GROWING PERPETUITY
- An annuity in which the cash flow continues forever, but grows at a constant rate of “g” per year.
- The stream of income looks like:
PV0 = C1 + C1(1+g) + C1(1+g)2 +…
1 + r (1+r)2 (1+r)3
- Fortunately, there is an easier way to present and calculate this value.
PVgrowing perpetuity = C1_ r – g
- This is a very important equation in the evaluation of common stock.
? FVt = PV * ( 1 + r )t
? PV = FVt * ___1____
( 1 + r )t
? PVA(r,t) = C * 1 – (1/(1 + r)t) r
? FVA(r,t) = C * ( 1 + r )t - 1
r
? PVperpetuity = C r
? PV growing perpetuity = __C1_ r – g
WHAT IS THE INTEREST RATE?
Your banker offers the following 5 choices:
? 7.75% compounded daily
? 7.75% compounded weekly
? 7.75% compounded monthly
? 7.75% compounded quarterly
? 8 % compounded annually
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WHICH OPTION DO YOU PREFER?
STATED or QUOTED INTEREST RATE
- the interest rate expressed in terms of the interest payment made each period.
? Example: a quoted rate of 7.75% per year, compounded weekly.
? Example: a stated rate of 7.75% per year, compounded monthly.
EFFECTIVE ANNUAL RATE: EAR: the interest rate expressed as if it were compounded once per year. EAR = ( 1 + QR/m)m – 1 Where: QR= quoted rate per year m = number of times compounded per year.
? Example: a quoted rate of 7.75% per year, compounded weekly.
EAR = ( 1 + .0775/52)52 – 1
EAR = ( 1 + .00149)52 – 1
EAR = 8.052%
EXAMPLE
At the local loan shark, you pay $1 per week compounded weekly for every $100 that you borrow. Actually sounds very lenient doesn’t it? What is the EAR?
EXAMPLE:
Mortgages are quoted in annual rates but are usually compounded semi-annually. What is the EAR for a 9% mortgage?
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DEFINITION
ANNUAL PERCENTAGE RATE (APR)
? The interest rate charged per period multiplied by the number of periods per year.
EXAMPLE
A credit card requiring monthly payments carries an APR of 14.4% and interest is calculated and compounded monthly. What is the EAR?
EXAMPLE:
? You are to receive $1,000 per month for 25 years.
? Interest is compounded monthly.
? The present value of this annuity is $155,206.86. (I) WHAT IS THE STATED OR QUOTED MONTHLY RATE? (II) WHAT IS THE EAR?
LOANS & BONDS
LOAN TYPES
? Pure Discount Loans
- e.g. Treasury bills
? Interest-only Loans
- e.g. secured loans from banks
? Amortized Loans
AMORTIZED LOANS
? Where the lender requires the borrower to repay parts of the loan amount over time.
? Two Basic Approaches:
? Simple method: pay the accrued interest plus some portion of the
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principal.
? Fixed payment per period, which is a blend of interest and
principal. There is a constant fixed payment every period.
SIMPLE AMORTIZATION
? 7% interest compounded annually; ? $8,000 loan ? 4-year term
Start of
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2
3
4
Interest Principal Total Ending Balance Year Period Paid Paid Paid 8,000 560 6,000 2,000 2,560 6,000 2,420 4,000 420 2,000 4,000 280 2,000 140 2,000 2,280 2,000 0 2,000 2,140 1,400 8,000 9,400 FIXED PAYMENT AMORTIZATION
? 7% interest compounded annually ? $8,000 loan ? 4-year term
This is a 4-year annuity with a present value of $8,000.
PVA(7%,4) = C * PVIFA(7%,4)
8,000 = C * 1 – (1/(1.07)4) .07
8,000 = C * 3.38721
C = $2,361.82
The annual payment at the end of each year would be $2,361.82.
Let’s call it $2,362.
AMORTIZATION SCHEDULE
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Start of Interest Principal Total Ending Year
1
2
3
4
Period Paid Paid 8,000 6,198 4,270 2,207 560 434 299 154 Paid Balance 1,802 2,362 6,198 1,928 2,362 4,270 2,063 2,362 2,207 2,208 2,362 0 1,447 8,000 9,448 MORTGAGES ? Many (most?) mortgages have a longer amortization period than the term.
? E.g. a $150,000 mortgage with a 25-year amortization, a 5-year term and a 7.25% interest rate. When presented in this manner, the 7.25% rate is an APR, and might be called a stated or quoted annual rate.
? Remember that mortgages almost always compound every 6 months.
Two questions: (i) What is the monthly mortgage payment with the 5-year term? (ii) What lump-sum payment (balloon payment) would be required to pay
off the mortgage after 5 years?
BONDS AND BOND EVALUATION
? What are the features of a bond?
? How do we find bond prices and the yield to maturity?
? Why do bond prices and yields move in opposite directions?
? What do meant by the terms “discount” and “premium” bonds?
? What do we mean by interest rate risk and default risk?
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武汉大学公司金融课件（二）
