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Chapter 5: Learning Objectives What is the Interest Rate? Different Interest Rate Measures: from YTM to STRIPS Nominal vs. Real Interest Rates Interest Rates and Taxes

Copyright (c) McGraw-Hill Ryerson Limited Payment Schemes Simple loan: one payment at maturity Fixed payment loan: payments at fixed intervals Coupon bond: fixed payouts & principal repayment Discounted bonds: Treasury bill STRIPS: Separate Trading of Registered Principal & Interest of Securities

The Present Value Concept PV = FV / (1 + R)ⁿ PV: the principal, the value on a given date of a future payment or series of future payments FV: the value of an asset at a specific date R: discount rate n: term to maturity Copyright (c) McGraw-Hill Ryerson Limited

The Basic Present Value Principle tt+1 $X$100 Present Value Future Value

Copyright (c) McGraw-Hill Ryerson Limited 2006 Present Value of $100

Debt Futures contract The difference b/w the amount to be repaid and the amount borrowed is expressed as the percentage of the debt, interest rate (R) R= (FV – PV)/ PV Copyright (c) McGraw-Hill Ryerson Limited

Ex: $100 to be delivered one year from now, what’s the size of the debt today? R= 10%= 0.10 FV= $100 PV in 1 year? PV= $100/(1.10)¹= $90.91 R = 12% = 0.12 PV= $100/(1.12)¹= $89.29 The present value of $100 at a 10% interest rate is $90.91 Copyright (c) McGraw-Hill Ryerson Limited

The present value of $100 At a 10% interest rate is $90.91 At a 12% interest rate is $89.29 The price of a financial asset is inversely related to the interest rate paid on the interest Copyright (c) McGraw-Hill Ryerson Limited

Fixed-Payment Loans Debt= A(1+R)¹ + A(1+R)²+….+ A(1+R)ⁿ A: Fixed-payments Annuity= A/R Copyright (c) McGraw-Hill Ryerson Limited

Coupon Bond Debt instruments Fixed interest payment (coupon payment) paid out regularly at a fixed coupon rate Principal is repaid at the maturity Provides a regular stream of income When a bond is traded at less than its face value, it’s trading at a discount When a bond is traded at more than its face value, it’s trading at a premium Copyright (c) McGraw-Hill Ryerson Limited

The Coupon Bond $C $FV RR R t t+1 t+n-1 t+n PV = [$C/(1+R)] + …+ [$C/(1+R) n ] + [$FV/(1+R) n ]

Copyright (c) McGraw-Hill Ryerson Limited Bond Prices and Interest Rates

PV = C/(1+R)¹ + C/(1+R)²+….+ (C+FV)/(1+R)ⁿ C: Annual coupon payment FV: Face value N: Years to maturity PV: Present value Copyright (c) McGraw-Hill Ryerson Limited

Ex: $1,000 bonds that promises to pay interest of $100 per year for 5 years with coupon rate of 10%. PV is the sum of total payment in 5 years and principal payment at the end of the maturity PV=(100/1.10)¹+(100/1.10)²+(100/1.10)³ +(100/1.10) ⁴ +(100+1,000)/(1.10) ⁵ = $1,000 Copyright (c) McGraw-Hill Ryerson Limited

Yield to Maturity (YTM) Actual rate of return from a stream of payments received when a debt is incurred today Average annual rate of return: rate of return at maturity divided by number of years Table 5.1/P.70 PV=FV/[1+(R/k)]^(k*n) k: number of times a year interest is compound Copyright (c) McGraw-Hill Ryerson Limited

Ex: $100 is compounded daily at 10% for a two-year period PV=[$100(1+(0.10/365)]^(365*2) =$ Copyright (c) McGraw-Hill Ryerson Limited

Discounted Bonds or Bills Debt instruments Issued at a discount rate and paid face value at maturity Ex: Canada’s Treasury Bills U.S’s Treasury Bills R=(FV-PD)/PD PD: Discounted price Copyright (c) McGraw-Hill Ryerson Limited

Discounted Bills: The Treasury Bill R = [($FV - $PD)/$PD]X (360/90) $PD$FV Maturity period

Ex: A discounted bond sold at the price of $975 today with face value of $1,000 paid at maturity R= ($1,000-$975)/$975=2.56% $25 or 2.56% gain from trading discounted bond Copyright (c) McGraw-Hill Ryerson Limited

Treasury Bills with maturity of n months R=(FV-PD)/(PD) * (360/n) Ex: T-Bills maturity in 3 months (90 days) R=($1,000-$975)/$975 * (360/90) = 10.24% Copyright (c) McGraw-Hill Ryerson Limited

STRIPS or Zero-Coupon Bonds Zero-coupon bonds created by removing the coupons and trading them separately The return to the investors comes from increases in price until maturity when the face value is paid One payment is paid out at a specific day in the future FV=X(1+R)ⁿ X: The amount invested today Copyright (c) McGraw-Hill Ryerson Limited

STRIPS $FV $C PtPt $C=% FV Investors wanting a lump-sum payment will prefer option I Investors wanting regular payments will want Option II I II P t (1 + R) n = FV

Ex: $5,000 STRIPS bought today will return $100,000 in 20 years $100,000=$5,000(1+R)^20  (1+R)^20=$100,000/$5,000=20  (1+R)=1.16  R=16% Copyright (c) McGraw-Hill Ryerson Limited

Capital Gains or Losses Changes in the market values of debt instruments (assets) between the time they are purchasing and the time they are sold Copyright (c) McGraw-Hill Ryerson Limited

Holding Period Yield The return for the period of ownership Could be different from the yield to maturity i=(C/PV)+(∆PV/PV) i: return from holding a bond for period of time ∆PV: change in the price of bond during the holding period Copyright (c) McGraw-Hill Ryerson Limited

Current Yield PtPt P t+1 $C R = $C/P t +  P t /P t Current Yield = one-period return + Capital gain/loss

Ex: C=$100, PV=$1,000, ∆PV=$10 i=($100/$1,000)+($10/$1,000) =11% Copyright (c) McGraw-Hill Ryerson Limited

Internal Rate of Return $NF 0 ={$NF 1 /(1+R)}+ {$NF 1 /(1+R) 2} +…+{$NF n /(1+R) n } Initial Outlay of $NF 0 t=0 NF 1 NF n Cash flows over n periods

The interest rate that equates the present value of outflows and inflows of cash over the anticipated life of a project Copyright (c) McGraw-Hill Ryerson Limited

Ex: Initial cash inflow is $10,000 today produces net flows of $4,000 per year over a four-year period $10,000=$4,000/(1+R)¹+$4,000/(1+R)²+ $4,000/(1+R)³+$4,000/(1+R) ⁴  R= 21% Copyright (c) McGraw-Hill Ryerson Limited

Nominal vs. Real Interest Rates R Nominal Interest rate fixed t+1 t+2 t+3  ee ee ee Ex ante Ex post R=  +  R=  +  e

Copyright (c) McGraw-Hill Ryerson Limited Nominal Interest Rate and Inflation

Copyright (c) McGraw-Hill Ryerson Limited Real Interest Rates in Canada

Copyright (c) McGraw-Hill Ryerson Limited Inflation and Nominal Interest Rates Across Countries:

Copyright (c) McGraw-Hill Ryerson Limited Financial Focus 5.2: Real Return Bonds Year Real return (%) LT Govt bond (%) Annual CPI Inflation Ex post real Interest rate Avg

Copyright (c) McGraw-Hill Ryerson Limited How the Real return bond works: An Example Reference vs Actual Date Need to interpolate because CPI is available monthly with a lag Purchase date: 20 JAN 2003; Selling date: 20 July 2003 (= 6 month holding period) Reference date is: 20 April 2003 CPI 20 July 2003= (130.1 [CPI 30/4/03] * (20/31=interpolation period) [CPI 31/3/03])= Inflation during holding period is: ( )/126.64=2.68%

Copyright (c) McGraw-Hill Ryerson Limited How the Real return bond works: An Example Amounts received for a $100,000 real return bond held for 6 months: compensation for inflation: $100,000*1.0268= $102, Compensation for real return: $102,677.94* (1/2 of 4.25% real return) = $ TOTAL: $102, $ = $104,859.83

Copyright (c) McGraw-Hill Ryerson Limited Summary Debt is like a futures contract The “price” of debt is the “interest rate” The PV of a stream of payments is discounted by the interest rate until the term to maturity The key mathematical relations are: X t = X 0 / (1+R) n PV t = [$C/(1+R)] + … + [$C/(1+R) n )] + [$FV/(1+R) n ] R = ($FV - $PD)/ $PD X (360/90) R c t = C/ P t Fisher equation: R =  + 