1 Math of Finance Texas Association of Counties January 2014
2 Money Market Securities Sample Treasury Bill Quote*: N Bid Ask Ask Yld *(Yields do not reflect current market conditions)
3 Bank Discount Rate Bid and Ask are quoted on a Bank Discount Rate basis: Where: 100= 100% of par P = Price (% of par) N= # of days until maturity Limitations: The BDR computes yield as percentage of par rather than on invested capital, plus annualizes using 360-day year. As a result, it understates the return on invested capital.
4 Bond Equivalent Yield The bond equivalent yield (BEY) computes yield as a percentage of invested capital (P) and annualizes using a 365-day year. The BEY remedies the bias inherent in the bank discount rate method.
5 Use BDR to Compute Price Computing the Price: Use BDR equation solve for P : P = 100 x [1 (BDR)(N/360)] = 100 x [1 (.0485)(126/360)] = 100 x = (% of par)
6 Convert to Bond Equivalent Yield Plugging P into the BEY equation: BEY BEY = or 5.00% (Reported as the Ask Yld in the Wall Street Journal)
7 Effective Annual Yield The bond-equivalent yield (BEY) ignores interest-on-interest that can be earned by reinvesting cash flows during the year. ] The effective annual yield assumes that the interim proceeds are reinvested at the current BEY. The compounding of interest-oninterest can add several basis points to your realized rate of return.
8 Bonds and Time Value of Money When a financial instrument generates cash flows over a period longer than 1 year, one must consider the time value of money and the compounding effect of interest-on-interest. In addition, one must recognize that a dollar received today is worth more than a future dollar, therefore one must discount future cash flows at an appropriate interest rate.
9 Future Value of a Lump Sum The future value (FV) of a lump sum invested at the beginning of an investment horizon compounding FV t = PV (1 + r) t = PV(FVIF r,t ) PV = present value of cash flow FV t = future value of cash flow (lump sum) received at the end of t periods r = interest rate per period t = number of periods in investment horizon FVIF r,t = future value interest factor of a lump sum
10 Present Value of a Lump Sum Discounting a future lump sum using current interest rates to find its present value (PV) PV = FV t [1/(1 + r)] t = FV t (PVIF r,t ) PVIF r,t = present value interest factor of a lump sum
11 Present Value of an Annuity The present value of a finite series of equal cash flows received on the last day of equal intervals throughout the investment horizon PV PMT t j 1 [1/(1 r)] t j ) PMT = periodic annuity payment t = number of annuity payments PVIFA r,t = present value interest factor of an annuity 1 (1 r PMT r
12 Future Value of an Annuity The future value of a finite series of equal cash flows received on the last day of equal intervals throughout the investment horizon FV t t 1 j (1 r) PMT (1 r) PMT j 0 r t 1 FVIFA r,t = future value interest factor of an annuity
13 Effective Annual Return Effective or equivalent annual return (EAR) is the return earned or paid over a 12-month period taking compounding into account EAR = (1 + r per period ) m 1 m = the number of compounding periods per year
14 Financial Calculators Key inputs/outputs (solve for one of five): N = number of compounding periods I/Y = interest rate per period PV = present value (i.e., current price) PMT = a constant payment every period FV = future value (i.e., future price)
15 Calculating Time Value: Computers and Spreadsheets Built-in functions similar to financial calculators Values for variables are entered in individual cells on spreadsheet Cells are linked by equations programmed to calculate time values Changes in variable values automatically update calculations
16 Future Value: An Extension of Compounding Future Value: the amount to which a current deposit will grow over a period of time when it is placed in an account paying compound interest Future Value Example: How much will $1,000 deposited into an account earning 8% compounded annually be worth in two years?
17 Future Value: Using an Excel Spreadsheet
18 Future Value of an Annuity Annuity: a stream of equal cash flows that occur at equal intervals over time Ordinary Annuity: an annuity where cash flows occur at the end of each period Future Value of an Annuity Example: What is the future value of $1,000 deposited at the end of each year for 8 years in an account earning 6% compounded annually?
19 Future Value of an Annuity: Using an Excel Spreadsheet
20 Present Value: An Extension of Future Value Present Value: the value today of a sum to be received at some future date Present Value Example: How much would need to be deposited today into an accounting earning 6% compounded annually to grow to $500 in 7 years?
21 Present Value: Using an Excel Spreadsheet
22 Present Value of an Annuity Annuity: a stream of equal cash flows that occur at equal intervals over time Present Value of an Annuity Example: What is the present value of $50 deposited at the end of each year for 5 years in an account earning 9% compounded annually?
23 Present Value of an Annuity: Using an Excel Spreadsheet
24 Bond Valuation The present value of a bond (V b ) can be written as (with semi-annual coupon payments): V b C 2 C 2 2N t 1 1 (1 ( r / 2)) 1 1 (1 (r (r 2) 2)) t 2N (1 ( Par r / 2)) 2N Par (1 (r/2)) 2N Par = the par or face value, usually $1,000 C = the annual interest (or coupon) payment (coupon rate x par) fixed for the life of the bond N = the number of years until the bond matures r = the annual interest rate (often called yield to maturity (ytm)) YTM changes with market interest rates.
25 Bond Valuation The key point to take from above equation: On a fixed-rate coupon bond: -When the required interest rate (YTM) increases (denominator), the bond price falls. -When the required interest rate (YTM) falls, the bond price increases.
26 The Pricing of Bonds (cont d) Bond Pricing Example: What is the market price of a $1,000 par value 20 year bond that pays 9.5 % compounded annually when the market rate is 10%? Since the required YTM is greater than the coupon rate, the bond trades at a discount.
27 Bond Pricing Rules A bond trades at a premium to par when the YTM is less than the coupon rate and the fair present value of the bond (V b ) is greater than the face or par value (Par) Premium bond: If YTM <C%, then V b > Par Discount bond: if YTM > C%, then V b < Par Par bond: if YTM = YTM, then V b = Par
28 Relation between Interest Rates and Bond Values (10% semi-annual coupon, 12 years to maturity, $1000 par) Yield to Maturity 12% 10% 8% ,000 1, Bond Value
29 Price at YTM = 12% (10% semi-annual, 12 years to maturity, $1000 par) Computing Price at required interest rate of 12% using a financial calculator: Inputs: N = 24 (12x2), I = 6 (12/2), PMT = 50 (100/2), FV = 1000 Output: PV = (negative number indicates you must pay to buy the bond)
30 Price Behavior of a Bond
31 Ways to Measure Bond Yield Current Yield Yield-to-Maturity Yield-to-Call (for bonds with call feature)
32 Current Yield Simplest yield calculation Only looks at current income Ignores difference between price and par
33 Yield-to-Maturity Most important and widely used yield calculation Accounts for difference between Price and Par Assumes all interest income is reinvested at rate equal to market rate at time of YTM calculation (realized return may differ due to interest rate risk price risk and reinvestment risk) Calculates value based upon PV of interest received and the appreciation/depreciation of the bond if held until maturity Difficult to calculate without a financial calculator (requires several iterations of trial-and-error)
34 Yield-to-Maturity (cont d) Yield-to-Maturity Example: Find the yield-to-maturity on a 7.5 % ($1,000 par value) bond that has 15 years remaining to maturity and is currently trading in the market at $809.50? Solution: YTM = 10% Concept Check: Since the bond is trading at a discount to par, we know that the YTM is greater than 7.5%.
35 Interest Rates and Bond Price Risk When interest rates change, bond prices change in the opposite direction. Interest rate price risk can seriously impact realized rate of return over short holding periods. What are the factors that determine how much price risk that a bond could suffer?
36 Impact of Maturity on Price Volatility (a) Absolute Value of Percent Change in a Bond s Price for a Given Change in Interest Rates Time to Maturity
37 Impact of Maturity on Price Volatility (b) Coupon 6.00% Par $ 1,000 yield rate old 7.00% yield rate change 0.50% yield rate new 7.50% Absolute Maturity Price old Price new Rate of change 1 $ $ % 5 $ $ % 10 $ $ % 15 $ $ % 20 $ $ % 25 $ $ % 30 $ $ % 35 $ $ % 40 $ $ % 45 $ $ % 50 $ $ % 55 $ $ % 60 $ $ % 65 $ $ % 70 $ $ % 75 $ $ % 80 $ $ % 85 $ $ % 90 $ $ % 95 $ $ % 100 $ $ % 105 $ $ % Predicted limit price change = 1 - (r old / r new) 6.67%
38 Impact of Coupon Rates on Price Volatility (a) Bond Value High-Coupon Bond Low-Coupon Bond Interest Rate
39 Impact of Coupon on Price Volatility (b) Coupon Varies Maturity Par $1, years rate old 7.00 % rate change % rate new 6.50% Absolute Coupon rate Price old Price new Rate of change 6.00% $ $ % 5.50% $ $ % 5.00% $ $ % 4.50% $ $ % 4.00% $ $ % 3.50% $ $ % 3.00% $ $ % 2.50% $ $ % 2.00% $ $ % 1.50% $ $ % 1.00% $ $ % 0.50% $ $ % 0.00% $ $ %
40 Duration Duration (Dur) for a fixed-income security that pays interest annually can be written as: Dur CF N t t t 1 (1 r) P P 0 = Current price of the security t = 1 to T, the period in which a cash flow is received T = the number of years to maturity CF t = cash flow received at end of period t r = yield to maturity or required rate of return PV t = present value of cash flow received at end of period t 0 t
41 What is Bond Duration? Duration measures the sensitivity of a fixedincome security s price to small interest rate changes. Duration captures the coupon and maturity effects on volatility.
42 Duration as a Measure of Price Risk Duration measures the interest rate elasticity of a bond s price: % Longer duration bonds exhibit greater price risk.
43 Duration and Volatility 9% Coupon, 4 year maturity annual payment bond with a 8% ytm Dur T t 1 CF t 0 t (1 r) P t Year (T) Cash Flow CF T /(1+r) T % of Value PV/Price Weighted % of Value (PV/Price)*T 1 $ 90 $ % % % $1090 $ % Totals $ % Duration = years
44 The Concept of Duration Generally speaking, bond duration possesses the following properties: Bonds with higher coupon rates have shorter durations Bonds with longer maturities have longer durations Bonds with higher YTM have shorter durations (see previous examples) Coupon-bearing bonds: D < #yrs. to maturity Zero-coupon bonds: D = # yrs. to maturity
45 Which Bonds Exhibit the Most Price Sensitivity? Long Maturity, Low Coupon
46 Bonds Between Coupon Dates Purchaser will pay accrued interest (portion of coupon due to seller for portion of coupon period. INT Accrued Interest 2 Actual# of days fromlast coupon payment Actual numberof daysin coupon period T-bonds Actual/Actual Corporates 30/360
47 MS Excel Example settlement 5/15/2014 maturity 11/15/2043 rate 0.04 yield 0.04 redemption 100 frequency 2 basis 0 price 100
48 Sample Treasury Bond Quote Maturity Coupon Bid Asked Chg Asked Yld 2017 Nov :26 112: Price = 112 and 27/32 % of Par YTM is % (YTM < 4.250%), so the bond trades at a premium to par.