Chapter Four Learning Objectives Valuing Monetary Payments Now and in the Future

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1 Chapter Four Future Value, Preset Value, ad Iterest Rates Chapter 4 Learig Objectives Develop a uderstadig of 1. Time ad the value of paymets 2. Preset value versus future value 3. Nomial versus real iterest rates Iterest rates lik the preset to the future. Tell the future reward for ledig today. Tell the cost of borrowig ow ad repayig later. Valuig Moetary Paymets Now ad i the Future We eed a set of tools: Future value Preset value 1

2 Future Value ad Compoud Iterest Future value: the value o some future date of a ivestmet made today. $100 ivested today at 5% iterest gives $105 i a year. So the future value of $100 today at 5% iterest is $105 oe year from ow. The $100 ivestmet yields $5, which is why iterest rates are called yields. This is a example of a simple loa of $100 for a year at 5% iterest. Future Value ad Compoud Iterest Formula If the preset value (PV) is $100 ad the iterest rate (i) is 5%, the the future value (FV) oe year from ow is: $105 = [$100 + $100(0.05)] = $100(1 +.05)= $100(1.05) The higher the iterest rate, the higher the future value. I geeral: FV = PV + PV x i = PV(1 + i) Future Value ad Compoud Iterest Most fiacial istrumets are ot this simple. We must cosider compoud iterest to compute the value repaid more tha oe year from ow. Compoud iterest is the iterest o the iterest. 2

Future Value ad Compoud Iterest Suppose you deposit $100 (PV) i a bak savig accout for two years at 5%(i) yearly iterest rate? The future value is: FV = $100 + $100(0.05) + $100(0.05) + $5(0.

3 Future Value ad Compoud Iterest Suppose you deposit $100 (PV) i a bak savig accout for two years at 5%(i) yearly iterest rate? The future value is: FV = $100 + $100(0.05) + $100(0.05) + $5(0.05) = $ I geeral: FV = $100(1.05)(1.05) = $100(1.05) 2 FV = PV(1 + i), where is time. Future Value ad Compoud Iterest Computig the future value of $100 at 5% aual iterest Future Value ad Compoud Iterest i ad must be i the same uits. If is aual, i must be aual If the aual iterest rate is 5%, what is the mothly rate? Assume i m is the oe-moth iterest rate ad is the umber of moths, the a deposit made for oe year will have a future value of PV(1 + i m ) 12. (NOTE: i ad are mothly) 3

4 Future Value ad Compoud Iterest We kow that i oe year the future value is $100(1.05) so we ca solve for i m : (1 + i m ) 12 = (1.05) (1 + i m ) = (1.05) 1/12 = , which is 0.407% Fractios of percetage poits are called basis poits. A basis poit is oe oe-hudredth of a percetage poit 1 percet is 100 basis poits 0.407% is 40.7 basis poits Preset Value Preset value is the value today (i the preset) of a paymet that is promised to be made i the future. Or, preset value is the amout that must be ivested today i order to realize a specific amout o a give future date. Preset Value Solve the Future Value Formula for PV: FV = PV x (1+i), so FV PV ( 1 i) This is just the future value calculatio iverted. 4

Preset Value We ca geeralize the process as we did for future value. Preset Value of paymet received years i the future: FV PV ( 1 i) Preset value is higher: 1.

5 Preset Value We ca geeralize the process as we did for future value. Preset Value of paymet received years i the future: FV PV ( 1 i) Preset value is higher: 1. The higher future value of the paymet, FV 2. The shorter time period util paymet,. 3. The lower the iterest rate, i. Preset value is the sigle most importat relatioship i our study of fiacial istrumets. Preset Value of $100 Paymet Higher iterest rates are associated with lower preset values, o matter what the size or timig of the paymet. At ay fixed iterest rate, a icrease i the time reduces its preset value. Computig Compoud Aual Returs We ca tur a mothly growth rate ito a compoud-aual rate. Ivestmet grows 0.5% per moth What is the compoud aual rate? (1.005) 12 = Compoud aual rate = 6.17% (Note: 6.17% > 12 x 0.05 = 6.0%) Mothly Rate = Com Aual Rate Simple PV = 100 Iterest

6 Computig Compoud Aual Rates We ca also use this to compute the percetage chage per year whe we kow how much a ivestmet has grow over a umber of years. Suppose a ivestmet has icreased 20 percet over five years: from $100 to $120. FV = PV(1 + i) 120 = 100(1 + i) 5 Solve for i i = Computig Compoud Aual Rates (1 i) FV PV FV PV ( 1 i) FV 1 i PV 120 i 100 (1/5) (1/ ) 1 FV i PV i = => i =3.71% (1/ ) 1 Iteral Rate of Retur Imagie that you ow a firm ad you are cosiderig purchasig a ew machie. Machie costs $1 millio ad ca produce 4000 uits of product per year. You sell the product for $30 per uit, geeratig $120,000 i added reveue per year. Keepig this simple, assume the machie is the oly iput ad you have certaity about the reveue (very simple), o maiteace (very, very simple) ad a 10 year lifespa. 6

7 Iteral Rate of Retur Questio: if you borrow $1 millio to buy the machie, is the 10 year reveue stream eough to make the paymets? We eed to compare iteral rate of retur (IRR) to the cost of buyig the machie. IRR is the iterest rate that equates the preset value of a ivestmet with its cost. Iteral Rate of Retur Balace the cost of the machie agaist the PV of the future stream of reveue. $1 millio today versus $120,000 a year for te years. Solve for i - the iteral rate of retur. Iteral Rate of Retur: Example $120,000 $120,000 $120,000 $120,000 $1,000, (1 i) (1 i) (1 i) (1 i) Solvig for i, i = or 3.46% So log as the iterest rate at which you borrow moey is less tha 3.46%, the you should buy the machie Or, if IRR is greater tha opportuity cost, you should buy the machie. 7

8 Iteral Rate of Retur: Example Suppose you are let go from your job ad your employer offers you two optios: A aual paymet of $8000 per year for 30 years or A lump sum of $50,000 today. Which do you take? I the 1990s, whe the Defese Departmet dowsized, they offered may persoel a similar deal. NOTE a fixed paymet for a fixed umber of years is called a auity Iteral Rate of Retur: Example Ecoomic theory suggests that a perso must compare their discout rate with the curret rate of borrowig or ledig. The military pamphlet gave the preset value of the auity usig a 7% discout rate, the iterest o moey markets at the time. The PV= $99,272. Iteral Rate of Retur: Example About 3/4 of military persoel took the lump sum, which was ½ the PV of the auity. Studyig other separatio packages, the study fids that people s discout rate varies from 17 to 20%. The discout rate that equates PV of the $8,000 aual auity to $50,000 is 15.8%. 8

9 Bod Basics A bod is a promise to make a series of paymets o specific future dates. Bods create obligatios, ad are therefore legal cotracts that: Require the borrower to make paymets to the leder, ad Specify what happes if the borrower fails to do so. Bod Basics The most commo type of bod is a coupo bod. Issuer is required to make aual paymets, called coupo paymets (C). The stated aual iterest the borrower pays is called the coupo rate (i c ). The date o which the paymets stop ad the loa is repaid (), is the maturity date or term to maturity. The fial paymet is called the pricipal, face value, or par value of the bod. Coupo Bod: the good-ole days Called a coupo bod as buyer would receive a certificate with a umber of dated coupos attached. Pricipal Coupos 9

10 Valuig the Pricipal Assume a bod has a pricipal (FV) paymet of $1000 ad its maturity date is years i the future. The preset value of the bod pricipal is: P BP FV $1000 i) ( 1 i) (1 Valuig the Coupo Paymets The preset value expressio gives the formula for the strig of yearly coupo paymets made over years. P CP C (1 i) 1 C (1 i) 2 C (1 i) 3 C... (1 i) The loger the paymets go, the higher their total value. The higher the iterest rate, the lower the preset value. Valuig the Coupo Paymets plus Pricipal We combie the previous two equatios to get the price of a coupo bod: P CB P P CP BP C C 1 (1 i) (1 i) 2 C (1 i) 3 C F... (1 i) (1 i) The value of the coupo bod, P CB, rises whe The yearly coupo paymets, C, rise ad The iterest rate, i, falls. 10

11 Real ad Nomial Iterest Rates Nomial Iterest Rates (i) The iterest rate expressed i curret-dollar terms. Real Iterest Rates (r) The iflatio adjusted iterest rate Borrowers care about the resources required to repay. Leders care about the purchasig power of the paymets they received. Neither cares solely about the umber of dollars, they care about what the dollars buy. Real ad Nomial Iterest Rates The omial iterest rate you agree o (i) must be based o expected iflatio ( e ) over the term of the loa plus the real iterest rate you agree o (r). i = r + e This is the Fisher Equatio. The higher expected iflatio, the higher the omial iterest rate. This equatio is a approximatio that works well whe expected iflatio ad the real iterest rate are low. Exact formula: (1 + i) = (1 + r)(1 + π e ) (1 + i) = 1 + r + π e +r π e Subtract 1 from each side ad igore the cross-term. Iflatio ad Nomial Iterest Rates Makiw 11

12 Iflatio ad Nomial Iterest rates Nomial Iterest Rate, Iflatio Rate ad Real Iterest Rate Real ad Nomial Iterest Rates Fiacial markets quote omial iterest rates. Whe people use the term iterest rate, they are referrig to the omial rate. The real iterest rate is estimated usig the Fisher equatio: r = i - e 12

13 Real ad Nomial Iterest Rates Ex ate real iterest rate is adjusted for expected chages i the price level (π e ) Ex post real iterest rate is adjusted for actual chages i the price level (π) Fisher Equatio: i = r + π e From this we get - r ex ate = i - π e r ex post = i - π Real ad Nomial Iterest Rates Real Iterest Rate - iterest rate that is adjusted for expected chages i the price level r = i π e if i = 5% ad π e = 3%; r = 5% - 3% = 2% if i = 8% ad π e = 10%; r = 8% - 10% = -2% A measure i iflatioary expectatios i = r + π e π e = i - r

14 Algebra - Auity To compute the paymet, we will use the preset-value formula. If we call the size of the mothly paymets C, the we eed to solve the followig formula: 14

Chapter Four 1/15/2018. Learning Objectives. The Meaning of Interest Rates Future Value, Present Value, and Interest Rates Chapter 4, Part 1.

Chapter Four 1/15/2018. Learning Objectives. The Meaning of Interest Rates Future Value, Present Value, and Interest Rates Chapter 4, Part 1. Chapter Four The Meaig of Iterest Rates Future Value, Preset Value, ad Iterest Rates Chapter 4, Part 1 Preview Develop uderstadig of exactly what the phrase iterest rates meas. I this chapter, we see that