# Measuring Interest Rates zBond Interest Rate is more formally called its Yield to Maturity zYield to Maturity -- the interest rate which equates the present.

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Measuring Interest Rates zBond Interest Rate is more formally called its Yield to Maturity zYield to Maturity -- the interest rate which equates the present value of all future payments with the current bond price

Present Value zPresent Value – an equation that converts future payments into their current dollar equivalent zExample 1 – Find the present value of payment received one year from now. zGiven P dollars today, with interest rate i, how much will you have one year from now (F)?

Answer to Example 1 zF = Repayment of principal + Payment of Interest zF = (P) + (i)(P) = (P)(1 + i) zTo obtain the present value of the future payment, solve for P zP = F/(1 + i) -- Present value of payment (F) received one year from now

Example 2 -- Present Value of Fixed Payment (F) Received n Years From Now zAfter One Year: F = P(1 + i) zTwo Years: F = [P(1 + i)](1 + i) F = P(1 + i) 2 zThree Years: F = P(1 + i) 3 … zn Years: F = P(1 + i) n

Obtaining The Present Value zTo convert to current dollars, solve previous equation for P zP = F/(1 + i) n zPresent Value of Payment Received n Years From Now

Example 3 -- Present Value of Annual Stream of Payments zSuppose one receives a payment of A 1 at the end of year 1, A 2 at the end of year 2, A 3 at the end of year 3, …, and A n at the end of year n. What is the present value (current dollar equivalent) of that series of payments?

Answer to Example 3 zPresent Value = Sum or the present values of each payment zP = A 1 /(1 + i) + A 2 /(1 + i) 2 + A 3 /(1 + i) 3 + … + A n /(1 + i) n

Present Value -- Applications zConsider formula (for simplicity, let A 1 = A 2 = A 3 = … = A n = A) zP = A/(1 + i) + A/(1 + i) 2 + A/(1 + i) 3 + … + A/(1 + i) n zGiven any 2 variables, we can solve for the third.

Application #1 -- Given A and i, Solve for P zExamples -- Multiyear Contracts, Lottery Winnings zExample -- You win \$100,000 for year 1 \$125,000 for year 2 and \$150,000 for year 3, with i = 0.08. P = \$100,000/(1 + 0.08) + \$125,000/(1 + 0.08) 2 + \$150,000/(1 + 0.08) 3 = \$318,834.78

Application #2 -- Given P and i, Solve for A zComputing Annual Loan Payments yP = Amount Borrowed yi = Interest rate on the loan

An Example zYou take out a 5 year loan of \$20,000 to buy a car, at a loan rate of 9% (0.09). What is your annual payment?

Answer to Car Loan Problem z\$20,000 = A/(1 + 0.09) + A/(1 + 0.09) 2 + A/(1 + 0.09) 3 + A/(1 + 0.09) 4 + A/(1 + 0.09) 5, zSolve for A zA = \$5141.85

Computing Monthly Loan Payments zExample -- Car Loan Problem zSame Present Value Formula -- Minor Adjustments i = 0.09/12 = 0.0075 (monthly interest rate) n = 5 x 12 = 60 months

Monthly Loan Payment \$20,000 = A/(1.0075) + A/(1.0075) 2 + A/(1.0075) 3 + … + A/(1.0075) 60 Solve for A (ugh!!)

A Compressed Formula for Computing Loan Payments  Consider again the present value formula. P = A/(1 + i) + A/(1 + i) 2 + A/(1 + i) 3 + … + A/(1 + i) n.  For loan payment, given P and i, solve for A.

Solution for A zBased upon the solution to a geometric series, one can show that the equation solves as: A = (i)(P)/[1 – 1/(1 + i) n ]. zMonthly loan payment: A = (0.0075)(\$20,000)/[1 – 1/(1.0075) 60 ] A = \$415.17

Application #3 -- Given P and A, Solve for i zExample: Yield to Maturity (interest rate) on Bonds zApply present value equation to determine bond interest rates z Based upon the series of future payments and the current bond price (P B )

Yield to Maturity: Long-Term Bonds zInformation printed on the face of the bond -- Coupon rate (i C ) -- Face value (F)

Structure of Repayment: Long-Term Bond zSeries of Future Payments: Coupon (interest) payment each year equal to C = (i C )(F) along with the face value (F) (or par value) at maturity. zThese payments are fixed, no matter what the bond sells for.

Long-Term Bonds: Bond Price and Interest Rate zBond price (P B ) -- determined by market conditions, constantly fluctuating. yP B < F -- the bond sells at a discount yP B > F -- the bond sells at a premium yP B = F -- the bond sells at par zInterest Rate (Yield to Maturity) -- solution to the present value equation, given future payments and bond price

A General Formula Yield to Maturity: Long-Term Bond P B = C/(1 + i) + C/(1 + i) 2 + C/(1 + i) 3 + … + C/(1 + i) n + F/(1 + i) n Solve for i (ugh!!)

An Example zFind the yield to maturity for a 20 year Corporate Bond, with a coupon rate of 7% (0.07), a face value of \$1000, which sells for \$975. zCoupon payment: C = (0.07)(\$1000) = \$70 per year zBond also pays \$1000 at maturity (year 20).

Solving the Problem \$975 = \$70/(1 + i) + \$70/(1 + i) 2 + \$70/(1 + i) 3 + … + \$70/(1 + i) 20 + \$1000/(1 + i) 20 Solve for i (ugh!!)

The Yield to Maturity and the Coupon Rate zOne can show the following properties. zIf P B = F (coincidentally) then i = i C. zIf P B i C. zIf P B > F, then i < i C.

Important Property: Bonds zBond Prices and Bond interest rates are inversely related, by definition. zIn other words, P B   i  zKey reason: future payments are fixed, no matter what price the bond sells for.

Special Cases: Yield to Maturity, Long-Term Bonds zConsol (Perpetuity) -- Pays fixed payment C each year, no maturity P B = C/(1 + i) + C/(1 + i) 2 + C/(1 + i) 3 + …, Solve for i P B = C/i, which implies that i = C/P B.

zZero Coupon Bond -- No annual payment, just face value (F) at maturity P B = F /(1 + i) n, Solve for i  i = (F/P B ) 1/n - 1

Yield to Maturity -- Money Market Bonds zMethod of repayment -- Holder just receives face value at maturity zFormula -- One year bond P B = F /(1 + i), Solve for i  i = (F - P B )/P B

Bonds With Maturities of Less Than One Year Simple Adjustment: Multiply the formula for the 1 year one by an annualizing factor. Formula: i = [(F - P B )/P B ][365/(# of days until maturity)]

An Example zSuppose that a 90-day Treasury-Bill has a face value of \$100000 and 59 days until maturity. It sells on the secondary market for \$99800. Find the Yield to Maturity (i). i = [(\$100000 - \$99800)/(\$99800)] x [365/59] = 0.0124 = 1.24%

Other Measures of Yield or Return on Financial Assets zCurrent Yield (i CUR ), i CUR = C/P B zYield on a Discount Basis (i DB ), or Discount Yield i = [(F - P B )/F][360/(# of days until maturity)]

Rate of Return zRate of Return (RET) -- Annual return based upon financial asset’s current value (bonds sold before maturity, stock) Formula for Rate of Return (bond) RET t = [C + (P Bt - P B,t-1 )]/P B,t-1

Rate of Return: An Example zSuppose that a long-term bond has a coupon rate of 5% and a face value of \$1000. It sold for \$990 last year and currently sells for \$975. Find the Rate of Return (RET). C = (0.05)(\$1000) = \$50 RET = [\$50 + (\$975 - \$990)]/\$990 = 0.0354 = 3.54%

Implications: Rate of Return zInvestors can lose money (RET < 0) holding bonds. zFormula also applies to stocks. zBonds and stocks are substitutes, existence of bond traders. zThe possibility of unknown capital gains or losses introduces uncertainty.

Another Inconvenience: Market Risk zMarket (Asset Price) Risk -- Uncertainty due to bond prices (and interest rates) changing, affecting rate of return zMarket Risk   i  zFactors affecting Market Risk yMaturity yInterest rate volatility (σ B ), or degree of interest rate fluctuation

Real Versus Nominal Interest Rates zNominal Interest Rate -- Observed, unadjusted yield to maturity zReal Interest Rate -- Interest Rate adjusted for inflation zKey issue -- Must align interest rate and inflation measure so that they cover the same time span.

The Ex-Post Real Interest Rate z Ex-Post Real Interest Rate (r) r = i PAST - , i PAST = past interest rate  = actual measured inflation rate (from past period to now)

The Ex-Ante Real Interest Rate z Ex-Ante Real Interest Rate (r e ) r e = i -  e, i = current interest rate  e = expected inflation rate (from now through the maturity of the bond) zThe most commonly used measure of the real interest rate

The Fisher Effect zFisher Effect -- The current nominal interest rate is constantly 2%-4% above the inflation rate expected over the life of the bond. zCrude initial theory of interest rate determination, shows important role of expected inflation in affecting nominal interest rates

Application: Inflation-Indexed Bonds zInflation-Indexed Bonds (I-Bonds) -- T-Bonds or Savings Bonds that pay a base rate (e.g. 2%) plus an adjustable interest rate based upon the existing rate of inflation (over a the given period from the most recent past). zSeeks to approximate a constant real interest rate, even though it’s actually neither the ex-ante nor ex-post measure.

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