Interest rates are among the most closely watched economic variables. Reported daily in the news. Why? Interest have a very powerful effect on the economy.  Interest rates affect stock market, borrowing decisions by households, firms, governments, etc.

Examples:
Stock Market - why?
Mortgage Market - Decision to buy or rent. Fixed rate vs. ARM?
Consumer debt - student loans, credit cards, auto loans.
Buy vs. Lease - cars.
Stock vs. Bonds

Understanding the basics of interest rates is very important for the rest of this course, for econ/fin/bus majors in general, and for our own personal benefit. Helps us in decisions like fixed vs. ARM, lock vs. float, invest in bonds vs stocks, 5 vs 7 year car loan?, refinance mortgage?, etc.

Interest rate often called by other names: yield, YTM, YTC, discount rate, rate of return, IRR, ROI, ROA, ROE, etc. Usually the same, mathematically.  CH 4 helps us understand the mechanics/mathematics of how interest rates (YTM in bond market, IRR for stock market/capital budgeting) are calculated, and the important difference between nominal interest rates and real interest rates (R = r + INF^A) or (R = r +  INF^E).

MEASURING INTEREST RATES

In CH 2, we looked at various credit instruments which fall into 4 general types of loans:

1. Simple loan. Borrow a certain amount. Pay it back at a future date with interest. Example: borrow $1000, pay back the $1000 principal in one year with 10% interest payment, or $100 = rental rate for the money.  (Interest rate = rental rate on money). Example: Commercial loans to businesses.

2. Fixed payment loan. Example: car loan, student loan or mortgage with monthly payments. Payments are part interest, part principal.  Fully amortized loans, so that the  balance at maturity is 0.  Formula for calculating the payment on this type of loan.

        PMT =       PV   *    i   
                          1   -     1    
                                 (1 + i)ⁿ

PV = loan amount in dollars
n = number of payments (periods)
i = interest rate per period.

Example: What are the monthly payments on a $10,000 car loan for five years at 12%.

PV = $10,000
n = 60 months (5 years x 12 months/year)
i = 1% per month (.12 / 12 = .01 per month)

PMT = 10,000  (.01)  / 1 - ( 1  /  (1.01)⁶⁰ )  = $222.44 / month

or PV = 10,000
i = 1% (not .01)
n = 60
FV = 0
PMT = ?

3. Coupon Bond - non-amortized loan, interest only. Corporation borrows $1m for 10 years, pays 10% interest (coupon rate) every year ($100,000) and then pays back the loan ($1m) at the end of ten years. Issued in increments of $1000 face value, par value bonds. Corporation would issue 1000 bonds of $1000 each to raise $1m.  Coupons are clipped off and sent in for interest payments of $100/year. Coupon payment ($) = coupon rate (%) x face value ($1000), or $100 = .10 x $1000.  Bonds are "fixed income securities" because of fixed maturity date, fixed coupon rate, fixed coupon payments, fixed maturity value. Treasury bonds and notes, municipal bonds and corporate bonds (including junk bonds) are all examples of coupon bonds.

4. Discount bond, zero coupon bond. Face value (maturity value) of $1000, sold at discount (P < $1000) and NO periodic interest is paid. Example: buy a one year Tbill for $900 with a face value of $1000. Or you buy a 10 year, zero coupon GM bond for $385, face value of $1000.  T-bills, savings bonds, long-term corporate discount bonds are examples of zero coupon bonds.
 

PRESENT VALUE / FUTURE VALUE / TIME VALUE OF MONEY:

Graph of simple interest vs. compound interest.

Simple Interest = PV x (1 + (i x n)) = FV
Compound Interest = PV (1 + i)ⁿ = FV (Compounded future value)

Example: $100 invested for three years at 10%. FV=?

Simple interest: $100 (1 + .1(3)) = 100 (1.3) = $130
Compound interest: $100 (1.1) = $110 (End of YR 1) x 1.1 = $121 (End of YR 2) x 1.10 = $133.10 (End of YR 3), see page 69, or $100 (1.1)³ = $133.10

Calculator: N = 3, I = 10, PV = 100, PMT = 0, FV = ?

We normally deal with compound interest only.

Rearranging the formula: PV (1 + i)ⁿ = FV, we can solve for PV:

PV = FV / (1 + i)ⁿ .

Given a lump sum to be received in the future, we can solve for the discounted present value.

Example: what is the PV of $1000 to be received in three years if the interest rate is 10%?

PV = $1000 / (1.1)³ = $751.31.  OR N = 3, I = 10, PMT = 0, FV = 1000, PV = ?

Given a lump sum in the future (FV), we can solve for the discounted present value today (PV), given i. Or given a lump sum today (PV), we can solve for the future compounded value (FV), invested at i.

Time Value of money calculations involve 5 variables: N, I, PV, PMT, FV. We always need to know at least three variables and we can solve for the fourth. Or we need to know 4 variables and we solve for the fifth. Caution: Enter 0 for PMT when the problem does not have a payment.

Application of PV: You win the "$1m" lottery. Is it really worth $1m?  Why or why not?
 

YIELD TO MATURITY

How to solve the interest rate (YTM) on a FIXED PAYMENT loan?

Example: (book, p. 71) $1000 loan for 25 years with fixed pmts of $126/year. What is the interest rate or YTM on this loan?

$1000 = $126 / (1 + i) + 126 / (1+i)² + 126 / (1+i)³.....  126 / (1+i)²⁵

Solve for i. Trial and error only....

Calculator solution: N = 25,  PV = $1000,  PMT = -$126,  FV = 0,  I = ?

i = 11.83%

Explain why 126 is negative. +/-CFs/PMTS.  If PV = +1000 and PMT = +126, you will get "NO SOLUTION" error message for HP-10B.
 

COUPON BONDS

How to solve for the interest rate (YTM) on COUPON BOND? Example: 10 year, 10% coupon rate bond, annual payments, FV = $1000, PV (Price) = $1000.

$1000 = $100 / ( 1+ i ) + 100 / ( 1+ i )² ....100 / ( 1+ i )¹⁰ + $1000 / ( 1+ i )¹⁰

Solve for the interest rate that makes the RHS = $1000.

N = 10, PV = -$1000, PMT = +$100, FV = +1000, I = ? (10%)

NOTE:  A bond's interest rate (I or YTM) is not usually the same as the bond's coupon rate.  However, when the YTM = coupon rate, then the bond WILL sell at PAR (P= $1000)

What if Price (PV) = $900?  Solve for Interest Rate (YTM = I):

N = 10, PV = -$900, PMT = $100, FV = 1000, I = ?  (I = 11.75%)
 

What if Price (PV) = $1100? Solve for I:

N = 10, PV = -$1100, PMT = $100, FV = $1000, I = ? (I = 8.48%)
 

See page 73, Table 1. Assume interest rates change - what happens to price. Or assume various prices, what is the YTM of the bond?

NOTE: A bond's coupon rate NEVER changes once it is issued, it is FIXED forever.  The original coupon rate is initially determined based on current, prevailing interest rates, the current YTMs for bonds, so the coupon rate initially reflects current yields in the bond market.  However, how often do bond yields/int rates change??   As market bond yields change, the price of the bond will change.  Two possibilities:

1. Bond yields are 10%, so a new Bond A is issued with a 10% coupon rate (PMT = $100, or 10% of $1000) to reflect current yields, and sells for $1000.  The next day  current interest rates INCREASE to 12%, and a new Bond B is issued with a 12% coupon rate (PMT = $120, or 12% of $1000), and it sells for $1000.  However, the Price of Bond A will now fall from $1000 to $887.00 (N = 10, I = 12, PMT = 100, FV = 1000, PV = ?).  Bond A will now sell at a DISCOUNT (P < $1000).

2. Bond yields are 10%, so a new Bond A is issued with a 10% coupon rate (PMT = $100, or 10% of $1000) to reflect current yields, and sells for $1000.  The next day  current interest rates DECREASE to 8%, and a new Bond B is issued with an 8% coupon rate (PMT = $80, or 8% of $1000), and it sells for $1000.  However, the Price of Bond A will now rise from $1000 to $1134.20 (N = 10, I = 8, PMT = 100, FV = 1000, PV = ?).  Bond A will now sell at a PREMIUM (P > $1000).

POINT: Interest rates (yield, YTM) are INVERSELY related to bond prices for OUTSTANDING bonds.  If market interest rates go UP, the price of outstanding bonds will go DOWN.  If market interest rates go DOWN, the price of outstanding bonds will go UP.  Bonds are typically issued at par (P = $1000), but then typically sell for either a discount or premium over the life of the bond.

Summary of the relationships between YTM, coupon rate and price.

1. When YTM = CR (coupon rate), bond sells at PAR ($1000).
2. The Price (PV) and YTM (I) are inversely related. When the YTM (I) falls, the price of the bond (PV) rises, when the YTM (I) rises, the bond price (PV) falls.
3. If current YTM > CR, the bond is selling at a discount (< 1000), since market interest rates have risen since the bond was issued.
If YTM < CR, the bond is selling at a premium, since market interest rates have fallen since the bond was issued.

What causes a bond to sell at a discount?  Current interest rates have risen.
What causes a bond to sell at a premium?  Current interest rates have fallen.
 

SEMIANNUAL COUPON BONDS

Bond payments can be made annually or semi-annually.  In practice, most coupon bonds make payments semi-annually.  Examples:

1. What is the price of a 5 year bond with semi-annual payments, a 10% coupon rate, face value = $1000, when current interest rates are 8%.
 
 
 

2. What is the annual YTM on a 12 year bond with semi-annual payments, an 8% coupon rate, FV = $1000, that sells for $925?
 
 
 

HINT: N, I, PMT ALWAYS HAVE TO AGREE!!  For example, it could be annual payments (PMT), the number of YEARS (N), and the annual interest rate (I).
Or it could be monthly payments, the number of months, and the monthly interest rate.

For example: 1. Calculate the monthly payments on a 30 year, $100,000 mortgage if interest rates are 12%?
 
 

2. Calculate the monthly payments on a 5 year, $20,000 car loan when interest rates are 12%?
 
 
 

Another Issue: We usually assume that Payments (PMTs) and Cash Flows (CFs) occur at the END OF THE PERIOD.  PMTs and CFs can occur at the beginning of the period, but that is not normal.  Which payments do occur at the beginning of the period?

To set PMTs/CFs to the BEGINNING of period: Yellow Key, BEG/END (above the 0 key).  Sample problem:

What is the present value of a "$10m" lottery that pays $250,000/year for 40 years?  Assume an interest rate of 10%.
 
 
 

Formula for Consols / Perpetuities:

PV = PMT / i  or   i = PMT / PV.
 

DISCOUNT BONDS

Solving for the interest rate (yield or YTM) on a Discount Bond (zero coupon bond).

Assume a T-bill, which matures in one year with a face value (FV) of $1000 and sells now for $900. (No coupon payments)

      PV = FV        so  PV(Price) = $1000 / (1 + i) , solve for i = 11.11%
             (1+i)ⁿ

OR N = 1, PV = -900, FV = +1000, PMT = 0, I = ? = 11.11%

Example: What is the yield on a 10 year zero coupon GM bond that sells for $350?

What if the price is $400?

Summary: Discount bonds show that a dollar in the future is worth less than a dollar today. You would always prefer a $1 today to $1 received in the future. Why?

1.
 

2.

Illustration: What is the present value of $1m to be received 100 years in the future if interest rates are 15%?

A bond represents fixed CFs to be received in the future (PMTS and FV). The PV or price is equal to the sum of the discounted CFs, discounted at the YTM, or current interest rate, current bond yield. Current interest rates and current bond prices are inversely related, because of the FIXED PAYMENTS of a bond.  Why are PMTs fixed?  How would the price of a bond behave if it had a variable rate of interest that was adjusted to reflect current interest rates?

PMTs vs CFs:  PMTs are fixed CFs, the same amount every period.    CFs are usually irregular payments.  If we have constant CFs, like bond coupon payments or loan payments, then we can use the PMT key on the calculator - it is reserved for CONSTANT CFS ONLY.  PMT = CONSTANT CF.  For irregular payments, we can use the CF key on the calculator.  For example:

What is the present discounted value of the following CFs: $500 in YR 1, $600 in YR 2 and $700 in YR 3 when the interest rate is 10%?  We cannot use PMT key, because the CFs are not constant.  Two approaches to solve for PV:

1. Treat the CFs like individual lump sum payments:
PV =  $500 / 1.1   +  $600 / (1.1)²  +  $700 / (1.1)³
=          $454.55    +    $495.87       +     $525.92   =   $1476.34

2. Calculator using CFj key:  0  CFj (for YR 0), 500  CFj (for YR1),  600 CFj (for YR 2), 700 CFj (for YR 3), I = 10, Yellow Key, NPV (above PRC).
Hint: The calculator is formatted starting with YR 0 (today), and you must always enter a 0 when there are no CFs at time 0 (YR 0), like in this problem.
 

OTHER MEASURES OF INTEREST RATES:

Current Yield = Coupon PMT / Price of the Bond. Represents the investor's current yield on investment, what their rate of return is during the first year of the investment.  Example: 12% coupon bond that sells for $800.  Current yield = $120 / $800 = .15 or 15%.  Investor pays $800 today and gets income of $120 one year from now, makes a 15% rate of return, or current yield, in YR 1.

Current Yield for Stock = ????

Investor will also get potential capital gain from their investment, and the current yield ignores that, it just looks at the return during the first year from INCOME.

Problem: What is the current yield on the following bond? 20 year, 8% coupon bond, annual payments, current market bond yields are 6.5%, face value = $1000.

Current Yield = PMT / PRICE, or $80 / ???
 
 

Rules for Current Yield:
1. The current yield will be a fairly close approximation of the YTM, especially a) when the bond price is selling close to par and b) when the maturity of the bond is longer.
2. The current yield will be a poor approximation of YTM when a) the bond is not selling close to par and b) when the maturity is short.
3. The current yield and the YTM always move in the same direction. 

Example: 10 year, 8% coupon bond sells at par (P = $1000), so YTM = Current Yield = 8%.  If YTM goes to 10%, P falls to $877, and Current Yield = $80 / 877 = 9.12%, so YTM and Current Yield BOTH increased. 
Intuition: CY = PMT / P, and PMT is FIXED.  If YTM goes UP, Price falls, and CY has to go UP.  If YTM goes down, Price goes UP, and CY goes DOWN.
 

READING THE WSJ BOND PAGE

See pages 79-82.  Notice:

a) Treasury Bond and Notes: Bid vs Asked quotes.  Bond dealers BUY at the BID price and SELL at the ASKED price, the difference is the commission.  They buy low and sell high.  As customers, we would buy at the ASKED price and sell to the dealer at the BID price.

The Rate is the Coupon Rate, so for T-Bond 3 the coupon rate is 6 1/2 or 6.5%, and would pay $65 ($1000 x 6.5%) in interest annually for every $1000 bond.
The bond matures in November of 2026, and sells for 96:01.  Bonds are quoted per $100 in 32nds, so the bond would sell for 96 1/32 = 96.031 for every $100 or $960.31 for a typical $1000 bond.  The yield on the bond for the investor/buyer would be 6.82%.  The bond is selling at a discount ( P < $1000) because market rates have risen from 6.5% to 6.82% since the bond was issued (probably 4 years ago as a 30 yr TBond).  The current yield is $65 / $960.31 = 6.76%, which is close to YTM (6.82%).  Why are they so close???

b) Tbills - Quoted as yields, not prices.

c) NYSE bonds - ATT bonds. 4 years to 31 years to maturity.

Notice: Bond 1 - ATT, 4 years to maturity in 2004. Coupon rate = 5 5/8 or 5.625%, PMT = $56.25/YR.  Price = 94 1/8 or $94.125 per $100, or $941.25 per bond (FV = $1000).  Current yield = $56.25 / $941.25 or 5.97%.  YTM = 7.37% (N = 4, PV = -1000, PMT = 56.26, FV = 1000, I = ?)

Notice: Current Yield is 5.97% and the current YTM is 7.37%.  Why so different??
 

Reason: You are buying the bond at ____________ and you get ___________ at maturity, resulting in a _______________ of _____________.

YTM = Current Yield  +/-  Capital Gain (loss).

All investments generate returns in two ways: Current Income and/or Capital Gain (Loss) on the sale of the asset from the change in value (price).

Return on any investment = Income + Cap Gain or (Loss)

Bonds:  PMT / Original Price  +  CHANGE IN PRICE / Original PRICE   (see page 83)
                 Current Yld (i_c )    +                   Capital Gain (g)

Stock:

Gold:

Mortgage:

Points: 1) Just because bond is a fixed income investment doesn't insulate it from cap losses and 2) The bond's initial interest rate or yield may not be your ACTUAL rate of return from holding the bond, especially if you have to sell it before maturity.

Example: Table 2, page 84. There are 6 bonds from 1 - 30 year maturity. Interest rates are initially at 10% and all bonds initially sell at par = $1000, and the
coupon rate = 10%, YTM = 10%, current yield = 10% for ALL 6 bonds.

Now assume that one year goes by, and interest rates have risen to 20%.  Prices of all bonds now falls, except the one year bond, which has now reached maturity and pays off $1000, giving the investor a 10% rate of return (they paid $1000, got a $100 interest payment and then got their $1000 back at maturity).  We now recalculate the new prices of the other 5 outstanding bonds.  For example the 30 year bond now has 29 years left: (N = 29, I = 20, PMT = $100, FV = $1000, P = ???)

Total Return on the 30 year bond:

Total Return =   $100 / $1000 (+10%)  +  (503 - 1000) / 1000 or (-49.7%)   =   -39.7%
Total Return = Current Yield    +        Capital loss (g)

The Rate of Return is calculated the same way for the other four bonds.  POINT: Your initial yield was 10%, but if you had to sell the bond after one year your rate of return would be -39.7%!  Current yield, the original YTM and the actual YTM are usually NOT EQUAL.
 

SUMMARY:

1. A rise (fall) in interest rates is always associated with a fall (rise) in existing bond prices, resulting in capital losses (gains) for existing bonds.

2. The longer (shorter) the term to maturity, the greater (smaller) the price change when interest rates change (in either direction). Longer (shorter) term bonds are more (less) price sensitive to interest rate changes, see page 84.

3. The initial yield and the rate of return could be the SAME for a bond held until maturity.

4. A bond's coupon rate (and current yield) are always positive, but its actualized return can be negative due to capital losses.
 

INTEREST RATE RISK - the uncertainty about future interest rates and the effect it will have.

Two types of interest rate risk for bonds:

1. Price risk or capital risk. The uncertainty about future value of the bond, due to interest rate changes, especially if you need to sell the bond before maturity. What exactly are you worried about? Interest rates RISING and bond prices falling.  Capital risk is most associated with LONG term bonds.

2. Reinvestment risk, Income risk. The risk that interest rates will FALL and you will have to reinvest interest payments (income) or proceeds from maturing bonds at LOWER interest rates.  Income Risk is more associated with SHORT term bonds whose proceeds will be reinvested, or HIGH INCOME bonds, or MORTGAGES.

Example: You have a two year planning horizon so you want to invest for two years. Choices: a two year bond at 10% or a one year bond at 10% and then another one year bond at ?%. If interest rates go to 5%, you have to reinvest at 5% during the second year instead of locking in at 10%.

Example: YTM on bond assumes ALL coupons are reinvested at the YTM. If rates fall below original YTM, your realized return will be lower.  Even a bond held until maturity will have some INCOME RISK, from reinvesting the interest payments at lower rates.  Example: 30 year Tbonds issued in the early 80s paid 15%, but now pay about 6%.  If you wanted to re-invest your bond payments in additional T-Bonds, you would have to reinvest @6% instead of 15%, which would lower your expected YTM of 15%.
 

DISTINCTION BETWEEN REAL AND NOMINAL INTEREST RATES

We want to distinguish between nominal (R) and real interest rates (r). We have so far talked about nominal, or stated interest rates, which is what we observe in the WSJ, etc. The nominal interest rate (R) actually has several components. Consider a risk-free T-bill, so that we don't have to consider any default risk premium, which we would for all other bonds.  In that case, the nominal interest rate (R) is composed of a real rate (r) plus an Inflation Premium (INF^e), where  INF^e =  Expected average inflation over the life of the debt/investment. If expected inflation over the next year is 3%, then the Inflation Premium on a one year TBill would be 3%.  The Inflation Premium compensates the lender for getting paid back in cheaper dollars.

Example: Expected inflation over the next year is 3%. You borrow $100. You will have to pay back $103 to the government just to maintain their purchasing power.  The nominal interest rate (R) for 1 Yr TBills is about 6% and if the expected Inflation Premium is 3%, then the real return (r) to the investor and the real cost of borrowing is 3%.

Fisher Equation:   R = r + INF^e, where R= Nominal interest rate, r = real interest rate, and INF^e is the expected inflation premium.

Rearranging, to solve for the real rate of interest.

r  =  R  -  INF^e

However, since actual inflation (INF^a) is not usually equal to expected inflation (INF^e) the ACTUALIZED real rate of interest will fluctuate.

RULES: 1. If actual inflation is greater than expected, the real rate (r) will ________________
2. If actual inflation is lower than expected, the real rate (r) will _________________
3. If actual inflation is greater than the nominal interest rate (R) the real rate (r) will be _____________
 

EXAMPLE:  Fixed Rate Mortgage of 6%.       R  =  r   +  INF^e
 
 
 
 
 

See graph page 88, shows the period of NEGATIVE actual real rates from about 1972 - 1982.  Why??

ISSUE: We easily and accurately observe nominal interest rates (R) every day in WSJ, e.g. two year Tnotes now yield 6.05% (R = 6.05%).  We also know that
R = r + INF^e,  but how to decompose R into the two component: r and INF^e?  That is, how to we precisely determine the market's expectation of INF, INF^e?  Once we know  INF^e,  we can easily solve for the real rate (r).  See page page 89 for discussion of Indexed Tbonds - Treasury Inflation Protection Securities.  These securities are sold to give an investor a guaranteed real rate (r), by adjusting both a) PMTS and b) FV for ACTUAL inflation.

Example: 5 year TIPS sold @$1000 with a real return (r) = 3.5%, Annual PMTS = $35, FV = $1000.  Assume actual inflation is 2% during YR 1. PMT becomes $35 + 2% = $35.70. Principal (FV) becomes $1000 + 2% = $1020. Full protection against inflation, guaranteeing investor a real return (r) of 3.5%.

We can look at regular Tbonds for the nominal yield (R) and TIPS for the real rate (r), and calculate the INF^e by taking the difference.  For example, current yields on 8 year regular Treasuries (R) are about 6.25% and the yield on an 8 year TIPS (r) is 3.625%.  Therefore INF^e =  _________________________ over the next 8 years.