its maturity. § For bonds with the same maturity and selling at the same yield, the lower the coupon rate, the greater are a bond’s dura+on and price sensi+vity. § For two bonds with the same coupon rate and selling at the same yield, the longer the maturity, the larger are the dura+on and price sensi+vity. § DuraIon can be used to approximate how the price of a security changes when interest rates changes (for small changes in yield). Aprox.%!P = "Duration # !y #100

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( )

FM: Dura+on and Price Sensi+vity QUANT TIME: CalculaFng duraFon and price sensiFvity

EXAMPLE 7: Calculate the effect of a 10 basis points increase in the interest base on the price of a 20-­‐year 5% plain vanilla bond that is trading at 63.1968 (to yield 9%).

Answer: _______ Hint: Use a calculator – hrp://www.investopedia.com

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FM: Macaulay Dura+on § Is named in honor of Frederick Macaulay who used the measure in a study published in 1938. § Is based on the idea that a coupon bond can be considered the equivalent to a set of zero-­‐coupon bonds.

§ Is computed as a weighted average of the maturiFes of the cash payments (the weights are the present values of the payments and the denominator is their sum, precisely the price of the bond).

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FM: Formulae of Macaulay Dura+on T

"t !
DurationMACAULAY = t=1 CFt

T

(

1+ y

)

t T

" t=1 CFt

DurationMACAULAY

(

1+ y

)

t

1+ y = #T ' $ 1# 1+ y ) M + C !& & ) y & ) 1+ y % ( t=1 "

t !C

(

1+ y

) ( ( )

t

+

T !M

)

T

(

)

T

1!C DurationMACAULAY

( =

1+ y

) (

1

+

2 !C 1+ y

)

2

+…+ P

T !C

(

1+ y

) (

T

+

T !M 1+ y

)

T

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FM: Macaulay Dura+on QUANT TIME: CalculaFng Macaulay duraFon

EXAMPLE 8: Determine the Macaulay dura+on of a $1000 10-­‐year 10% plain vanilla if the interest rate is: (a) 10% / (b) 20%. Answer: ____ Answer: ____

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FM: Key Facts of Macaulay Dura+on § All else equal, when the maturity of a bond lengthens, the dura+on rises as well. § All else equal, when interest rates rise, the dura+on of a coupon bond fall. § The higher is the coupon rate on the bond, the shorter is the dura+on of the bond

Source: www.investopedia.com

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FM: Macaulay Dura+on QUANT TIME: CalculaFng Macaulay duraFon

EXAMPLE 9: Bruce holds a five-­‐year bond with a par value of 1.000 EUR at a moment when the current interest rate is 8%. Determine the Macaulay dura+on of the bond if the coupon is paid annually and the coupon rate is: (a) 6.5% / (b) 8%. Answer: _______ Hint: Use a calculator – hrp://www.investopedia.com

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FM: Macaulay Dura+on QUANT TIME: CalculaFng Macaulay duraFon

EXAMPLE 10: An investor is holding a por|olio consis+ng of two bonds: § A – dura+on of 4.5-­‐year and weight of 30%; § B -­‐ 7-­‐year dura+on. Determine the dura+on of the bond por|olio. Answer: _______ Hint: DuraIon is addiIve. The dura+on of a por|olio of securi+es is the weighted-­‐ average of the dura+ons of the individual securi+es.

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FM: Modified Dura+on § Is determined as the ra+o of Macaulay dura+on and (1+y). § It gives approximately the same result as dura+on determined as the approximate percentage price change.

DurationMODIFIED

DurationMACAULAY = 1+ y
1!C + 1 2 !C
2

DurationMODIFIED

(1+ y ) (1+ y ) =

(1+ y ) (1+ y ) P ! (1+ y )

+…+

T !C
T

+

T !M
T

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FM: Macaulay Dura+on and Interest Rate Risk

§ Macaulay Dura+on can be used in prac+ce to measure interest-­‐ rate risk. § Macaulay Dura+on provides a very good approxima+on (par+cularly when interest-­‐changes are small) for how much the security price changes for a given change in interest rates.

!P !y = "DurationMACAULAY # P 1+ y

!P = "DurationMODIFIED # !y P

§ The greater is the dura+on of a security, the greater is the percentage change in the market value of the security for a given change in interest rates (interest-­‐rate risk).

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FM: Price Sensi+vity and Dura+on QUANT TIME: Price sensiFvity and duraFon

EXAMPLE 11: Bruce holds a 5-­‐year 6.5% coupon rate bond with a par value of 1.000 EUR at a moment when the current interest rate is 8%. What is expected in to happen with the price of the bond if the interest rate rises to 8.75% tomorrow? Answer: _______ Hint: Use a calculator – hrp://www.investopedia.com

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FM: Price Sensi+vity and Dura+on QUANT TIME: Price sensiFvity and duraFon

EXAMPLE 12: Bruce holds a 5-­‐year 8% coupon rate bond with a par value of 1.000 EUR at a moment when the current interest rate is 8%. What is expected in to happen with the price of the bond if the interest rate rises to 8.75% tomorrow? Answer: _______ Hint: Use a calculator – hrp://www.investopedia.com

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FM: Price Sensi+vity and Dura+on QUANT TIME: Price sensiFvity and duraFon

EXAMPLE 13: (a) Calculate the dura+on of a $1,000 6% coupon bond with 3 years to maturity. Assume that all market interest rates are 7%. (b) Calculate the expected price change if interest rates drop to 6.75% using the dura+on approxima+on. Answer: _______ Hint: Use a calculator – hrp://www.investopedia.com

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FM: Price Sensi+vity and Dura+on QUANT TIME: Price sensiFvity and duraFon

EXAMPLE 14: A bank has two commercial loans with a present value of $70 million and a maturity of 3 years: § A -­‐ $30 million loan that requires a single payment of $37.8 million in 3 years, with no other payments +ll then; § B -­‐ $40 million loan that requires an annual interest payment of $3.6 million and is due in 3 years. (a) Calculate the dura+on of the bank’s loan por|olio. (b) Explain what happens to the value of its por|olio if the general level of interest rates increased from 8% to 8.5%. Answer: _______ Hint: Use a calculator – hrp://www.investopedia.com

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FM: Measuring Price Vola+lity -­‐ Convexity § Convexity is another measure of price vola+lity to be used in conjunc+on with dura+on to improve the es+mate for price vola+lity for large changes in interest rates. Convexity = V+ +V! ! 2 "V0 2 "V0 " #y

( )

2

Approximate percentage price change = Approximate %!P due to duration + Approximate %!P not explained by duration

%!P = -DUR " !y "100 + CON " !y

( )

( )

2

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FM: Dura+on and Convexity QUANT TIME: CalculaFng DuraFon and Convexity

EXAMPLE 15: I. Calculate (a) dura+on and (b) convexity of a 20-­‐year 9% coupon bond trading to yield 9% it its face value (equal to maturity value) is 100 CHF. II. Calculate the expected price change if interest rates increase by 300 bps using the dura+on and convexity approxima+on. Answer: _______ Hint: Use a calculator – hrp://www.investopedia.com

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