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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.

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Presentation on theme: "Measuring Interest Rates zBond Interest Rate is more formally called its Yield to Maturity zYield to Maturity -- the interest rate which equates the present."— Presentation transcript:

1 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

2 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)?

3 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

4 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

5 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

6 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?

7 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

8 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.

9 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 = P = $100,000/( ) + $125,000/( ) 2 + $150,000/( ) 3 = $318,834.78

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

11 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?

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

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

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

15 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.

16 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

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 )

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

19 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.

20 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

21 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!!)

22 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).

23 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!!)

24 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.

25 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.

26 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.

27 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

28 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

29 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)]

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

31 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)]

32 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

33 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 = = 3.54%

34 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.

35 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

36 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.

37 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)

38 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

39 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

40 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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