FINC313 Exam 2
Present Value (PV) Definition
- The value today of some future amount - To find the present value of a future payment, run the process you used to get the Future Value in reverse, i.e. divide by (1 + r)t.
Zero-Coupon Bond
-A bond paying no coupons that sell at a discount and provides only a payment of par value at maturity
Bond Ratings Chart
*Look at slides 21-24*
Why is D(T) useful? Why not just use r(T)?
- The discount function only depends on maturity "T" and NOT on any compounding frequency. In other words, the discount rate whether computed using semiannual rates or annual rates is the same. - Recall from the previous slide that with interest rates, the rates for annual and semi-annual compounding are typically different. - We will see later that computing bond prices using the discount function is easier than having to deal with interest rates and compounding frequencies.
Future Value (FV) Definition
- The value in the future of some present amount - To work out how much you will have in the future if you invest for t periods at an interest rate r, multiply the initial investment by (1 + r)t.
Exchangeable Bond
-Allows bondholder to exchange the issue for a number of shares of stock of a corporation different from the issuer
Annuity
-An Annuity is a stream of constant cash flows that lasts for a fixed number of periods --Ordinary Annuity: payments at end of period (what we will consider in this course) --Annuity Due: payments at beginning of period -A 10-year Annuity is simply the difference between two perpetuities, one beginning in year 1, the other in year 11
What does the equal sign mean in Finance
-An economic statement
Treasury (Federal Government)
-Bills: zero-coupon, maturity 1 year or less -Notes: coupons paid twice per year (semi-annually), maturity 2 to 10 years -Bonds: coupons paid twice per year, maturity greater than 10 years -Strips: zero-coupon, maturity typically greater than 1 year
Put Feature
-Bondholder has the right to sell the issue back to the issuer at par value on designated dates
Asset-Backed Sector
-Corporate issuer pools loans or receivables and uses the pool of assets as collateral for issuance of a security -Mortgage sector = securities backed by mortgage loans
Discount Bond
-Coupon/Face Value < Discount Rate -Sell for less than face value Price < Face Value YTM > Coupon Rate
Par Bond
-Coupon/Face Value = Discount Rate -Sell for face value Price = Face value YTM = Coupon Rate
Premium Bond
-Coupon/Face Value > Discount Rate -Sell for more than face value Price > Face Value YTM < Coupon Rate
Deferred-Coupon Bonds
-Defers the coupons and pays them at maturity of the bond -Issuer avoids using cash to pay coupon payments for a number of years
Effective Annual Rate Definition
-EAR is useful to compare different investments. If two investments are equivalent, they have the same EAR -You can find equivalent APRs (annual percentage rate) for different compounding frequencies by setting investment A equal to Investment B and solving for the APR that is not given
Convertibility
-Exchange a bond for a pre-specified number of shares of stock
Fixed vs. Floating Coupons
-Fixed Rate Bonds --Pre-specify the coupon as a fixed rate -Floating Rate Bonds --Pre-specify the coupon as a fixed spread to a floating short-term benchmark rate such as T-bill, Libor, Fed funds, Prime rate, AA commercial paper rate --Coupon rate resets periodically (coupon reset rate) according to coupon reset formula --- Reference rate + quoted margin e.g. 1-month Libor + 150 basis points ---Example: Inverse Floater- coupon rises when index/benchmark falls and vice versa
Compounding More Than One Time Per Year
-Interest may be paid semi-annually, quarterly, monthly, weekly, or daily. -Returns, interest, yields are quoted with different compounding frequencies -- Generally given in APR form
Continuous Compounding
-Intuitively, continuous compounding is when interest is compounded every even so small instant. It is as if interest is compounded "all the time" (continuous) instead of at discrete points in time such as each year -Formally, if interest is paid "n" times per year the investment will grow to = (1 + (r/n))^nt -It can be shown, that as "n" becomes very large then that formula =e^rt -Future Value of $1 under continuous compounding: FV= e^rt -Present Value of $1 under continuous compounding: PV= e^-rt -The discount factor under continuous compounding: D(T)= e^-rt
Accrued Interest
-Is added to the price buyer B pays to seller S for the bond -The price buyer B actually pays is referred to as the "all-in", "full", or "dirty" price. The "clean", or "flat" price is the bond price without accrued interest --When you read a quote for a bond price, the quoted price will typically be the "clean" price. However, when you buy the bond, your "invoice" will typically be higher because you are responsible for accrued interest as well -All-in Price = "Price you pay" = Clean Price + Accrued Interest --The price calculation is made as of the bond's settlement date, the date on which it actually changes hands after being traded -"Sawtooth-pattern" - Even if the clean price does not change, the dirty price for the same bond will increase steadily as the coupon interest accrues from one coupon payment date until the next coupon date, when it falls by the present value of the amount of the coupon payment. On the coupon date, the accrued interest is zero, so the clean and dirty prices are the same.
Corporations
-Issued by U.S. corporations and non-U.S. corporations in the U.S. ("Yankee Bonds") -Bonds, medium-term notes, structured notes, commercial papers (investment grade, non-investment grade)
Call Feature
-Issuer has the right to retire the debt fully or partially before the scheduled maturity date -Allows issuer to replace bond issue with one with lower interest rates when interest rates in market decline
Multiple Cash Flows and Multiple Periods: -Value Additivity-
-PV= C1/1+r + C2/(1+r)^2 + ... + Ct/(1+r)^t -FV= C0*(1+r) + C1*(1+r)^2 + ...+ Ct/(1+r)^t -This formula assumes that the interest rate is constant for different maturities and that payments occur at the end of the period -In the real world, interest rates change with the maturity according to the term structure of interest rates. *Look at slide 22 for abbreviated formula, can't type on here because of the sum sign*
Amortizing Securities
-Principal is paid over the lifetime of the bond instead of the total principal being paid at maturity
Credit Risk
-Promised cash flows are specified, but debtor may not pay (default risk) -- Default Risk (credit risk) may be significant; US Treasuries perceived as having almost zero default risk -Spread -- Yield of bond= yield of treasury with same maturity + risk premium -Credit Spread -- Part of spread associated with default risk -Credit Spread Risk -- Risk that price of bond will decline due to increase in credit spread -Rating Agencies -- Moody's Investor Services (Aaa, Aa, A, Baa, etc.) -- Standard & Poor's (AAA,AA,A,BBB, etc.) -- Fitch --Investment Grade: BBB (Baa) or above --Junk Bonds: BB (Ba) or below --Highly distressed (CCC or below) --Default: D
Agency Issues
-Securities issued by federally related institutions and government-sponsored enterprises (Fannie Mae, Freddie Mac, Ginnie Mae, etc.) -Securities not backed by any collateral ("agency debenture securities")
International Bonds
-Sovereign Bonds -Foreign Bonds --A UK firm selling its bonds in the US market ($)- Yankee bonds [SEC regulated], Bulldog Bonds (pound), Samurai Bonds (Yen)
Municipal Sector
-State and local governments -General Obligation sector -Revenue Sector -Securities exempt from federal income tax ("tax exempt sector")
Index-Linked
-TIPS (treasury Inflation-Protected Securities) since 1997 in the US -Coupon payments increase with inflation
Annualizing Yields
-The EAR is the amount a dollar grows by after a year
Bond Pricing
-The cash flows from a bond consist of coupon payments until maturity plus the final payment of par value (received at maturity) Therefore, Bond Value = Present Value of coupons + Present Value of par value
Coupon Bond
-The payments received from a coupon bond consist of two parts -- 1) Coupon Payments: Payments of interest called coupons that occur in regular intervals -- 2) Payment of Principal: At maturity the bond pays a principal or face value in addition to the coupon
History Lesson in Finance
-To finance Napoleonic Wars, the British gov't borrowed by issuing British Consol Bonds. -Instead of repaying their loans, the British gov't committed to paying a fixed annual payment in perpetuity (forever).
Market Sectors/ Issuers
-Treasury (Federal Government) -Agency Issues -Corporations -Asset-Backed Sector -International Bonds -Municipal Sector
Excel equations for PV and FV
=FV(RATE,NPER,PMT,[PV],[Type]) -FV represents the future value of that same series of cash flows, compounded at a constant rate of interest. =PV(RATE,NPER,PMT,[FV],[Type]) -PV represents the value today of a series of future cash flows (Cƒt) occurring at regular intervals from t = 1 to t = Τ and discounted at a constant rate of interest. *NOTE* -Note that to use the FV and PV functions in Excel, the periodic cash flows must be constant (i.e., a level-payment annuity); otherwise, each cash flow must be separately valued in Excel and the results summed to arrive at a final answer. • It's also important to input the interest rate on a basis that matches the payment period (monthly rate for monthly payments, annual rate for annual payments,...)
Perpetuity
A Perpetuity is an investment which makes constant payments "C" forever
What is a Bond?
A bond is a debt instrument requiring the issuer ("debtor") to repay to the lendor/investor the amount borrowed plus interest over a specified period of time.
Yield to Maturity "Sleeping Beauty" Bond Example
Bond Maturity - Example "Sleeping Beauty" Bond - On July 21, 1993, Disney issued a 100-year bond. - They sold $300,000,000 worth of debt at an annual yield of 7.55%. - For reference, the 30-year Treasury bond yield was approximately 6.6% at that time. - Their bond was graded AA. - Disney has an option to call or redeem the bonds beginning July 15, 2023 at 103.02% of their face value. - A lot of interest from pension funds, insurance companies and other institutional investors
Yield to Maturity - Zero Coupon Bonds Formula
Bond Price= Face Value of Bond / (1 + YTMt)^t OR YTM= ((face value / Bond Price)^(1/t)) -1 -YTMt: the yield to maturity of the zero coupon bond with maturity of T years We can compute a YTM for each bond with a known price - The internal rate of return (IRR) on the bond - IRR discounts future promised payment stream - It is known from the current bond price Difference between discount rate "r" and YTM? - r comes from prevailing returns on alternative investments - r discounts expected cash flows to get fair market price - YTM comes from a bond for which we already know the price - When is YTM the same as r?
Bond Value Formula
Bond Value = Coupon*(1/r) * (1-(1/(1+r)^t)) + Par value*(1/(1+r)^t)
Bond Value for zero coupon bond Formula
Bond Value zero= Face Value/((1+r)^t)
Effective Annual Interest Rate (EAR)
Effective Interest Rate= (1 + r/m)^m - 1
Excel Functions for Present/Future Value Calculations
Excel "NPV" Function: =NPV)RATE,SUM) Note the difference between the Excel functions PV and NPV (for "net present value"). - PV calculates the present value of a stream of constant future cash flows. - NPV measures the present value of all future cash inflows and outflows minus the initial investment outlay. - In Excel, the NPV function calculates the present value of a stream of future cash inflows/outflows at the rate you specify, but it does not reflect the initial investment (unless it is included in the specified cell range).
Future Value Formula
FV= PV*(1+r)^t
LIBOR
London - Interbank Offered Rate - ICE Benchmark Administration Fixing for US Dollar. - The fixing is conducted each day at 11am & released at 11.45am (London time). - The rate is an average derived from the quotations provided by the banks determined by the ICE Benchmark Administration. • The top and bottom quartile is eliminated and an average of the remaining quotations calculated to arrive at fixing. • The fixing is rounded up to 5 decimal places where the sixth digit is five or more. - ICE Libor day count follows normal market convention: 365 days for GBP, 360 days for the other currencies and for value two business days after the fixing.
Other Discount Function Formulas
Look on slide 49 to see 4 formulas
Growing Perpetuity Equation
PV growing perpetuity= PMT/(r-g)
Perpetuity Equation
PV perpetuity= PMT/r
Annuity PV Formula
PV= C/r * (1 - (1/(1+r)^t)
Present Value Formula
PV= FV/(1+r)^t
Properties of Discount Function- D(T)
Properties of D(T) - D(T) will usually be less than 1, since the present value of $1 to be received in the future cannot in general be greater than 1. • Although there are certain circumstances where it will be greater than 1. This will be the case if interest rates are negative. We actually observe this phenomenon in a number of markets today (e.g. Germany). - D(T) usually decreases with T - As T becomes very large (T→∞), the D(T) will become very small (D(T)→0). • This is intuitive since the value to you today of $1 received in 1 million years is likely very small, even if you are still around by the time the $1 is paid. Under continuous compounding: D(T)=e^−rT
Yield to Maturity "Sleeping Beauty" Bond News:
Who is the better bet: Uncle Sam or the mouse? - New York Post, 7/22/93 • However, some argued that the issue was a Mickey Mouse offering -- since the company can call the bonds after 30 years. "It's just a gimmick," said Rick Meckler, chief investment advisor at Liberty Views and former manager of the CBS pension fund. • An official at a large insurance company who spoke on the condition of anonymity said: "We weren't interested. [We might have been had there] been a more attractive yield spread and if they were non-callable." - Los Angeles Times, 7/25/93 • Things are getting really goofy in the financial markets. Here is the Walt Disney Co., one of the world's shrewdest borrowers, scarfing up $300 million of eternal money by issuing 100-year bonds. But at the same time Disney is nailing down every long-term dollar it can get, the U.S. Treasury, under orders from the Clinton Administration, is cutting back long-term borrowings and increasing short-term borrowings.
Bond Equivalent Yield (BEY)
• In the example of the 5-year Strips with maturity in August 2018, we computed a YTM of 1.391%. • To interpret this result, we may want to compare it to yields on other bonds. • But if the other bonds are semiannual coupon bonds, we need to compute yields for both on the same semi-annual compounding basic • This YTM is referred to as "Bond Equivalent Yield" (BEY). To compute the BEY of the five-year ZCB, we treat it as a ten-period bond (2T = 2*5 = 10) and compute the semiannual IRR, then double it. ^WHY????
Bond Pricing using Discount Factors
• In this course we will typically NOT use the same interest rate "r" to discount cash flows that arrive at different times (year 1, year 2, etc.). • Instead, we will use an interest rate for the time between today and when the cash flow comes in. We will write these interest rates as "r(T)", for example r(1) for the 1-year rate, r(2) for the two-year rate and so on. • Example: Coupon Bond with annual coupon rate of 10%. Write "rA(T)" for the interest rate where the subscript "A" means annual compounding. Look at slide 52 to see formula
Discount Function
• Recall that "Present Value" is simply the value today of $1 to be received at sometime in the future. • We will use D(T) to denote the present value of $1 to be received at time T. • We will also use D(t, T) to represent the present value of $1 to be received at time T, but computed at time t. - Note that D(T) is just short-hand notation for D(t=0,T) where t=0 is today. - We will compute the D(T) function from bonds issued by the U.S Government. Hence, we will consider the D(T) as risk free. • Examples - D(2) is the present value "now" (or at time 0) of $1 to be received in 2 years from now. - D(0.5, 2) is the present value "6-months from now" of $1 to be received in 2 years from now. - D(1,3) is the present value "1-year from now" of $1 to be received in 3 years from now.
Yield to Maturity
• The YTM is the return an investor will earn by buying a bond at the stipulated price now and holding it until maturity (assuming no default by the issuer). • The yield to maturity is simply the discount rate implied by a bond's market price and its promised cash flows. - If we already have a market price for a bond, we can use the present value formula to infer the bond's yield. - In other words, the YTM is the discount rate that makes the present value of a bond's payments equal to its price. • To calculate the yield to maturity, we solve the bond price equation for the interest rate (YTM) given the bond's prices. - The YTM is simply the internal rate of return (IRR) of the bond's market price and its promised future cash flows. *Look at slide 28 for formula*
