Chapter 4. Present and Future Value Future Value Present Value Applications  IRR  Coupon bonds Real vs. nominal interest rates Future Value Present Value Applications  IRR  Coupon bonds Real vs. nominal interest rates

Present & Future Value time value of money $100 today vs. $100 in 1 year  not indifferent!  money earns interest over time,  and we prefer consuming today time value of money $100 today vs. $100 in 1 year  not indifferent!  money earns interest over time,  and we prefer consuming today

example: future value (FV) $100 today interest rate 5% annually at end of 1 year: (100 x.05) = 100(1.05) = $105 at end of 2 years: (1.05) 2 = $ $100 today interest rate 5% annually at end of 1 year: (100 x.05) = 100(1.05) = $105 at end of 2 years: (1.05) 2 = $110.25

future value of $100 in n years if annual interest rate is i: = $100(1 + i) n with FV, we compound cash flow today to the future of $100 in n years if annual interest rate is i: = $100(1 + i) n with FV, we compound cash flow today to the future

Rule of 72 how long for $100 to double to $200? approx. 72/i at 5%, $100 will double in  72/5 = 14.4  $100(1+i) 14.4 = $201.9 how long for $100 to double to $200? approx. 72/i at 5%, $100 will double in  72/5 = 14.4  $100(1+i) 14.4 = $201.9

present value (PV) work backwards if get $100 in n years, what is that worth today? work backwards if get $100 in n years, what is that worth today? PV= $100 (1+ i) n

exampleexample receive $100 in 3 years i = 5% what is PV? receive $100 in 3 years i = 5% what is PV? PV= $100 (1+.05) 3 =$86.36

With PV, we discount future cash flows  Payment we wait for are worth LESS With PV, we discount future cash flows  Payment we wait for are worth LESS

i = interest rate = discount rate = yield annual basis i = interest rate = discount rate = yield annual basis About i

n i PV

PV, FV and i given PV, FV, calculate I example: CD initial investment $1000 end of 5 years $1400 what is i? given PV, FV, calculate I example: CD initial investment $1000 end of 5 years $1400 what is i?

is it 40%? is 40%/5 = 8%? No…. i solves is it 40%? is 40%/5 = 8%? No…. i solves i = 6.96%

ApplicationsApplications Internal rate of return (IRR) Coupon Bond Internal rate of return (IRR) Coupon Bond

Application 1: IRR Interest rate  Where PV of cash flows = cost Used to evaluate investments  Compare IRR to cost of capital Interest rate  Where PV of cash flows = cost Used to evaluate investments  Compare IRR to cost of capital

ExampleExample Computer course  $1800 cost  Bonus over the next 5 years of $500/yr. We want to know i where PV bonus = $1800 Computer course  $1800 cost  Bonus over the next 5 years of $500/yr. We want to know i where PV bonus = $1800

Solve the following: Solve for i? Trial & error Spreadsheet Online calc. Solve for i? Trial & error Spreadsheet Online calc. Answer? 12.05%

ExampleExample Bonus: 700, 600, 500, 400, 300 Solve Bonus: 700, 600, 500, 400, 300 Solve i = 14.16%

ExampleExample Bonus: 300, 400, 500, 600, 700 Solve Bonus: 300, 400, 500, 600, 700 Solve i = 10.44%

Example: annuity vs. lump sum choice:  $10,000 today  $4,000/yr. for 3 years which one? implied discount rate? choice:  $10,000 today  $4,000/yr. for 3 years which one? implied discount rate?

i = 9.7%

purchase price, P promised of a series of payments until maturity  face value at maturity, F (principal, par value)  coupon payments (6 months) purchase price, P promised of a series of payments until maturity  face value at maturity, F (principal, par value)  coupon payments (6 months) Application 2: Coupon Bond

size of coupon payment  annual coupon rate  face value  6 mo. pmt. = (coupon rate x F)/2 size of coupon payment  annual coupon rate  face value  6 mo. pmt. = (coupon rate x F)/2

what determines the price? size, timing & certainty of promised payments assume certainty size, timing & certainty of promised payments assume certainty P =PV of payments

i where P = PV(pmts.) is known as the yield to maturity (YTM)

example: coupon bond 2 year Tnote, F = $10,000 coupon rate 6% price of $9750 what are interest payments? (.06)($10,000)(.5) = $300  every 6 mos. 2 year Tnote, F = $10,000 coupon rate 6% price of $9750 what are interest payments? (.06)($10,000)(.5) = $300  every 6 mos.

what are the payments? 6 mos. $300 1 year $ yrs. $300 ….. 2 yrs. $300 + $10,000 a total of 4 semi-annual pmts. 6 mos. $300 1 year $ yrs. $300 ….. 2 yrs. $300 + $10,000 a total of 4 semi-annual pmts.

YTM solves the equation i/2 is 6-month discount rate i is yield to maturity i/2 is 6-month discount rate i is yield to maturity

how to solve for i?  trial-and-error  bond table*  financial calculator  spreadsheet how to solve for i?  trial-and-error  bond table*  financial calculator  spreadsheet

price between $9816 & $9726 YTM is between 7% and 7.5% (7.37%) price between $9816 & $9726 YTM is between 7% and 7.5% (7.37%)

P, F and YTM P = F then YTM = coupon rate P coupon rate  bond sells at a discount P > F then YTM < coupon rate  bond sells at a premium P = F then YTM = coupon rate P coupon rate  bond sells at a discount P > F then YTM < coupon rate  bond sells at a premium

P and YTM move in opposite directions interest rates and value of debt securities move in opposite directions  if rates rise, bond prices fall  if rates fall, bond prices rise P and YTM move in opposite directions interest rates and value of debt securities move in opposite directions  if rates rise, bond prices fall  if rates fall, bond prices rise

Maturity & bond price volatility

YTM rises from 6 to 8%  bond prices fall  but 10-year bond price falls the most Prices are more volatile for longer maturities  long-term bonds have greater interest rate risk YTM rises from 6 to 8%  bond prices fall  but 10-year bond price falls the most Prices are more volatile for longer maturities  long-term bonds have greater interest rate risk

Why?  long-term bonds “lock in” a coupon rate for a longer time  if interest rates rise -- stuck with a below-market coupon rate  if interest rates fall -- receiving an above-market coupon rate Why?  long-term bonds “lock in” a coupon rate for a longer time  if interest rates rise -- stuck with a below-market coupon rate  if interest rates fall -- receiving an above-market coupon rate

Real vs. Nominal Interest Rates thusfar we have calculated nominal interest rates  ignores effects of rising inflation  inflation affects purchasing power of future payments thusfar we have calculated nominal interest rates  ignores effects of rising inflation  inflation affects purchasing power of future payments

exampleexample $100,000 mortgage 6% fixed, 30 years $600 monthly pmt. at 2% annual inflation, by 2037  $600 would buy about half as much as it does today $600/(1.02) 30 = $331 $100,000 mortgage 6% fixed, 30 years $600 monthly pmt. at 2% annual inflation, by 2037  $600 would buy about half as much as it does today $600/(1.02) 30 = $331

so interest charged by a lender reflects the loss due to inflation over the life of the loan

real interest rate, i r nominal interest rate = i expected inflation rate = π e approximately: i = i r + π e The Fisher equation ori r = i – π e [exactly: (1+i) = (1+i r )(1+ π e )] nominal interest rate = i expected inflation rate = π e approximately: i = i r + π e The Fisher equation ori r = i – π e [exactly: (1+i) = (1+i r )(1+ π e )]

real interest rates measure true cost of borrowing why?  as inflation rises, real value of loan payments falls,  so real cost of borrowing falls real interest rates measure true cost of borrowing why?  as inflation rises, real value of loan payments falls,  so real cost of borrowing falls

inflation and i if inflation is high… lenders demand higher nominal rate, especially for long term loans long-term i depends A LOT on inflation expectations if inflation is high… lenders demand higher nominal rate, especially for long term loans long-term i depends A LOT on inflation expectations