Biz Finance Week 3 - Discounted Cash Flow Valuation - Chapter 5
Annuity - finite series of equal payments that occur at regular intervals
- If the first payment occurs at the end of the period, it is called an ordinary annuity - If the first payment occurs at the beginning of the period, it is called an annuity due
annuity
A level stream of cash flows for a fixed period of time.
Amortized Loans
A loan in which principal as well as interest is payable in periodic installments over the term of the loan. --The process of paying off a loan by making regular principal reductions is called amortizing the loan.
annuity due is an annuity for which the cash flows occur at the beginning of each period.
Almost any type of arrangement in which we have to prepay the same amount each period is an annuity due.
annuity due
An annuity for which the cash flows occur at the beginning of the period.
perpetuity
An annuity in which the cash flows continue forever.
There are two ways of calculating present and future values when there are multiple cash flows.
Both approaches are straightforward extensions of our earlier analysis of single cash flows.
must use sign convention + = cash inflow - = cash outflow
Clear the Cash Flow Memory by pushing CF, 2nd and then the CE/C button. Press the CF button this should display CF0 on the TIBAII Plus.
Interest rates can be quoted in a variety of ways. For financial decisions, it is important that any rates being compared first be converted to effective rates. The relationship between a quoted rate, such as an annual percentage rate, or APR, and an effective annual rate, or EAR, is given by:
EAR = (1+Quoted rate/m)^m-1 where m is the number of times during the year the money is compounded, or, equivalently, the number of payments during the year.
Interest Rates
Effective Annual Rate (EAR) - The interest rate expressed as if it were compounded once per year .- Used to compare two alternative investments with different compounding periods • Annual Percentage Rate (APR) "Nominal" - The annual rate quoted by law - APR = periodic rate X number of periods per year - Periodic rate = APR / periods per year
8. Calculating Annuity Values For each of the following annuities, calculate the future value.
FVA = C{[(1 + r)t - 1]/r} FVA = $2,100[(1.0710 - 1)/.07] FVA = $29,014.54 FVA = C{[(1 + r)t - 1]/r} FVA = $6,500[(1.0840 - 1)/.08] FVA = $1,683,867.37 FVA = C{[(1 + r)t - 1]/r} FVA = $1,100[(1.099 - 1)/.09] FVA = $14,323.14 FVA = C{[(1 + r)t - 1]/r} FVA = $5,000[(1.1130 - 1)/.11] FVA = $995,104.39
12. Calculating EAR Find the EAR in each of the following cases.
For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR/m)]m - 1 EAR = [1 + (.078/4)]4 - 1 = .0803, or 8.03% EAR = [1 + (.153/12)]12 - 1 = .1642, or 16.42% EAR = [1 + (.124/365)]365 - 1 = .1320, or 13.20% To find the EAR with continuous compounding, we use the equation: EAR = eq - 1 EAR = e.114 - 1 EAR = .1208, or 12.08%
14. Calculating EAR First National Bank charges 13.8 percent compounded monthly on its business loans. First United Bank charges 14.1 percent compounded semiannually. As a potential borrower, which bank would you go to for a new loan?
For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR/m)]m - 1 So, for each bank, the EAR is: First National: EAR = [1 + (.138/12)]12 - 1 = .1471, or 14.71% First United: EAR = [1 + (.141/2)]2 - 1 = .1460, or 14.60% Notice that the higher APR does not necessarily result in the higher EAR. The number of compounding periods within a year will also affect the EAR.
16. Calculating Future Values What is the future value of $5,500 in 17 years assuming an APR of 8.4 percent compounded semiannually?
For this problem, we need to find the FV of a lump sum using the equation: FV = PV(1 + r)t It is important to note that compounding occurs semiannually. To account for this, we will divide the interest rate by two (the number of compounding periods in a year), and multiply the number of periods by two. Doing so, we get: FV = $5,500[1 + (.084/2)]17(2) FV = $22,277.43
46. Calculating Annuities Due Suppose you are going to receive $14,500 per year for five years. The appropriate discount rate is 7.1 percent. -What is the present value of the payments if they are in the form of an ordinary annuity? What is the present value if the payments are an annuity due? -Suppose you plan to invest the payments for five years. What is the future value if the payments are an ordinary annuity? What if the payments are an annuity due? -Which has the higher present value, the ordinary annuity or annuity due? Which has the higher future value? Will this always be true?
If the payments are in the form of an ordinary annuity, the present value will be: PVA = C({1 - [1/(1 + r)t]}/r) PVA = $14,500[{1 - [1/(1 + .071)5]}/ .071] PVA = $59,294.01 If the payments are an annuity due, the present value will be: PVAdue = (1 + r)PVA PVAdue = (1 + .071)$59,294.01 PVAdue = $63,503.89 b. We can find the future value of the ordinary annuity as: FVA = C{[(1 + r)t - 1]/r} FVA = $14,500{[(1 + .071)5 - 1]/.071} FVA = $83,552.26 If the payments are an annuity due, the future value will be: FVAdue = (1 + r)FVA FVAdue = (1 + .071)$83,552.26 FVAdue = $89,484.47 c. Assuming a positive interest rate, the present value of an annuity due will always be larger than the present value of an ordinary annuity. Each cash flow in an annuity due is received one period earlier, which means there is one period less to discount each cash flow. Assuming a positive interest rate, the future value of an annuity due will always be higher than the future value of an ordinary annuity. Since each cash flow is made one period sooner, each cash flow receives one extra period of compounding.
1. Present Value and Multiple Cash Flows Mendez Co. has identified an investment project with the following cash flows. If the discount rate is 10 percent, what is the present value of these cash flows? What is the present value at 18 percent? At 24 percent?
PV = FV/(1 + r)t PV@10% = $470/1.10 + $610/1.102 + $735/1.103 + $920/1.104 = $2,111.99 PV@18% = $470/1.18 + $610/1.182 + $735/1.183 + $920/1.184 = $1,758.27 PV@24% = $470/1.24 + $610/1.242 + $735/1.243 + $920/1.244 = $1,550.39
5. Calculating Annuity Cash Flows For each of the following annuities, calculate the annual cash flow.
PVA = C({1 - [1/(1 + r)t]}/r) $12,000 = C{[1 - (1/1.11)6]/.11} C = $12,000/4.23054 C = $2,836.52 PVA = C({1 - [1/(1 + r)t]}/r) $19,700 = C{[1 - (1/1.07)8]/.07} C = $19,700/5.97130 C = $3,299.11 PVA = C({1 - [1/(1 + r)t]}/r) $134,280 = C{[1 - (1/1.08)15]/.08} C = $134,280/8.55948 C = $15,687.87 PVA = C({1 - [1/(1 + r)t]}/r) $300,000 = C{[1 - (1/1 .06)20]/.06} C = $300,000/11.46992 C = $26,155.37
6. Calculating Annuity Values For each of the following annuities, calculate the present value.
PVA = C({1 - [1/(1 + r)t]}/r) PVA = $1,750{[1 - (1/1.05)7]/.05} PVA = $10,126.15 PVA = C({1 - [1/(1 + r)t]}/r) PVA = $1,390{[1 - (1/1.10)9]/.10} PVA = $8,005.04 PVA = C({1 - [1/(1 + r)t]}/r) PVA = $17,500{[1 - (1/1.08)18]/.08} PVA = $164,008.02 PVA = C({1 - [1/(1 + r)t]}/r) PVA = $50,000{[1 - (1/1.14)28]/.14} PVA = $348,033.11
47. Annuity and Perpetuity Values Mary is going to receive a 30-year annuity of $11,500 per year. Nancy is going to receive a perpetuity of $11,500 per year. If the appropriate discount rate is 4.3 percent, how much more is Nancy's cash flow worth?
PVA = C({1 - [1/(1 + r)t]}/r) PVA = $11,500{[1 - (1/1.043)30]/.043} PVA = $191,810.81 And the present value of the perpetuity is: PVP = C/r PVP = $11,500/.043 PVP = $267,441.86 So, the difference in the present values is: Difference = $267,441.86 - 191,810.81 Difference = $75,631.05 There is another common way to answer this question. We need to recognize that the difference in the cash flows is a perpetuity of $11,500 beginning 31 years from now. We can find the present value of this perpetuity and the solution will be the difference in the cash flows. So, we can find the present value of this perpetuity as: PVP = C/r PVP = $11,500/.043 PVP = $267,441.86 This is the present value 30 years from now, one period before the first cash flows. We can now find the present value of this lump sum as: PV = FV/(1 + r)t PV = $267,441.86/(1 + .043)30 PV = $75,631.05
Perpetuity
Perpetuity formula: PV = PMT / r• Current required return: r = PMT/PV• Look for the keyword "forever"• Used to value preferred stock
Interest rate and time period must match! - Annual periods annual rate - Monthly periods monthly rate
The Sign Convention - Cash inflows are positive - Cash outflows are negative
annual percentage rate (APR)
The interest rate charged per period multiplied by the number of periods per year.
effective annual rate (EAR)
The interest rate expressed as if it were compounded once per year.
stated interest rate
The interest rate expressed in terms of the interest payment made each period. Also quoted interest rate.
quoted interest rate
The interest rate expressed in terms of the interest payment made each period. Also stated interest rate.
Many loans are annuities.
The process of paying off a loan gradually is called amortizing the loan, and we discussed how amortization schedules are prepared and interpreted.
10. Calculating Perpetuity Values The Maybe Pay Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $30,000 per year forever. If the required return on this investment is 5.6 percent, how much will you pay for the policy?
This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation: PV = C/r PV = $30,000/.056 PV = $535,714.29
Preferred stock (or preference stock) is an important example of a perpetuity. When a corporation sells preferred stock, the buyer is promised a fixed cash dividend every period (usually every quarter) forever.
This dividend must be paid before any dividend can be paid to regular stockholders, hence the term preferred.
2. Present Value and Multiple Cash Flows Investment X offers to pay you $5,300 per year for eight years, whereas Investment Y offers to pay you $7,300 per year for five years. Which of these cash flow streams has the higher present value if the discount rate is 5 percent? If the discount rate is 15 percent?
To find the PVA, we use the equation: PVA = C({1 - [1/(1 + r)t]}/r) At a 5 percent interest rate: X@5%: PVA = $5,300{[1 - (1/1.05)8]/.05} = $34,255.03 Y@5%: PVA = $7,300{[1 - (1/1.05)5]/.05} = $31,605.18 And at a 15 percent interest rate: X@15%: PVA = $5,300{[1 - (1/1.15)8]/.15} = $23,782.80 Y@15%: PVA = $7,300{[1 - (1/1.15)5]/.15} = $24,470.73 Notice that the cash flow of X has a greater PV at a 5 percent interest rate, but a lower PV at a 15 percent interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the total cash flow is more important since the cost of waiting (the interest rate) is not as great. At a higher interest rate, Y is more valuable since it has larger cash flows. At the higher interest rate, these larger cash flows early are more important since the cost of waiting (the interest rate) is so much greater.
52. Calculating Present Value of a Perpetuity Given a discount rate of 4.6 percent per year, what is the value at Date t = 7 of a perpetual stream of $7,300 payments with the first payment at Date t = 15?
To find the value of the perpetuity at t = 14, we first need to use the PV of a perpetuity equation. Using this equation, we find: PV = $7,300/.046 PV = $158,695.65 Remember that the PV of perpetuity (and annuity) equations give the PV one period before the first payment, so, this is the value of the perpetuity at t = 14. To find the value at t = 7, we find the PV of this lump sum as: PV = $158,695.65/1.0467 PV = $115,835.92
20. Calculating Loan Payments You want to buy a new sports coupe for $84,500, and the finance office at the dealership has quoted you a loan with an APR of 4.7 percent for 60 months to buy the car. What will your monthly payments be? What is the effective annual rate on this loan?
We first need to find the annuity payment. We have the PVA, the length of the annuity, and the interest rate. Using the PVA equation: PVA = C({1 - [1/(1 + r)t]}/r) $84,500 = C[1 - {1/[1 + (.047/12)]60}/(.047/12)] Solving for the payment, we get: C = $84,500/53.3786 C = $1,583.03 To find the EAR, we use the EAR equation: EAR = [1 + (APR/m)]m - 1 EAR = [1 + (.047/12)]12 - 1 EAR = .0480, or 4.80%
consol
a type of perpetuity, particularly in Canada and the UK
A series of constant cash flows that arrive or are paid at the end of each period is called an ordinary annuity,
and we described some useful shortcuts for determining the present and future values of annuities.
Interest-Only Loans
borrower pays interest each period and repays the entire principal at some point in the future -has a repayment plan that calls for the borrower to pay interest each period and to repay the entire principal (the original loan amount) at some point in the future. Such loans are called interest-only loans. Notice that if there is just one period, a pure discount loan and an interest-only loan are the same thing.
N and I/y need to be same
if n is months, then monthy interest, if n in years, yearly interest
Perpetuity -
infinite series of equal payments.
a series of constant, or level, cash flows that occur at the end of each period for some fixed number of periods is called an
ordinary annuity; or, more correctly, the cash flows are said to be in ordinary annuity form.
Change PMT to begin:
press 2nd then PMT, then 2nd and enter, then 2nd CPT- BGN is only for ordanry due, typically use end mode
three basic types of loans are
pure discount loans, interest-only loans, and amortized loans
pure discount loan
the borrower receives money today and repays a single lump sum at some time in the future -With such a loan, the borrower receives money today and repays a single lump sum at some time in the future. A one-year, 10 percent pure discount loan, for example, would require the borrower to repay $1.1 in one year for every dollar borrowed today. --Pure discount loans are very common when the loan term is short, say, a year or less. In recent years, they have become increasingly common for much longer periods.
Decisions, Decisions• Which savings accounts should you choose:- 5.25% with daily compounding. - 5.30% with semiannual compounding.
• First account:• EAR = (1 + .0525/365)365 - 1 = 5.39%• ICONV: NOM=5.25; C/Y=365 EFF=5.3899 • Second account:• EAR = (1 + .053/2)2 - 1 = 5.37%• ICONV: NOM=5.3; C/Y=2 EFF=5.3702
Pure Discount Loans
• Treasury bills are excellent examples of pure discount loans. - Principal amount is repaid at some future date- No periodic interest payments• If a T-bill promises to repay $10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market?- 1 N; 10,000 FV; 7 I/Y; CPT PV = -9345.79
Things to Remember
• You ALWAYS need to make sure that the interest rate and the time period match. - Annual periods annual rate. - Monthly periods monthly rate. • If you have an APR based on monthly compounding, you have to use monthly periods for lump sums or adjust the interest rate accordingly.