PRICING SECURITIES Chapter 6 Bank Management, 5th edition. Timothy W. Koch and S. Scott MacDonald Copyright © 2003 by South-Western, a division of Thomson Learning PRICING SECURITIES Chapter 6

Future value and present value PV(1+i) = FV1 Example: 1,000 PV and 1,080 FV means: i = $80 / $1,000 = 0.08 = 8% If we invest $1,000 at 8% for 2 years: $1,000 (1+0.08) (1+0.08) = $1,080 (1.08) = $1,166.40

In general, the future value is… FVn = PV(1+i)n Alternatively, the yield can be found as: i = [FVn / PV](1/n) - 1 Example: $1,000 invested for 6 years at 8%: FV6 = $1,000(1.08)6 = $1,586.87 Example: Invest $1,000 for 6 years, Receive $1,700 at the end of 6 years. What is the rate of return? i = [$1,700 / $1,000](1/6) - 1 = 0.0925

Future value and present value …multiple payments The cumulative future value of a series of cash flows (CFVn) after n periods is: CFVn =CF1(1+i)n +CF2(1+i)n-1 +...+CFn(1+i) The present value of a series of n cash flows: PV = [CF1 / (1+i)] + [CF2 / (1+i)2] + [CF3 / (1+i)3] . . . +[CFn / (1+i)n] Example: rate of interest = 10%, what is the PV of a security that pays $90 at the end of the next three years plus $1,000 at the end of three years? PV = 90/(1.1) + 90/(1.1)2 + 1090/(1.1)3 = $975.13

Simple interest versus compound interest Simple interest is interest that is paid only on the initial principal invested: simple interest = PV x (i) x n Example, simple interest: if i = 12% per annum, n = 1 and principal of $1,000: simple interest = $1,000 (0.12) 1 = $120 Example, interest is paid monthly: monthly simple interest = $1,000 (0.12 / 12) 1= $10 Compounded interest is interest that is paid on the interest: PV (1 + i/m)nm = FVn and PV = FVn / (1 + i/m)nm

Compounding frequency Example: $1,000 invested for 1 year at 8% with interest compounded monthly: FV1 = 1,000 (1 + 0.08/12)12 = 1083.00 The effective annual rate of interest, i* can be calculated from: i* = (1 + i/m)m - 1 In this example: i* = (1 + 0.08/12)12 - 1 = 8.30%

The effect of compounding on future value and present value

Bond prices and interest rates vary inversely Consider a bond which pays semi-annual interest payments of $470 with a maturity of 3 years. If the market rate of interest is 9.4%, the price of the bond is: If the market rates of interest increases to 10%, the price of the bond falls to $9,847.72:

Price and yield relationships for option-free bonds that are equivalent except for the feature analyzed

In general Par Bond Discount Bond Premium Bond Yield to maturity = coupon rate Discount Bond Yield to maturity > coupon rate Premium Bond Yield to maturity < coupon rate

Relationship between price and interest rate on a 3-year, $10,000 option-free par value bond that pays $270 in semiannual interest For a given absolute change in interest rates, the percentage increase in a bond’s price will exceed the percentage decrease. 10,155.24 10,000.00 9,847.73 8.8 9.4 10.0 Interest Rate % $’s D = +$155.25 D = -$152.77 This asymmetric price relationship is due to the convex shape of the curve-- plotting the price interest rate relationship. Bond Prices Change Asymmetrically to Rising and Falling Rates

The effect of maturity on the relationship between price and interest rate on fixed-income, option free bonds $’s For a given coupon rate, long-term bonds price changes proportionately more in price than do short-term bonds for a given rate change. 10,275.13 10,155.24 10,000.00 9,847.73 9,734.10 8.8 9.4 10.0 Interest Rate % 9.4%, 3-year bond 9.4%, 6-year bond

The effect of coupon on the relationship between price and interest rate on fixed-income, option free bonds % change in price For a given change in market rate, the bond with the lower coupon will change more in price than will the bond with the higher coupon. + 1.74 + 1.55 0 - 1.52 - 1.70 8.8 9.4 10.0 Interest Rate % 9.4%, 3-year bond Zero Coupon, 3-year bond

Duration and price volatility Maturity simply identifies how much time elapses until final payment. It ignores all information about the timing and magnitude of interim payments. Duration is a measure of effective maturity that incorporates the timing and size of a security's cash flows. Duration captures the combined impact of market rate, the size of interim payments and maturity on a security’s price volatility.

Duration versus maturity 1.) 1000 loan, principal + interest paid in 20 years. 2.) 1000 loan, 900 principal in 1 year, 100 principal in 20 years. 1000 + int |-------------------|-----------------| 0 10 20 900+int 100 + int |----|--------------|-----------------| 0 1 10 20 What is the maturity of each? 20 years What is the "effective" maturity? 2.) = [(900/100) x 1]+[(100/1000) x 20] = 2.9 yrs Duration, however, uses a weighted average of the present values. 1 2

Duration …approximate measure of the price elasticity of demand Price elasticity of demand = % in quantity demanded / % in price Price (value) changes Longer duration  larger changes in price for a given change in i-rates. Larger coupon  smaller change in price for a given change in i-rates.

Duration …approximate measure of the price elasticity of demand Solve for DPrice: DP @ -Duration x [Di / (1 + i)] x P Price (value) changes Longer maturity/duration larger changes in price for a given change in i-rates. Larger coupon smaller change in price for a given change in i-rates.

Measuring duration In general notation, Macaulay’s duration (D): Example: 1000 face value, 10% coupon, 3 year, 12% YTM

Measuring duration If YTM = 5% 1000 face value, 10% coupon, 3 year, 5% YTM

If YTM = 20% 1000 face value, 10% coupon, 3 year, 20% YTM Measuring duration If YTM = 20% 1000 face value, 10% coupon, 3 year, 20% YTM

Measuring duration If YTM = 12% and Coupon = 0 1000 face value, 0% coupon, 3 year, 12% YTM 1000 |-------|-------|-------| 0 1 2 3

Compare price sensitivity Duration allows market participants to estimate the relative price volatility of different securities: Using modified duration: modified duration = Macaulay’s duration / (1+i) We have an estimate of price volatility: %change in price = modified duration x change in i

Comparative price sensitivity indicated by duration DP = - Duration [Di / (1 + i)] P DP / P = - [Duration / (1 + i)] Di where Duration equals Macaulay's duration.

Valuation of fixed income securities Traditional fixed-income valuation methods are too simplistic for three reasons: Investors do not hold securities until maturity Present value calculations assumes all coupon payments are reinvested at the calculated Yield to Maturity Many securities carry embedded options, such as a call or put, which complicates valuation since it is unknown if the option will be exercised.

Total return analysis Market participants attempt to estimate the actual realized yield on a bond by calculating an estimated total return = [Total future value / Purchase price](1/n) - 1

Total return for a 9-year 7. 3% coupon bond purchased at $99 Total return for a 9-year 7.3% coupon bond purchased at $99.62 per $100 par value and held for 5-years. Assume: semiannual reinvestment rate = 3% after five years; a comparable 4-year maturity bond will be priced to yield 7% (3.5% semiannually) to maturity Coupon interest: 10 x $3.65 = $36.50 Interest-on-interest: $3.65 [(1.03)10 -1] / 0.03 - $36.50 = $5.34 Sale price after five years: Total future value: $36.50 + $5.34 + $101.03 = $142.87 Total return: [$142.87 / $99.62]1/10 - 1 = 0.0367 or 7.34% annually

Money market yields Interest rates for most money market yields are quoted on a different basis. In particular, some money market instruments are quoted on a discount basis, while others bear interest. Some yields are quoted on a 360-day year rather than a 365 or 366 day year.

Interest-bearing loans with maturities of one year or less The effective rate of interest depends on the term of the loan and the compounding frequency. A one year loan which requires monthly interest payments at 12% annually, carries an effective yield to the bank of 12.68%: i* = (1 + 0.12/12)12 - 1 = 12.68% If the same loan was made for 90 days: i* = [1 + 0.12 / (365/90)](365/90) - 1 = 12.55% In general: i* = [1 + i / (365 / h)](365/h) - 1

360-day versus 365-day yields Some securities are reported using a 360 year rather than a full 365 day year. This will mean that the rate quoted will be 5 days too small on a standard annualized basis of 365 days. To convert from a 360-day year to a 365-day year: i365 = i360 (365/360) Example: one year instrument at 8% nominal rate on a 360-day year is actually an 8.11% rate on a 365-day year: i365 = 0.08 (365/360) = 0.0811

Discount yields Some money market instruments, such as Treasury Bills, are quoted on a discount basis. This means that the purchase price is always below the par value at maturity. The difference between the purchase price and par value at maturity represents interest. The pricing equation for a discount instrument is: idr = [(Pf - Po) / Pf] (360 / h) where idr = discount rate Po = initial price of the instrument Pf = final price at maturity or sale, h = number of days in holding period.

The bond equivalent rate on discount securities The problems of a 360-day year for a rate quoted on a discount basis can be handled by converting the discount rate to a bond equivalent rate: (ibe) ibe = [(Pf - Po) / Po] (365 / h) Example: consider a $1 million T-bill with 182 days to maturity, price = $964,500. The discount rate is 7.02%, idr = [(1,000,000 - 964,500) / 1,000,000] (360 / 182) = 0.072 The bond equivalent rate is 7.38%: idr = [(1,000,000 - 964,500) / 964,500] (365 / 182) = 0.0738 The effective annual rate is 7.52%: i* = [1 + 0.0738 / (365 / 182)](365/ 182) - 1 = 0.0752

Yields on single-payment interest-bearing securities Some money market instruments, such as large negotiable CD’s, Eurodollars, and federal funds, pay interest calculated against the par value of the security and make a single payment of interest and principal at maturity. Example: consider a 182-day CD with a par value of $1,000,000 and a quoted rate of 7.02%. Actual interest paid at maturity is: (0.0702)(182 / 360) $1,000,000 = $35,490 The 365 day yield is: i365 = 0.0702(365 / 360) = 0.0712 The effective annual rate is 7.31%: i* = {1 + [0.0712 / (365 / 182)]}(365/182) - 1 = 0.0724

Summary of money market yield quotations and calculations Simple Interest is: Discount Rate idr: Money Mkt 360-day rate, i360 Bond equivalent 365 day rate, i365 or ibe: Effective ann. interest rate, Definitions Pf = final value Po = initial value h=# of days in holding period Discount Yield quotes: Treasury bills Repurchase agreements Commercial paper Bankers acceptances Interest-bearing, Single Payment: Negotiable CDs Federal funds

PRICING SECURITIES Chapter 6 Bank Management, 5th edition. Timothy W. Koch and S. Scott MacDonald Copyright © 2003 by South-Western, a division of Thomson Learning PRICING SECURITIES Chapter 6