³ ² ± Termiology for Bods ad Loas Pricipal give to borrower whe loa is made Simple loa: pricipal plus iterest repaid at oe date Fixed-paymet loa: series of (ofte equal) repaymets Bod is issued at some price Face Value is repaymet at maturity date Zero coupo bod pays oly face value at maturity Coupo bod also makes periodic coupo paymets, equal to coupo rate times face value
Compoudig Assume that the iterest rate is 10% p.a. What this meas is that if you ivest $1 for oe year, you have bee promised $1*(1+10/100) or $1.10 ext year Ivestig $1 for yet aother year promises to produce 1.10 *(1+10/100) or $1.21 i 2-years
Value of $5 Ivested More geerally, with a ivestmet of $5 at 10% we obtai 1 Year $5*(1+0.10) $5.5 2 years $5.5*(1+0.10) $6.05 3 years $6.05*(1+0.10) $6.655 4 Years $6.655*(1+0.10) $7.3205
Future Value of a Lump Sum FV PV * (1 + FV with growths from -6% to +6% 3,500 Future Value of $1000 3,000 2,500 2,000 1,500 1,000 500 0 0 2 4 6 8 10 12 14 16 18 20 Years 6% 4% 2% 0% -2% -4% -6%
Geeralizig the method Geeralizig the method requires some defiitios. Let i be the iterest rate be the life of the lump sum ivestmet PV be the preset value FV be the future value
Example: Future Value of a Lump Sum Your bak offers a CD with a iterest rate of 3% for a 5 year ivestmet. You wish to ivest $1,500 for 5 years, how much will your ivestmet be worth? FV PV * (1 + $1500 * (1 + 0.03) $1738.1111145 5 i 3% PV 1,500 FV? Result 1738.911111 5
Preset Value of a Lump Sum FV PV *(1 + Divide both sides by (1 + PV FV (1 + FV *(1 + to obtai :
Example: Preset Value of a Lump Sum You have bee offered $40,000 for FV PV your pritig (1 + busiess, payable i 2 years. Suppose 40,000 2 the iterest rate is (1 + 0.08) 8%. What is the preset value of the 34293.55281 offer? $34,293.55 today
Solvig Lump Sum Cash Flow for Iterest Rate FV FV PV (1 + i PV *(1 + (1 + FV PV FV PV 1
³ ² ± Calculatios for Bods ad Loas Iterest rate i is yield to maturity. is time to maturity Simple Loas: use lump sum formula; PV pricipal; FV pricipal plus iterest. Zero Coupo Bods: use lump sum formula; PV price; FV face value. Fixed-Paymet Loas: use auity formula; loa paymet; PV pricipal Coupo Bods: combie auity ad lump sum formulas. With
³ ² ± face value F ad coupo paymet C, the price p is equal to P C 1+i + C (1 + 2 +... + C (1 + + C (1 + + F (1 +
Example: Iterest Rate o a Lump Sum Ivestmet If you ivest $15,000 for te years, you receive $30,000. What is your aual retur? Other iterpretatio: if price of 10 year bod with face value 30,000 is 15,000, the iterest rate (yield to maturity) o bod is 7.18% i FV 1 PV 30000 1 10 10 10 1 2 1 2 1 15000 0.071773463 7.18% (to the earest basis poit)
Yield to Maturity: Loas Yield to maturity iterest rate that equates today s value with preset value of all future paymets 1. Simple Loa (i 10%) $100 $110/(1 + $110 $100 $10 i 0.10 10% $100 $100 2. Fixed Paymet Loa (i 12%) $126 $126 $126 $126 $1000 + + +... + (1+ (1+ 2 (1+ 3 (1+ 25 FP FP FP FP LV + + +... + (1+ (1+ 2 (1+ 3 (1+
Yield to Maturity: Bods 3. Coupo Bod (Coupo rate 10% C/F) $100 $100 $100 $100 $1000 P + + +... + + (1+ (1+ 2 (1+ 3 (1+ 10 (1+ 10 C C C C F P + + +... + + (1+ (1+ 2 (1+ 3 (1+ (1+ C C P i i P 4. Discout Bod (P $900, F $1000), oe year $900 $1000 (1+ $1000 $900 i 0.111 11.1% $900 i Cosol: Fixed coupo paymets of $C forever F P P
Relatioship Betwee Price ad Yield to Maturity Three Iterestig Facts i Table 1 1. Whe bod is at par, yield equals coupo rate 2. Price ad yield are egatively related 3. Yield greater tha coupo rate whe bod price is below par value 2004 Pearso Addiso-Wesley. All rights reserved 4-5
³ ² ± Bod Yields ad Bod Prices yield to maturity o a coupo bod caot be calculated aalytically; eed computer or fiacial calculator YTM high if bod price is low YTM equals coupo rate if bod price equals face value (bod trades at par) YTM higher (lower) tha coupo rate if bod price lower (higher) tha face value curret yield (coupo rate/price) is approximatio to YTM; works well if log maturity, bod trades close to par.
The Problem How much will I have available i my retiremet accout if I deposit $2,000 per year ito a IRA that pays o average 16% per year if I pla to retire i 40 years time?
Auity Stream of Cash Flows, such that the first cash flow will occur exactly oe period form ow all subsequet cash flows are separated by exactly oe period all periods are of equal legth the iterest rate is costat all cash flows have the same value
Auity Formula Notatio PV the preset value of the auity i iterest rate to be eared over the life of the auity the umber of paymets the periodic paymet (cash flow)
Derivatio of PV of Auity Formula PV ( ) 1 1+ i ( 1+ + Λ + + ( ) 3 ( ) 1 1+ i 1+ i ( 1+ 2 + +
PV of Auity Formula PV *{1 i i * 1 1 ( 1+ 1 } ( + ) 1 i
PV Auity Formula: Paymet PV i i * 1 ( 1+ ( ( 1+ ) * 1 PV * i ( 1 ( 1+ ) 1
Derivatio of FV of Auity Formula: Algebra PV FV FV i PV i i * * 1 ( 1+ ( 1+ * 1 * 1 1 ( 1+ ( 1+ 1) (lump sum) * (reg. auity) ( 1+
FV Auity Formula: Paymet FV i * ( 1+ 1) FV * i ( 1+ 1)
Data for Problem I 16% 40 Pmt $2,000 FV?
Solutio F i 2000 0.16 ( ) ) 1 + i 1 ( ) ) 40 1 + 0.16 1 $4,721,514.481