Finance in Debt Management Holland

Finance in Debt Management: 

Finance in Debt Management

Yield / Pricing Formulae: 

Yield / Pricing Formulae Money market securities Discounting Bonds

Treasury bills: 

Treasury bills Short-term debt instruments – maturity usually between overnight and 1 year No coupon, or interest payment Issued at a discount to face value Yield calculation: where F is the face value; P is the purchase or issue price; B is the day-count year basis; and n is the number of days to maturity

Example: Treasury bill: 

Example: Treasury bill 90-day T-bill; Face value $10,000; current price $9,901

Example: Treasury bill: 

5 Example: Treasury bill 90-day T-bill; face value $10,000; current yield 3.96% Price can be calculated by re-arranging the formula:

Discounting: 

6 Discounting So discount yield relates value of a future cash flow to its value today In previous example, today’s value (or present value) of $10,000 was $9,901 at a discount rate / yield of 3.96% Valuing any financial asset is simply the process of finding the present value of all its future cash flows and adding them Requires relevant discount rates for each time period, i.e. the rate that captures the opportunity cost of waiting for the funds

Discounting – multiple periods: 

7 Discounting – multiple periods Suppose the money is to be paid back in two year’s time and that the money could be deposited for 5% for this 2 –year period: FV = PV  (1+r)  (1+r) £110.25 = £100  (1.05)  (1.05) or £100  (1.05)2 = £110.25 So, FV = PV  (1+r)n Rearranging, the present value from the future value: PV = FV/(1+r)n £100 = £110.25/(1.05)2

Bond Basics: 

Bond Basics Bonds issuer pays investors regular interest payment (coupon) when bond matures issuer pays back borrowed amount, known as nominal/principal/face value Coupon and maturity are the two key bond characteristics Recall, present value (price) is equal to sum of PV of all future cash flows

Special case: Valuing a zero coupon bond: 

9 Special case: Valuing a zero coupon bond Assume Holland corporation issues 10-year zero coupon bond today Face/notional value of each bond £100 Required yield/discount rate 8.0% = zero rate Applying formula:

The PV formula for a bond: 

10 The PV formula for a bond Plain vanilla bonds have more payments to consider and to discount Formula for pricing a bond which matures in n years, paying an annual coupon of C and face value P is given as follows: where r is the discount rate, internal rate of return, required yield, yield to maturity or gross redemption yield on the bond.

The PV formula for a bond: 

11 The PV formula for a bond Breaking out the formula – (bond with three year’s to maturity, annual coupon of 6% and required yield of 5%):

Adjusting the PV formula: 

12 Adjusting the PV formula In practice the bond formula has to be adjusted for a variety of factors, such as: Payment frequency (annual, semi annual) Day count convention. Some bonds define a year as consisting of 360 days (corporates), others as having 365 (US Treasuries), others the actual # of days in payment period (UK gilts) But the basic principle still holds

Settlement value: 

13 Settlement value In practice, don’t always trade at start of coupon period Trading convention – prices / yields quoted are on a “clean basis” Accrued interest accounts for the difference between ‘clean’ and ‘dirty’ prices Clean price is the price of bond excluding accrued interest Dirty price is clean price plus accrued interest Prices are always quoted clean, but the price one pays for a bond is always the dirty price Why?

Answer... information: 

Answer... information NB: assuming underlying bond price does not change

Market conventions: 

Market conventions Care needed for price / yield calculations Day-count conventions Payment frequency But can express everything in common form Example, which is cheaper to issue: a one year Treasury bill issued at a discount of 5% or a one year zero coupon bond issued at a yield of 5%?

Example: Discount yield to bond equivalent yield: 

Example: Discount yield to bond equivalent yield So price of $10,000 face value 1-year Treasury bill issued at a discount of 5% is: $10,000*(1 - (0.05)) = $9,500 Price of zero coupon bond? $10,000/ (1.05) = $9,524 Therefore, cheaper to issue zero coupon bond What zero rate would make you indifferent? z1 = $10,000/$9,500 -1 z1 = 5.3% Recall

Calculating yields: 

17 Calculating yields We know that to calculate the present value of a bond that we can use this formula: where r is the yield to maturity, gross redemption yield on the bond r is therefore often interpreted as the ‘expected return’ on the bond But there are other yield calculations …

The current yield: 

18 The current yield The ‘current yield’, or ‘running yield’ on a bond, is the ratio of the coupon to the present value of the bond

Current yield, coupon & the YTM: 

19 Current yield, coupon & the YTM Relationship between coupon current yield and gross redemption yield is straightforward

Yield Curves: 

Yield Curves

Spot and Forward Rates : 

Spot and Forward Rates Spot Rate: is the annual rate of interest that one can earn on an investment (or loan) made today which will be repaid with interest in the future Equivalent to a zero rate Forward Rate: is the rate of interest implicit in the quoted spot or zero rates, that would be applicable from one point of time in the future to another point of time in the future. Single-year forward rates can be extracted from a schedule of spot rates by using the following formula: Fn= [(1+Sn)n /(1+Sn-1)n-1] - 1

Quick Example: 

Quick Example 2-year zero bond, trading at a yield of 8% 1-year zero bond, trading at a yield of 5% What is the implied 1-year – 1-year forward zero rate? To be indifferent: (1+z2)2 = (1+z1)*(1+f1) Therefore: f1 = (1+z2)2 / (1+z1) -1 i.e. 11% Implication: expect rates to rise?

Yield Curves: 

Yield Curves The Yield Curve: Charts the relationship between the YTM and the time to maturity for bonds with identical default risk characteristics. Where bonds pay coupons, have a coupon curve Special case of coupon curve is par curve Bonds priced at par so coupon rate is equivalent to YTM Where all bonds are zero bonds then have spot or zero curve. Captures the relevant discount rates for each period; allows any collection of cash flows to be valued From par curve, derive spot curve From spot curve, derive forward curve Can repackage forward curve as forward par curve

Bootstrapping?: 

Bootstrapping? Par to spot / zero y1 = z1 (assume pay coupon annually) z2? Make returns the same from investment YTM: 100=y1/(1+y1)+(y1+100)/(1+y1)^2 PV: 100 = y1/(1+z1) + (y1+100)/(1+z2)^2 So, 100-(y1/(1+y1))=(y1+100)/(1+z2)^2 Rearrange and solve for z2

We are family ...: 

We are family ... Maturity Yield Forward Spot Par

Term Structure Shapes: 

Term Structure Shapes Maturity Maturity Maturity Maturity Y i e l d Y i e l d Y i e l d Y i e l d Normal Inverse Humped Flat

Some Theory: 

Some Theory Expectations Hypothesis: Yield on a long-term bond should be based on the expectations of investors about the future yields on a sequence of short-term bonds. For example: (1+i20)2 = [(1+i10)(1+i11)] Recall, z2>z1, i.e. yield curve rising, f1>z1 – expectation is that interest rates will keep rising. Similarly, downward sloping curve consistent with expectations that rates will fall

But, what about term premium... : 

But, what about term premium... Term Premium: investors demand a premium for holding long-term bonds: (1+i20) 2 = [(1+i10)(1+i11+LP)] This premium is induced by investor risk aversion

Anything else?: 

Anything else? The Market Segmentation Theory: Both supply and demand determine the YTM. Investors have an affinity for certain securities whose maturity structure match their liabilities. For example, insurance companies demand long-term bonds to match their long-term obligations, whereas commercial banks prefer short-term securities, given their liabilities. Preferred Habitat Theory: Investors can be lured out of their habitat by paying them a premium.

Interest Rate Risk: Duration: 

Interest Rate Risk: Duration Maturity does not reflect the price variability of a security that provides cash flows before its maturity date. Bonds with identical maturity dates, but with differing cash flows, will show differing sensitivities of their prices to changes in interest rates. Duration measures the exposure and sensitivity of an asset and its price to interest rate risk.

Duration : 

Duration Duration: weighted average of the times to arrival of all scheduled future payments of a bond, where the weight attached to each payment reflects the relative contribution of that payment to the value of the bond. Example: A 3 year coupon bond, priced at par $10,000, i = 9.4% Or D = 5.36/2 = 2.68 vs. 3 years

Duration and Price Sensitivity: 

Duration and Price Sensitivity Sensitivity of a bond’s price to changes in yields is: Modified Duration: When yields are small, then one can approximate D as: D= % DP Otherwise, it is more accurate to use Modified Duration MD = D/(1+y) = % DP

Duration: 

Duration It reflects the number of years that a bondholder has to wait before recouping his investment. It measures the exposure and sensitivity of an asset and its price to interest rate risk. The shorter is the duration the lower is the price variability. What about a zero-coupon bond? D = T

Factors Affecting Duration : 

Factors Affecting Duration Duration and Coupon Duration declines with higher coupon payments, since cash payments are made earlier. Duration Coupon

Factors Affecting Duration : 

Factors Affecting Duration Duration and Yield Duration declines with higher yield, since early CFs and their time measures are given higher weight. Duration Yield

Duration: Who cares?: 

Duration: Who cares? Duration important for fund managers, particularly those managing combined asset and liability portfolios Duration matching of assets and liabilities helps protect the net worth of an entity (banks, insurance companies, etc) against unexpected changes in interest rates. Bond portfolio managers can “immunize” their portfolios from interest rate changes by setting duration to zero. Issuers of debt? Care about price (and yield) on issue; but only care about price movements if considering re-financing

Some Practical Implications: 

Some Practical Implications

US yield curve: October 20, 2005: 

US yield curve: October 20, 2005

Choices, choices,...: 

Choices, choices,... Choice: issue 2 year bond at par with a YTM of 4.66% or issue a 1 year bond with a YTM of 4.58% and rollover in a year? How do you decide? What coupon should you set on the bonds?

Further finance?: 

Further finance? IMF Institute Course Fabozzi – “Bond Markets, Analysis and Strategies” Tuckman – “Fixed Income Securities” Benninga – “Financial Modelling”