Lecture 3 Understanding Interest Rate  Future Value & Present Value  Credit market instruments Simple Loan Fixed Payment Loan Coupon Bond Discount Bond  Real vs. Nominal interest rates

1.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

example: future value (FV)  $100 today  interest rate 5% annually  at the end of 1 year: (100% x 5%) = 100(1.05) = $105  at the 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

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

example  receive $100 in 3 years  i = 5%  what is PV?  With PV, we discount future cash flows Payment we wait for are worth LESS PV= $100 (1+.05) 3 =$86.36

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

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

1.Simple Loan  Yield to maturity = interest rate that equates today ’ s value with present value of all future payments  Interest rate Where PV of cash flows = cost example ： $100 = $110/(1 + i) 

2.Fixed Payment Loan  A fixed payment loan in which the lender providers the borrower with an amount of funds, which must be repaid by making the same payment every period, consisting of part of the principal and interest for a set number of years.  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 Spreadsheet  Online calc. Online calc. Solve for i?  Trial & error  Spreadsheet Spreadsheet  Online calc. Online calc. Answer?  12.05% Answer?  12.05%

LV ： loan value FP ： fixed yearly payment n ： number of year payment

3.Coupon Bond  A coupon bond pays the owner of the bond a fixed interest payment(coupon payment) every year until the maturity date, when a specified final amount (face value or par value) is repaid.

Example (Coupon rate = 10% = C/F)  F: face value.  Consol: Fixed coupon payments of $C forever

3.Coupon Bond  purchase price, P  promised of a series of payments until maturity face value at maturity, F (principal, par value) coupon payments (6 months)

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

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.

what are the payments?  6 mos. $300  1 year $300  1.5 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

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 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

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

4. Discount Bond  Discount Bond( also called a zero-coupon bond) is bought at a price below its face value (at a discount), and the face value is repaid at a maturity date.

Example (P = $900, F = $1000), one year

Relationship Between Price and Yield to Maturity Three Interesting Facts in Table 1 1.When bond is at par, yield equals coupon rate 2.Price and yield are negatively related 3.Yield greater than coupon rate when bond price is below par value

Distinction Between Interest Rates and Returns Rate of Return C + P t+1 – P t RET == i c + g P t C where: i c = = current yield P t P t+1 – P t g == capital gain P t

Key Facts about Relationship Between Interest Rates and Returns

Maturity and the Volatility of Bond Returns Key Findings from Table 2 1.Only bond whose return = yield is one with maturity = holding period 2.For bonds with maturity > holding period, i  P  implying capital loss 3.Longer is maturity, greater is % price change associated with interest rate change 4.Longer is maturity, more return changes with change in interest rate 5.Bond with high initial interest rate can still have negative return if i  Conclusion from Table 2 Analysis 1.Prices and returns more volatile for long-term bonds because have higher interest-rate risk 2.No interest-rate risk for any bond whose maturity equals holding period

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

example  $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 )]

 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