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 the cocept of yield to maturity is the most accurate measure of iterest rate. 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. 1
Valuig Moetary Paymets Now ad i the Future We eed a set of tools: Future value Preset value 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 sometimes called a yield. The above is a example of a simple loa of $100 for a year at 5% iterest. Future Value ad Compoud Iterest If the preset value is $100 ad the iterest rate is 5%, the the future value oe year from ow is: $105 = $100 + $100(0.05) = $100(1.05) This also shows that the higher the iterest rate, the higher the future value. I geeral: FV = PV + PV x i = PV(1 + i) 2
Future Value ad Compoud Iterest Most fiacial istrumets are ot this simple. Whe usig oe-year iterest rates to compute the value repaid more tha oe year from ow, we must cosider compoud iterest. Compoud iterest is the iterest o the iterest. Future Value ad Compoud Iterest What if you leave your $100 i the bak for two years at 5% yearly iterest rate? The future value is: I geeral $100 + $100(0.05) + $100(0.05) + $5(0.05) = $110.25 $100(1.05)(1.05) = $100(1.05) 2 FV = PV(1 + i) Future Value ad Compoud Iterest Computig the future value of $100 at 5% aual iterest 3
Future Value ad Compoud Iterest i ad must be i the same uits. 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 $100(1 + i m ) 12. (NOTE: i ad are mothly) 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 = 1.00407, which is 0.407% These fractios of percetage poits are called basis poits. A basis poit is oe oe-hudredth of a percetage poit, 0.01 percet: 0.407% is 40.7 basis poits Note:.05/12 =.00417 >.00407 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. 4
Preset Value Solve the Future Value Formula for PV: FV = PV x (1+i) FV PV ( 1 i) This is just the future value calculatio iverted. 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) We ca see that 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. 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 = 1.0617 Compoud aual rate = 6.17% (Note: 6.17 > 12x0.05 = 6.0) 5
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. 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 = 0.0371 Computig Compoud Aual Rates PV FV ( 1i ) FV PV FV 1 i PV 120 i 100 (1/5) (1/ ) 1 FV i PV i = 1.0371-1=> i =3.71% (1/ ) 1 Iteral Rate of Retur Imagie that you ru 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, geeratig $120,000 i reveue per year. Assume the machie is the oly iput (keepig this simple), you have certaity about the reveue (very simple), o maiteace (very, very simple) ad a 10 year lifespa. 6
Iteral Rate of Retur If you borrow $1 millio, 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 reveue. $1 millio today versus $120,000 a year for te years. To fid the iteral rate of retur, we take the cost of the machie ad equate it to the sum of the preset value of each of the yearly reveues. Solve for i - the iteral rate of retur. Iteral Rate of Retur: Example $120,000 $120,000 $120,000 $120,000 $1,000,000... 1 2 3 10 Solvig for i, i = 0.0508 or 5.08% So log as your iterest rate at which you borrow the moey is less tha 5.08%, the you should buy the machie Or, if IRR is greater tha opportuity cost, you should buy the machie. 7
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. 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 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 8
Valuig the Pricipal Assume a bod has a priciple 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 These resemble loa paymets. The loger the paymets go, the higher their total value. The higher the iterest rate, the lower the preset value. The preset value expressio gives us a geeral formula for the strig of yearly coupo paymets made over years. C C PCP 1 2 C 3 C... Valuig the Coupo Paymets plus Pricipal We ca just combie the previous two equatios to get: C C C C F PCB PCP PBP... 1 2 3 (1 i) Auity + Face Value The value of the coupo bod, P CB, rises whe The yearly coupo paymets, C, rise ad The iterest rate, i, falls. 9
Bod Pricig The relatioship betwee the bod price ad iterest rates is very importat. Bods promise fixed paymets o future dates, so the higher the iterest rate, the lower their preset value. The value of a bod varies iversely with the iterest rate used to calculate the preset value of the promised paymet. 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 called the Fisher Equatio. The higher expected iflatio, the higher the omial iterest rate. This equatio is a approximatio that works well oly 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. 10
Iflatio ad Nomial Iterest Rates Makiw Iflatio ad Nomial Iterest rates Nomial Iterest Rate, Iflatio Rate ad Real Iterest Rate 11
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. We caot directly observe the real iterest rate; we have to estimate it. r = i - e Real ad Nomial Iterest Rates Nomial iterest rate (i) makes o allowace for iflatio Real iterest rate (r) is adjusted for chages i price level so it more accurately reflects the cost of borrowig 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) Real ad Nomial Iterest Rates Fisher Equatio: i = r + π e From this we get - r ex ate = i - π e r ex post = i - π 12
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 http://www.bloomberg.com/markets/rates-bods/govermetbods/us/ http://research.stlouisfed.org/fred2/ 13