I have a test coming up on Sept 22. It will be 65 min long with 10 multiple choice questions. I will need to someone to answer the questions while I take it. I have provided sample test problems and solutions along with the formula sheet.
FNAN 303 Formulas and Notes (p. 1 of 6)
Value in t periods with simple interest:
C0 × [1 + (simple interest rate per period × t)] = C0 + (C0 × simple interest rate per period × t)
FVt = C0 × (1+r)t
Financial calculator: In either BEGIN or END mode, FV is the future value in N periods from the reference point
(time 0) of a cash flow equal to -PV at the reference point with an interest rate, return, etc. of I% per period
FVt = Ck × (1+r)(t-k)
PV0 = PV = Ct / (1 + r)t
Financial calculator: In either BEGIN or END mode, PV is the opposite of the present value as of the reference
point (time 0) of a cash flow equal to FV that takes place in N periods from the reference point, with a discount rate
of I% per period
PV0 = PV = C0 + [C1/(1+r)1
] + [C2/(1+r)2
] + … + [Ct-1/(1+r)t-1
] + [Ct/(1+r)t
]
PV for a fixed perpetuity = [C/(1+r)] + [C/(1+r)2
] + [C/(1+r)3
] + … = C / r
Rate of return for a fixed perpetuity = r = C /PV
Cash flow for a fixed perpetuity = C = PV × r
PV for a growing perpetuity = C1/(1+r) + [C1(1+g)]/(1+r)2 + [C1(1+g)2
]/(1+r)3 + … = C1 / (r – g)
Rate of return for a growing perpetuity = r = [C1 / PV] + g
First cash flow for a growing perpetuity = C1 = PV × (r – g)
Growth rate for a growing perpetuity = g = r – [C1 / PV]
Ck = C1 × (1 + g)k – 1 which is the same as Ct = C1 × (1 + g)(t – 1)
Also, Cb = Ca × (1 + g)(b-a)
so g = [(Cb / Ca)
[1/(b-a)]
] – 1
PV for an annuity = [C/(1+r)] + [C/(1+r)2
] + … + [C/(1+r)t
]
= C × [{1 – 1/(1+r)t } / r ] = C × [(1/r) – 1/{r(1+r)t}] = (C/r) × [1 – (1/{(1+r)t})]
Financial calculator: In END mode, PV is the opposite of the present value as of the reference point (time 0) of a
series of N regular cash flows equal to PMT per period where the first regular cash flow takes place 1 period from
the reference point, the last cash flow takes place N periods from the reference point, and the discount rate is I% per
period
PV for an annuity due = (1+r) × PV for an annuity = C + [C/(1+r)] + [C/(1+r)2
] + [C/(1+r)3
] + … + [C/(1+r)t-1
]
= (1+r) × C × [{1 – 1/(1+r)t } / r ] = (1+r) × C × [(1/r) – 1/{r(1+r)t}] = C + (C/r) × [1 – (1/{(1+r)t-1})])
Financial calculator: In BEGIN mode, PV is the opposite of the present value as of the reference point (time 0) of a
series of N regular cash flows equal to PMT per period where the first regular cash flow takes place at the reference
point, the last cash flow takes place N-1 periods from the reference point, and the discount rate is I% per period
PV0 = PV = PVk / (1+r)k
FVt = [C0 × (1+r)t
] + [C1 × (1+r)t-1
] + [C2 × (1+r)t-2
] + … + [Ck × (1+r)t-k
] + … + [Ct-1 × (1 + r)1
] + [Ct]
FV for an annuity = [C1 × (1+r)t-1
] + [C2 × (1+r)t-2
] + … + Ct
= (1+r)t × C × [{1 – 1/(1+r)t} / r] = C × [{(1+ r)t – 1} / r] = (1+r)t × C × [(1/r) – 1/{r(1+r)t}]
Financial calculator: In END mode, FV is the future value in N periods from the reference point (time 0) of a series
of N regular cash flows equal to -PMT per period where the first regular cash flow takes place 1 period from the
reference point, the last cash flow takes place N periods from the reference point, and the interest rate, return, etc. is
I% per period
FV for an annuity due = (1+r) × FV for an annuity = [C0 × (1+r)t
] + [C1 × (1+r)t-1
] + … + [Ct-1 × (1+r)1
]
= (1+r)t+1 × C × [{1 – 1/(1+r)t} / r] = (1+r) × C × [{(1+r)t – 1} / r] = (1+r)t+1 × C × [(1/r) – 1/{r(1+r)t}]
Financial calculator: In BEGIN mode, FV is the future value in N periods from the reference point (time 0) of a
series of N regular cash flows equal to -PMT per period where the first regular cash flow takes place at the
reference point, the last cash flow takes place N-1 periods from the reference point, and the interest rate, return, etc.
is I% per period
FNAN 303 Formulas and Notes (p. 2 of 6)
APR = annual percentage rate = # periods in a year × periodic rate = # periods in a year × [(1 + EAR)1/# periods in a year – 1]
EAR = Effective annual rate = [(1 + periodic rate)# of periods in a year] – 1 = [1 + (APR/# periods per year)]# periods per year – 1
Periodic rate = [APR / # periods per year] = [(1 + EAR)(1 / # of periods in a year) – 1]
EAR with continuous compounding = (e
APR) – 1
Bond value = [cpn/(1+r)1
] + [cpn/(1+r)2
] +…+ [cpn/(1+r)t
] + [face/(1+r)t
]
= {cpn × [{1 – 1/(1+r)t} / r]} + {face/(1+r)t} = {cpn × [(1/r) – 1/{r(1+r)t}]} + {face/(1+r)t}
Financial calculator: Bond value equals -PV, where PV is the opposite of the present value as of the reference point
(time 0) of N coupon payments equal to PMT per period where each coupon equals the coupon rate multiplied by the
face value divided by the number of coupons per year, the first coupon is paid 1 period from the reference point (END
mode), N is the number of coupons paid before maturity and equals number of coupons per year multiplied by the
number of years to maturity, the discount rate is I% per period, where I% equals the bond’s yield-to-maturity divided
by the number of coupons per year, and FV is the face (or par) value of the bond
r for a bond = discount rate per period, where a period equals 1 year divided by the number of coupons per year
Coupon payment = (coupon rate × face value) / number of coupons per year
= total aggregate dollar amount of coupons per year / number of coupons per year
Total aggregate dollar amount of coupons per year = coupon rate × face value = coupon rate × par value
= coupon payment × number of coupons per year
Coupon rate = annual coupon rate = total aggregate dollar amount of coupons per year / face value
YTM = yield-to-maturity = expected annual return for a bond (as an APR)
= r × the number of coupon payments per year
= discount rate per period × the number of coupon payments per year
Current yield = total aggregate dollar amount of coupons per year / bond value
= (coupon rate × face value) / bond value
Total dollar return = cash flows from investment + capital gain
= initial value × percentage return = initial value × return
Capital gain = ending value – initial value
Percentage return = return
= total dollar return ÷ initial value
= (cash flows from investment + capital gain) ÷ initial value
= (cash flows from investment + ending value – initial value) ÷ initial value
Percentage return for a bond = return for a bond
= (coupons + capital gain) ÷ initial bond value
= (coupons + ending bond value – initial bond value) ÷ initial bond value
FNAN 303 Formulas and Notes (p. 3 of 6)
Return for a stock = (dividends + capital gain) ÷ initial stock value
= (dividends + ending stock value – initial stock value) ÷ initial stock value
= (D1 + P1 – P0) / P0 when time 1 is today or earlier (so all relevant time periods have taken place)
= dividend yield + capital appreciation yield
Dividend yield = dividends / initial value
Capital appreciation yield = capital gain / initial value = (ending value – initial value) / initial value
Ending value = initial value × (1 + capital appreciation yield)
Expected total dollar return = expected cash flows from investment + expected capital gain
= expected initial value × expected percentage return = expected initial value × expected return
Expected capital gain = expected ending value – expected initial value
Expected percentage return = expected return
= expected total dollar return ÷ expected initial value
= (expected cash flows from investment + expected capital gain) ÷ expected initial value
= (expected CFs from investment + expected ending value – expected initial value) ÷ expected initial value
Expected return for a stock = (expected dividends + expected capital gain) ÷ expected initial stock value
= (expected dividends + expected ending stock value – expected initial stock value) ÷ expected initial stock value
= (D1 + P1 – P0) / P0 when time 0 is today or later (so not all relevant time periods have taken place)
= expected dividend yield + expected capital appreciation yield
Expected dividend yield = expected cash flows from investment / expected initial value
= expected dividends / expected initial value
Expected capital appreciation yield = expected capital gain / expected initial value
= (expected ending value – expected initial value) / expected initial value
Expected ending value = expected initial value × (1 + expected capital appreciation yield)
Stock value
P0 = [(D1 + P1)/(1 + R)]
= [D1/(1 + R)] + [(D2 + P2)/(1 + R)2
] = [D1/(1 + R)] + [D2/(1 + R)2
] + [P2/(1 + R)2
]
= [D1/(1 + R)] + [D2/(1 + R)2
] +…+ [(DN + PN)/(1 + R)N
] = [D1/(1+R)] + [D2/(1+R)2
] +…+ [DN/(1+R)N
] + [PN/(1+R)N
]
= [D1/(1 + R)] + [D2/(1 + R)2
] + …
R for a stock is the annual expected return for the stock divided by the number of possible dividends per year
Expected stock value
Pt = [(Dt+1 + Pt+1) / (1 + R)]
= [Dt+1/(1 + R)] + [(Dt+2 + Pt+2)/(1 + R)2
] = [Dt+1/(1 + R)] + [Dt+2/(1 + R)2
] + [Pt+2/(1 + R)2
]
= [Dt+1/(1 + R)] + [Dt+2/(1 + R)2
] + … + [(Dt+N + Pt+N)/(1 + R)N
]
= [Dt+1/(1 + R)] + [Dt+2/(1 + R)2
] + …
Constant dividend (no-growth) model
P0 = D / R
R = D / P0 and D = R × P0
Constant dividend (no-growth) model
Pt = D / R
R = D / Pt and D = R × Pt
Constant dividend growth model
P0 = D1 / (R – g)
Dk = D1 × (1 + g)k – 1 which is the same as Dt = D1 × (1 + g)t – 1
Also, Db = Da × (1 + g)b-a
so g = [(Db / Da)
1/(b-a)] – 1
R = (D1 / P0) + g and D1 = P0 × (R – g) and g = R – (D1 / P0)
Constant dividend growth model
Pt = Dt+1 / (R – g)
R = (Dt+1 / Pt) + g and Dt+1 = Pt × (R – g) and g = R – (Dt+1 / Pt)
Non-constant dividend growth model
P0 = [D1/(1 + R)] + [D2/(1 + R)2
] + … + [(DN + PN)/(1 + R)N
] where PN = DN+1 / (R – g)
FNAN 303 Formulas and Notes (p. 4 of 6)
Net present value = NPV = C0 + [C1/(1+r)1
] + [C2/(1+r)2
] + … + [Ct/(1+r)t
]
Financial calculator: npv(discount rate, C0, {C1, C2, …, last non-zero expected cash flow}) produces NPV
Internal rate of return = IRR = discount rate such that the present value of a project’s expected cash flows is zero
0 = C0 + [C1/(1+IRR)1
] + [C2/(1+IRR)2
] + … + [Ct/(1+IRR)t
]
Financial calculator: irr(C0, {C1, C2, …, last non-zero expected cash flow}) produces IRR
The payback period is the length of time that it takes for the cumulative expected cash flows produced by a
project to equal the initial investment
If payback period is between t and t+1 years, then the portion of year t+1 needed to produce the cash for payback
= expected CF needed for payback after t years / expected CF in year t+1
= [investment – cumulative expected CFs produced through time t] / expected CF in year t+1
= [investment – (C1 + C2 + … + Ct)] / Ct+1
The discounted payback period is the length of time that it takes for the cumulative discounted expected cash
flows produced by a project to equal the initial investment
If discounted payback period is between t and t+1 years, then the portion of year t+1 needed to produce the
discounted cash for discounted payback
= expected DCF needed for discounted payback after t years / expected DCF in year t+1
= [investment – cumulative expected DCFs produced through time t] / expected DCF in year t+1
= [investment – {PV(C1) + PV(C2) + … + PV(Ct)}] / PV(Ct+1)
Relevant cash flow for a project = incremental expected cash flow
= expected cash flow with project – expected cash flow without project
Relevant cash flow for a project = operating cash flow + cash flow effects from changes in net working capital +
cash flow from capital spending + terminal value
Operating cash flow = OCF = net income + depreciation
Project net income = EBIT – taxes
Project EBIT = project earnings before interest and taxes = project taxable income
= revenue – costs – depreciation
Project taxes = taxable income × tax rate = EBIT × tax rate
Revenue = sales = number of units sold × average price per unit
Total costs = costs = total expenses = expenses = fixed costs + variable costs
Variable costs = number of units sold × average variable cost per unit
Annual straight-line depreciation = (investment – amount item is depreciated to) / depreciable life
= (investment – amount item is depreciated to) / useful life
Note: depreciation expense can only be taken during useful life
Annual MACRS depreciation = investment × relevant rate
Net working capital = NWC = current assets – current liabilities = CA – CL
For project analysis, CA = cash + securities + receivables + inventories and CL = payables
Change in NWC = ΔNWC = NWC at end of period – NWC at start of period, except for the initial change in
NWC at time 0, which equals NWC at time 0
ΔNWCt = NWCt – NWCt-1 and ΔNWCt+1 = NWCt+1 – NWCt
Cash flow effect from change in NWC = opposite of the change in NWC = -ΔNWC
Cash flow from asset sale = sale price of asset – taxes paid on sale of asset
Taxes paid on sale of asset = (sale price of asset – book value of asset) × tax rate
= taxable gain on asset × tax rate
Book value of asset = initial investment – accumulated depreciation
Accumulated depreciation = cumulative sum of all depreciation taken for an asset
FNAN 303 Formulas and Notes (p. 5 of 6)
Average annual return = arithmetic average return = arithmetic average annual return = arithmetic mean return
= arithmetic mean annual return = arithmetic return = arithmetic annual return = mean return = mean annual return
= (return in year 1 + return in year 2 + … + return in year n) / n
= (1/n)(return in year 1) + (1/n)(return in year 2) + … + (1/n)(return in year n)
Compound return = compound annual return = geometric average return = geometric average annual return
= geometric mean return = geometric mean annual return = geometric return = geometric annual return
= (1 + return in year 1) × (1 + return in year 2) × … × (1 + return in year n) – 1
= (ending value / starting value) – 1 with no interim CFs or with reinvestment of any interim CFs
(1 + compound annual return)
n
= (1 + return in year 1) × (1 + return in year 2) × … × (1 + return in year n)
= (ending value / starting value) with no interim CFs or with reinvestment of any interim CFs
(1 + real rate) = (1 + nominal rate) ÷ (1 + inflation rate)
Real rate = [(1+nominal rate) ÷ (1+inflation rate)] – 1
(1 + nominal rate) = (1 + real rate) × (1 + inflation rate)
Nominal rate = [(1+real rate) × (1+inflation rate)] – 1
= the “regular” or “normal” return, expected return, discount rate, etc. used throughout the course
(1+inflation rate) = (1+nominal rate) ÷ (1+real rate)
Inflation rate = [(1+nominal rate) ÷ (1+real rate)] – 1
Variance of R based on past returns = sample variance
= {[1/(n-1)] × [R1 – mean(R)]2} + {[1/(n-1)] × [R2 – mean(R)]2} + … + {[1/(n-1)] × [Rt – mean(R)]2} + … + {[1/(n-1)]
× [Rn – mean(R)]2}
Standard deviation of R based on past returns = sample standard deviation
= √variance of R based on past returns = √sample variance
Expected return based on possible future outcomes = E(R) = [p(1) × R(1)] + [p(2) × R(2)] + … + [p(S) × R(S)]
p(s) is the probability of state (or outcome) s occurring, where the sum of all probabilities equals 1, which is 100%
R(s) is the return in state (or outcome) s
Variance of returns based on future outcomes
= {p(1) × [R(1) – E(R)]2} + {p(2) × [R(2) – E(R)]2} + … + {p(S) × [R(S) – E(R)]2}
Standard deviation of returns based on future outcomes = √variance of returns based on future outcomes
Portfolio return = Rp = [x1 × R1] + [x2 × R2] + … + [xn × Rn]
Expected portfolio return = E(Rp) = [x1 × E(R1)] + [x2 × E(R2)] + … + [xn × E(Rn)]
xi = (value of holdings of asset i in the portfolio) / (total value of the portfolio) where the sum of all weights equals 1,
which is 100%, and the total value of the portfolio is the sum of the holdings of all the assets in the portfolio
Expected return = return on risk-free asset + risk premium = risk-free rate + risk premium
= required return for financial asset like a stock or bond
Risk premium = expected return – return on risk-free asset = expected return – risk-free rate
Actual return = E(R) + U = expected return + unexpected return
= E(R) + systematic portion of unexpected return + unsystematic portion of unexpected return
Total risk = systematic risk + unsystematic risk
= (risk from macroeconomic surprises + risk from individual surprises) in FNAN 303
β for a portfolio = βp = x1β1 + x2β2 + … + xnβn
CAPM: E(Ri) = Rf + (βi × [E(RM) – Rf]) for asset i and E(Rp) = Rf + (βp × [E(RM) – Rf]) for portfolio p
E(Ri) = Rf + (βi × market premium) for asset i and E(Rp) = Rf + (βp × market premium) for portfolio p
Market risk premium = market premium = expected market return – risk-free rate = E(RM) – Rf
After-tax expected cost of debt = pre-tax cost of debt × (1 – Tc) = RD × (1 – Tc) = YTM × (1 – Tc) for a bond
With 3 capital structure items (common stock, preferred stock, and debt):
Weighted average cost of capital = WACC = RA = [(E/V) × RE] + [(P/V) × RP] + [(D/V) × RD × (1 – Tc)]
where V = E + P + D = value of firm’s assets = value of firms’ capital structure
E, P, and D = value of all of firm’s common equity, preferred equity, and debt respectively
Where E, P, and D each = (number of that particular type of shares or bonds × price per that type of share or bond) ÷ V
FNAN 303 Formulas and Notes (p. 6 of 6)
For exam problems:
Round values less than $10,000 to nearest penny and values equal to or greater than $10,000 to nearest dollar
Round rates (and figures based on rates) to nearest hundredth of a percent
Round variance and (weight for variance multiplied by the squared deviation) terms to 6 decimal places
Round (asset weight multiplied by asset β) terms to 3 decimal places
Round portfolio and asset β figures to 2 decimal places
Assume a rate (such as interest or return) is a compound rate unless told otherwise.
Assume a given rate is for a year unless told otherwise. More specifically, assume rate is an APR unless told
otherwise.
Note that a quarter is 3 months, a semi-annual period (or half year) is 6 months, and a year is 12 months
Assume “in x periods” is equivalent to “in x periods from today” unless told otherwise.
Assume there is a “flat yield curve.” Therefore, the rate (interest rate, discount rate, expected return, etc.)
associated with any set of cash flows is not influenced by the timing of the cash flows.
The terms “value” and “market value” refer to present value unless it is explicitly indicated that some other
value (like future value, book value, or face value) is relevant and being referred to.
For a given source of cash flows, assume that the discount rate is constant and the same for each individual
time period.
Assume markets for financial investments and securities like stocks and bonds are well-functioning and that
markets for projects and business activities are not well-functioning. However, assume that markets for
assets such as buildings, plants, stores, etc. are well-functioning in questions that involve comparisons of risk
and/or value.
Assume that the rate of return is greater than the growth rate with a growing perpetuity or a stock with a
constantly growing dividend.
If a question asks for a "payment" or a “contribution” then the payment or contribution equals the magnitude
of the cash flow and is a positive number.
Assume the term bond refers to a “typical” bond that pays regular, fixed coupon payments and all principal at
maturity; par (or face) value is $1,000; and amount initially borrowed is par (or face) value unless explicitly
noted otherwise.
The terms “return” and “rate of return” refer to percentage return, not total dollar return.
When valuing a bond or stock at the same time that a coupon or dividend is paid, assume that the coupon or
dividend is paid just before the bond or stock is valued. Also, if a bond or stock is sold at the same time that a
coupon or dividend is paid, assume the security is sold just after the coupon or dividend is paid
Assume that the expected return for stocks, bonds, and other financial investments remains constant over
time. It is the same for each individual time period.
Assume that a project has conventional cash flows if its expected cash flows are consistent with conventional
cash flows. In other words, if the cash flows look conventional, assume they are conventional.
Assume any cash flows from terminal values are after-tax values
Assume expected returns, discount rates, and costs of capital are positive, unless explicitly noted as negative
or given information that indicates rate is or may be negative.
Unless explicitly noted or told otherwise, use the arithmetic average return when computing an average (or a
mean, which is the same thing). The terms “average” and “mean” refer to the arithmetic average.
Unless noted otherwise, assume the returns of all the assets in a portfolio (with more than 1 asset) do not
move perfectly together in the same direction by the same relative amount.
Unless noted otherwise, assume that any information that is “announced” is information that was previously
private and made public by the announcement.
Assume the expected return of the market portfolio is greater than the risk-free return and that both are greater
than 0.
Assume that the expected cash flows and level of risk associated with a particular project are the same for all
firms.
Lecture Problem 1-a
If Eric invests $500 today in an account that earns 8.00% per year in simple interest, how much will he have in 15 years?
With simple interest, C0 today becomes the following in t periods:
C0 × (1 + (simple interest rate per period × t))
So, Eric would have 500 × (1 + (.0800 × 15)) = 500 × (1 + 1.200) = $1,100
Eric would earn $600 in interest as .0800 × 500 = $40 per year for 15 years, so he would have $500 + $600 = $1,100 in 15 years
---------------------------------------------------------
With simple interest,
Ct = C0 × [(1 + (r)(t)]
1,100 = 600 x [1+(r)(12)]
1,100/600 = 1+ 12(r)
1.8333 = 1+12(r)
1.8333-1 = 12(r)
r = 0.8333/12 = 0.0694 = 6.94%
-------------------------------------------------
Lecture Problem 1-c
If Ben can earn simple interest of 8.6% per year, then how much would Ben need to invest today to have $1,100 in 12 years?
C0 = Ct / [(1 + (r)(t)] = 1,100/[1+(0.86)(12)] = 541.34
=============================
Lecture Problem 2
If Martha invests $500 today in an account that earns 12.34% per year in compound interest, how much will she have in 12 years?
With compound interest, FVt = C0 × (1 + r)t
In this case:
C0 = 500
r = .1234
t = 12
FV12 = 500 × (1.1234)12 = $2,020.15
Martha would have $2,020.15 in 12 years
================
Lecture Problem 3
How much will you have in 9 years if you invest $1,234 in 2 years and your account has a return of 2.36% per year for each of the next 14 years?
FVt = Ck × (1 + r)t-k
| Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Re-time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
| Investment amt | 1,234 | |||||||||
| Future value | ? |
Note: in 9 years means in 9 years from today
t = 9
k = 2
t – k = 9 – 2 = 7
r = .0236
C2 = 1,234
FVt = Ck × (1 + r)t-k
FV9 = C2 × (1 + r)7
FV9 = 1,234 × (1.0236)7 = $1,452.87
Mode is not relevant, since PMT = 0
Enter 7 2.36 -1,234 0
N I% PV PMT FV
Solve for 1,452.87
You would have $1,452.87
=============================
Lecture Problem 4
If Maria plans to invest $800 in 3 years in an account that has an expected return of 7.46% per year and JoJo plans to invest $1,100 in 5 years in an account that has an expected return of 4.52% per year, then who is expected to have more money in 11 years?
Maria
| Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| Re-time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |||
| Maria investment | 800 | |||||||||||
| Maria future value | ? |
FVt = Ck × (1 + r)t-k
t = 11; k = 3; t – k = 11 – 3 = 8
r = .0746
C3 = 800
FV11 = C3 × (1 + r)8
= 800 × (1.0746)8 = $1,422.54
Mode is not relevant, since PMT = 0
Enter 8 7.46 -800 0
N I% PV PMT FV
Solve for 1,422.54
Maria is expected to have $1,422.54 in 11 years
JoJo
| Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| Re-time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |||||
| JoJo investment | 1,100 | |||||||||||
| JoJo future value | ? |
FVt = Ck × (1 + r)t-k
t = 11; k = 5; t – k = 11 – 5 = 6
r = .0452
C5 = 1,100
FV11 = C5 × (1 + r)6
= 1,100 × (1.0452)6 = $1,434.13
Mode is not relevant, since PMT = 0
Enter 6 4.52 -1,100 0
N I% PV PMT FV
Solve for 1,434.13
JoJo is expected to have $1,434.13 in 11 years
Answer: JoJo is expected to have more money in 11 years
=============================
Lecture Problem 5
If Maria plans to invest $800 in 3 years in an account that has an expected return of 7.46% per year and JoJo plans to invest $1,100 in 5 years in an account that has an expected return of 4.52% per year, then who is planning to invest in a riskier account?
The expected return or interest rate associated with a given investment or asset is related to the risk of the investment’s or asset’s cash flows
Recall that in FNAN 303, we impose the simplifying assumption of a “flat yield curve” which means that the rate (expected return interest rate, etc.) associated with any set of cash flows depends only on the risk of those cash flows.
Maria’s investment has an expected return of 7.46%
JoJo’s investment has an expected return of 4.52%
Answer: Maria is planning to invest in a riskier account
==================================
Lecture Problem 6-a
I Scream Ice Cream Company just bought 1 ton of sugar from Sweet Cane, I Scream has been offered the following 2 options, and the discount rate is 7.20% per quarter.
Option A: I Scream Ice Cream will pay $1,260 to Sweet Cane in 2 quarters
Option B: I Scream Ice Cream will pay $1,340 to Sweet Cane in 3 quarters
Which option, A or B, is better for I Scream Ice Cream?
To answer this question, we need to find the present value of options A and B from the perspective of I Scream Ice Cream, which would be making a payment to Sweet Cane
Option A:
PV0 = Ct ÷ (1+r)t
r = .0720
t = 2
C2 = -1,260
PV0 = C2 ÷ (1+r)2
= -1,260 ÷ (1.0720)2
= -1,096.43
Option B:
PV0 = Ct ÷ (1+r)t
r = .0720
t = 3
C3 = -1,340
PV0 = C3 ÷ (1+r)3
= -1,340 ÷ (1.0720)3
= -1,087.73
The better option for I Scream Ice Cream is the one with the higher present value, which is option B: -1,087.73 > -1,096.43
The better option for I Scream Ice Cream is B, which is equivalent in value to paying $1,087.73 today. This is better than option A, which is equivalent in value to paying $1,096.43 today.
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Lecture Problem 6-b
I Scream Ice Cream Company just bought 1 ton of sugar from Sweet Cane, I Scream has been offered the following 2 options, and the discount rate is 7.20% per quarter.
Option A: I Scream Ice Cream will pay $1,260 to Sweet Cane in 2 quarters
Option B: I Scream Ice Cream will pay $1,340 to Sweet Cane in 3 quarters
Which option, A or B, is better for Sweet Cane?
To answer this question, we need to find the present value of options A and B from the perspective of Sweet Cane, which would be receiving a payment from I Scream Ice Cream
Option A:
PV0 = Ct ÷ (1+r)t
r = .0720
t = 2
C2 = 1,260
PV0 = C2 ÷ (1+r)2
= 1,260 ÷ (1.0720)2
= 1,096.43
Option B:
PV0 = Ct ÷ (1+r)t
r = .0720
t = 3
C3 = 1,340
PV0 = C3 ÷ (1+r)3
= 1,340 ÷ (1.0720)3
= 1,087.73
The better option for Sweet Cane is the one with the higher present value, which is option A: 1,096.43 > 1,087.73
The better option for Sweet Cane is A, which is equivalent in value to receiving $1,096.43 today. This is better than option B, which is equivalent in value to receiving $1,087.73 today.
======================
Lecture Problem 7-a
I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a discount rate is 6%, for an expected cash flow of $800,000 in 3 years and plant B, which has a discount rate is 8%, for an expected cash flow of $800,000 in 3 years. What is the value of plant A? The plants are expected to produce no cash flows other than the cash produced when they are sold.
PV0 = Ct / (1 + r)t
| Time | 0 | 1 | 2 | 3 |
| Cash flow | $800,000 | |||
| Present value | ? |
t = 3
r = .06
C3 = 800,000
PV0 = 800,000 / (1.06)3 = 671,695
Mode is not relevant, since PMT = 0
Enter 3 6 0 800,000
N I% PV PMT FV
Solve for -671,695
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Lecture Problem 7-b
I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a discount rate is 6%, for an expected cash flow of $800,000 in 3 years and plant B, which has a discount rate is 8%, for an expected cash flow of $800,000 in 3 years. What is the value of plant B? The plants are expected to produce no cash flows other than the cash produced when they are sold.
PV0 = Ct / (1 + r)t
t = 3
r = .08
C3 = 800,000
PV0 = 800,000 / (1.08)3 = 635,066
Mode is not relevant, since PMT = 0
Enter 3 8 0 800,000
N I% PV PMT FV
Solve for -635,066
-------------------------------------
Lecture Problem 7-c
I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a discount rate is 6%, for an expected cash flow of $800,000 in 3 years and plant C, which has a discount rate is 6%, for an expected cash flow of $800,000 in 5 years. What is the value of plant C? The plants are expected to produce no cash flows other than the cash produced when they are sold.
PV0 = Ct / (1 + r)t
| Time | 0 | 1 | 2 | 3 | 4 | 5 |
| Cash flow | $800,000 | |||||
| Present value | ? |
t = 5
r = .06
C5 = 800,000
PV0 = 800,000 / (1.06)5 = 597,807
Mode is not relevant, since PMT = 0
Enter 5 6 0 800,000
N I% PV PMT FV
Solve for -597,807
=================================
Lecture Problem 8-a
I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 3 years and plant B, which has a cost of capital is 8%, for an expected cash flow of $800,000 in 3 years. Which plant, A or B, is riskier, or are they equally risky? The plants are expected to produce no cash flows other than the cash produced when they are sold.
Plant B is riskier than plant A, because plant B has a higher cost of capital than plant A. Recall that riskier cash flows are associated with higher discount rates, as investors demand greater reward for bearing more risk.
-------------------------------------------------------------
Lecture Problem 8-b
I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 3 years and plant B, which has a cost of capital is 8%, for an expected cash flow of $800,000 in 3 years. Which plant, A or B, is worth more today? The plants are expected to produce no cash flows other than the cash produced when they are sold.
Plant A: PV0 = 800,000 / (1.06)3 = 671,695
Plant B: PV0 = 800,000 / (1.08)3 = 635,066
The present value of plant A (671,695) is greater than the present value of plant B ($635,066), so plant A is worth more despite the fact that both plants are expected to be sold in 3 years for $800,000. Recall that a higher cost of capital leads to a lower present value, all else equal (cash flow amount and timing).
---------------------------------------------------
Lecture Problem 8-c
I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 3 years and plant C, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 5 years. Which plant, A or C, is riskier, or are they equally risky? The plants are expected to produce no cash flows other than the cash produced when they are sold.
Plants A and C are equally risky, because they have the same cost of capital. Recall that riskier cash flows are associated with higher discount rates, as investors demand greater reward for bearing more risk. Since the costs of capital are the same, the cash flows, and thus the plants, are equally as risky.
----------------------------------------------------------
Lecture Problem 8-d
I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant A, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 3 years and plant C, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 5 years. Which plant, A or C, is worth more today? The plants are expected to produce no cash flows other than the cash produced when they are sold.
Plant A: PV0 = 800,000 / (1.06)3 = 671,695
Plant C: PV0 = 800,000 / (1.06)5 = 597,807
The present value of plant A (671,695) is greater than the present value of plant C (597,807), so plant A is worth more despite the fact that both plants have the same cost of capital (and level of risk) and are expected to be sold for $800,000. Recall that a longer time leads to a lower present value, all else equal (cash flow amount and discount rate).
-----------------------------------------------------------------
Lecture Problem 8-e
I Scream Ice Cream Company is considering selling several of its plants. The firm expects to sell plant C, which has a cost of capital is 6%, for an expected cash flow of $800,000 in 5 years and plant D which has a cost of capital is 6%, for an expected cash flow of $900,000 in 5 years. Which plant, C or D, is worth more today? The plants are expected to produce no cash flows other than the cash produced when they are sold.
Plant C: PV0 = 800,000 / (1.06)5 = 597,807
Plant D: PV0 = 900,000 / (1.06)5 = 672,532
The present value of plant D (672,532) is greater than the present value of plant C (597,807), so plant D is worth more despite the fact that both plants have the same cost of capital (and level of risk) and are expected to be sold at the same time (in 5 years). Recall that a larger expected cash flow leads to a higher present value, all else equal (time until cash flow and discount rate).
=====================================
Lecture Problem 9
What is the cost of capital of the I Scream Ice Cream Company plant in Texas if it is worth $1,000,000 and is expected to produce no cash flows other than the cash produced when it is sold in 4 years for an expected cash flow of $1,400,000?
PV0 = Ct / (1 + r)t
t = 4
PV0 = 1,000,000
C4 = 1,400,000
Can be solved algebraically, but much easier with financial calculator
Mode is not relevant, since PMT = 0
Enter 4 -1,000,000 0 1,400,000
N I% PV PMT FV
Solve for 8.78
Confirm
PV0 = Ct / (1 + r)t
= 1,400,000 / (1.0878)4
= $999,843.02 ≈ $1,000,000 ☺
(Difference due to rounding)
====================================
Lecture Problem 10
In how many years from today is the I Scream Ice Cream Company plant in Florida expected to be sold if it is worth $1,000,000, has a cost of capital of 8.20 percent, and is expected to produce no cash flows other than the cash produced when it is sold for an expected cash flow of $1,300,000?
PV0 = Ct / (1 + r)t
PV0 = 1,000,000
Ct = 1,300,000
r = .0820
Can be solved algebraically, but much easier with financial calculator
Mode is not relevant, since PMT = 0
Enter 8.20 -1,000,000 0 1,300,000
N I% PV PMT FV
Solve for 3.33
The plant will be sold in 3.33 years
Confirm
PV0 = Ct / (1 + r)t
= 1,300,000 / (1.0820)3.33
= $999,923.04 ≈ $1,000,000 ☺
(Difference due to rounding)
===========================================
Lecture Problem 11-a
Three years ago, Pablo invested $2,000. In 2 years, he expects to have $2,850. If Pablo expects to earn the same annual rate of return after 2 years from today as the annual rate implied from the past and expected values given in the problem, then how much does he expect to have in 5 years from today?
To solve:
1) Find the implied return over the 5 year period from 3 years ago to 2 years from today
2) Use the implied return to determine how much he’ll have 5 years from today
1) Find the implied return over the 5 year period from 4 years ago to 1 year from today
| Time | -3 | -2 | -1 | 0 | 1 | 2 |
| Re-time | 0 | 1 | 2 | 3 | 4 | 5 |
| Invest | 2,000 | |||||
| Future value | 2,850 |
Mode is not relevant, since PMT = 0
Enter 5 -2,000 0 2,850
N I% PV PMT FV
Solve for 7.34
2) Use the implied return to determine how much he’ll have 5 years from today
| Time | 0 | 1 | 2 | 3 | 4 | 5 |
| Re-time | 0 | 1 | 2 | 3 | ||
| Invest | 2,850 | |||||
| Future value | ? |
Mode is not relevant, since PMT = 0
Enter 3 7.34 -2,850 0
N I% PV PMT FV
Solve for 3,524.76
Pablo would have $3,524.76 in 5 years from today, which is 3 years from 2 years from today
(Solutions may differ somewhat due to rounding annual rate of return)
Alternatively
| Time | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
| Re-time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Invest | 2,000 | ||||||||
| Future value | ? |
Mode is not relevant, since PMT = 0
Enter 8 7.34 -2,000 0
N I% PV PMT FV
Solve for 3,524.70
Pablo would have $3,524.70 in 5 years from today, which is 8 years from 3 years ago
(Solutions may differ somewhat due to rounding annual rate of return)
------------------------------------------------
Lecture Problem 11-b
Three years ago, Pablo invested $2,000. In 2 years, he expects to have $2,850. If Pablo expects to earn the same annual rate of return after 2 years from today as the annual rate implied from the past and expected values given in the problem, then in how many years from today does he expect to have exactly $4,000?
To solve:
1) Find the implied return over the 5 year period from 3 years ago to 2 years from today
2) Use the implied return to determine when goal will be reached relative to one of the given values and then relative to today
1) Find the implied return over the 5 year period from 4 years ago to 1 year from today
| Time | -3 | -2 | -1 | 0 | 1 | 2 |
| Re-time | 0 | 1 | 2 | 3 | 4 | 5 |
| Invest | 2,000 | |||||
| Future value | 2,850 |
Mode is not relevant, since PMT = 0
Enter 5 -2,000 0 2,850
N I% PV PMT FV
Solve for 7.34
2) Use the implied return to determine when goal will be reached relative to one of the given values and then relative to today
| Time | 0 | 1 | 2 | 3 | … | ? |
| Re-time | 0 | 1 | … | ? – 2 | ||
| Invest | 2,850 | |||||
| Future value | 4,000 |
Mode is not relevant, since PMT = 0
Enter 7.34 -2,850 0 4,000
N I% PV PMT FV
Solve for 4.79
Pablo would have $4,000 in 4.79 years from 2 years from today
Therefore, Pablo would have $4,000 in 6.79 years from today
(Solutions may differ somewhat due to rounding annual rate of return)
Alternatively
| Time | -3 | -2 | -1 | 0 | … | ? |
| Re-time | 0 | 1 | 2 | 3 | … | ? + 3 |
| Invest | 2,000 | |||||
| Future value | 4,000 |
Mode is not relevant, since PMT = 0
Enter 7.34 -2,000 0 4,000
N I% PV PMT FV
Solve for 9.79
Pablo would have $4,000 in 9.79 years from 3 years ago
Therefore, Pablo would have $4,000 in 6.79 years from today
(Solutions may differ somewhat due to rounding annual rate of return)
===============================================
Lecture Problem 1-a
How much is a building worth that is expected to produce cash flows of $200,000 in 1 year and $500,000 in 2 years if the cost of capital is 10.0 percent? Since there’s no information on expected cash flows at times other than in 1 and 2 years, assume those are the only times with non-zero expected cash flows.
| Time | 0 | 1 | 2 |
| Expected cash flow | 0 | $200,000 | $500,000 |
| Present value | ? |
PV = C0 + [C1/(1+r)1] + [C2/(1+r)2]
C0 = 0 C1 = 200,000 C2 = 500,000 r = .100
PV = C0 + [C1/(1+r)1] + [C2/(1+r)2]
= 0 + [200,000/(1.100)] + [500,000/(1.100)2]
= 0 + 181,818 + 413,223
= 595,041
Answer: $595,041
====================================
Lecture Problem 1-b
A building that is worth $500,000 and has a cost of capital of 10.0% is expected to produce cash flows of $200,000 in 1 year and X in 3 years. What is X?
| Time | 0 | 1 | 2 | 3 |
| Expected cash flow | 0 | $200,000 | $0 | X |
| Present value | $500,000 |
PV = C0 + [C1/(1+r)1] + [C2/(1+r)2]
PV = 500,000 C0 = 0 C1 = 200,000 C2 = 0 C3 = X r = .100
PV = C0 + [C1/(1+r)1] + [C2/(1+r)2] + [C3/(1+r)3]
500,000 = 0 + [200,000/(1.10)] + [0/(1.100)2] + [X/(1.100)3]
500,000 = 0 + [200,000/(1.10)] + [X/(1.100)3]
500,000 = 0 + 181,818 + [X/(1.100)3]
So 500,000 – 181,818 = [X/(1.100)3]
So 318,182 = [X/(1.100)3]
The present value of the cash flow expected in year 3 is $318,182
The value of the cash flow expected in year 3 is X where 318,182 = [X/(1.100)3]
So X = [318,182 × (1.100)3] = $423,500
Confirm:
PV = C0 + [C1/(1+r)1] + [C2/(1+r)2] + [C3/(1+r)3]
PV = 0 + [200,000/(1.100)1] + [0/(1.100)2] + [423,500/(1.100)3]
= 0 + 181,818 + 0 + 318,182
= 500,000 ☺
========================================
Lecture Problem 2
You own a building that is expected to pay annual cash flows forever. What is the value of the building if the cost of capital is 8.0% and annual cash flows of $500,000 are expected with the first one in 1 year?
| Time | 0 | 1 | 2 | 3 | 4 | … |
| Cash flow | $0 | $500,000 | $500,000 | $500,000 | $500,000 | … |
| Present value | ? |
The cash flows reflect a fixed perpetuity
PV = C / r
C = $500,000 r = .080
PV = $500,000 / .080 = $6,250,000
==============================
Lecture Problem 3-a
You own a building that is expected to pay annual cash flows forever. What is the amount of the annual cash flow produced by a building expected to be if the building is worth $2,300,000, the cost of capital is 5.0%, and annual fixed cash flows are expected with the first one due in one year?
| Time | 0 | 1 | 2 | 3 | 4 | … |
| Cash flow | $0 | C | C | C | C | … |
| Present value | $2,300,000 |
The cash flows reflect a fixed perpetuity
C = PV × r
PV = $2,300,000 r = .050
C = $2,300,000 × .050
= $115,000
==============================
Lecture Problem 3-b
What is the present value (as of today) of the expected cash flow produced by a building in 3 years if the building is worth $2,300,000, the cost of capital is 5.0%, and annual fixed cash flows are expected with the first one due in one year?
Approach
1) Find the cash flow expected in 3 years
2) Find the present value of the cash flow expected in 3 years
1) Find the cash flow expected in 3 years
| Time | 0 | 1 | 2 | 3 | 4 | … |
| Cash flow | $0 | C | C | C | C | … |
| Present value | $2,300,000 |
The cash flows reflect a fixed perpetuity
C = PV × r
PV = $2,300,000
r = .050
C = $2,300,000 × .050
= $115,000
2) Find the present value of the cash flow expected in 3 years
PV0 = C3 / (1+r)3
C3 = $115,000 (note that all cash flows are $115,000)
r = .050
PV0 = 115,000 / 1.0503
= $99,341
===============================
Lecture Problem 3-c
What is the cost of capital for the building if its value is $1,700,000 and annual cash flows of $100,000 are expected with the first one in 1 year?
| Time | 0 | 1 | 2 | 3 | 4 | … |
| Cash flow | $0 | $100,000 | $100,000 | $100,000 | $100,000 | … |
| Present value | 1,700,000 |
The cash flows reflect a fixed perpetuity
r = C / PV
C = $100,000 PV = $1,700,000
r = $100,000 / $1,700,000 = .0588 = 5.88%
=========================================
Lecture Problem 4
An investment is expected to generate annual cash flows forever. The first annual cash flow is expected in 1 year and all subsequent annual cash flows are expected to grow at a constant rate annually. We know that the cash flow expected in 2 years from today is expected to be $50.00 and the cash flow expected in 5 years from today is expected to be $55.62. What is the cash flow expected to be in 4 years from today?
Approach:
1) Find the annual growth rate
2) Find the cash flow expected in 4 years
| Time | 0 | 1 | 2 | 3 | 4 | 5 | … |
| CF | 0 | C1 | C2 | C3 | C4 | C5 | … |
| CF | 0 | C1 | 50.00 | C3 | C4 | 55.62 | … |
| CF | 0 | C1 | C2 | C2 × (1+g) | C2 × (1+g)2 | C2 × (1+g)3 | … |
| CF | 0 | C1 | C2 | C3 | C4 | C4 × (1+g) | … |
| CF | 0 | C1 | C2 | C3 | C4 | C5 | … |
1) Find the annual growth rate
We know that Cb = Ca × (1+g)b-a so C5 = C2 × (1+g)3
C2 = 50.00 C5 = 55.62
So 55.62 = 50.00× (1+g)3
(55.62/50.00) = (1+g)3
(55.62/50.00)(1/3) = [(1+g)3](1/3) = 1 + g
= 1.0361
So g = 1.0361 – 1 = .0361
2) Find the cash flow expected in 4 years
We know that Cb = Ca × (1+g)b-a so C4 = C2 × (1+g)4-2 = C2 × (1+g)2
C2 = 50.00 and g = .0361
So C4 = C2 × (1+g)4-2 = 50.00× (1.0361)2 = $53.68
We also know that Cb = Ca × (1+g)b-a so C5 = C4 × (1+g)5-4 = C4 × (1+g)1
So 55.62 = C4 × 1.0361
C5 = 55.62 / 1.0361 = $53.68
=================================
Lecture Problem 5-a
You own a building that is expected to pay annual cash flows forever. What is the value of the building if the cost of capital is 8.3% and annual cash flows are expected to grow by 2.3% per year forever with the first one expected to be $500,000 in 1 year?
| Time | 0 | 1 | 2 | 3 | 4 | … |
| Cash flow | $0 | $500,000 | $500,000 × 1.023 | $500,000 × 1.0232 | $500,000 × 1.0233 | … |
| Cash flow | $0 | $500,000 | $511,500 | $523,265 | $535,300 | … |
| Present value | ? |
The cash flows reflect a growing perpetuity
PV = C1 / (r – g)
C1 = $500,000 r = .083 g = .023
PV = $500,000 / (.083 – .023) = $500,000 / .060 = $8,333,333
====================================
Lecture Problem 5-b
You own a building that is expected to pay annual cash flows forever. If the building is worth $2,300,000, the cost of capital is 5.0%, and annual cash flows are expected with the first one due in one year and all subsequent ones growing annually by 2.2%, then what is the amount of the cash flow produced by the building in 1 year expected to be?
| Time | 0 | 1 | 2 | 3 | 4 | … |
| Cash flow | $0 | ? | ? × (1.022) | ? × (1.022)2 | ? × (1.022)3 | … |
| Present value | $2,300,000 |
The cash flows reflect a growing perpetuity
C1 = PV × (r – g)
PV = $2,300,000 r = .050 g = .022
C1 = $2,300,000 × (.050 – .022)
= $2,300,000 × .028
= $64,400
======================================
Lecture Problem 5-c
You own a building that is expected to pay annual cash flows forever. If the building is worth $2,300,000, the cost of capital is 5.0%, and annual cash flows are expected with the first one due in one year and all subsequent ones growing annually by 2.2%, then what is the amount of the cash flow produced by the building in 3 years expected to be?
We know that C1 = $64,400
Therefore, C2 = C1 × (1.022)
And C3 = C2 × (1.022) = C1 × (1.022)2 = 64,400 × (1.022)2 = 67,265
| Time | 1 | 2 | 3 |
| Cash flow | C1 | C2 | C3 |
| Cash flow | C1 | C1 × (1.022) | C2 × (1.022) |
| Cash flow | C1 | C1 × (1.022) | C1 × (1.022)2 |
| Cash flow | 64,400 | 65,816.80 | 67,264.77 |
===============================
Lecture Problem 5-d
You own a building that is expected to pay annual cash flows forever. If the building is worth $850,000, the cost of capital is 8.30%, annual cash flows are expected with the first one due in one year and equal to $62,000, and all subsequent cash flows are expected to grow annually by a constant rate, then what is the expected annual growth rate of the expected cash flows?
| Time | 0 | 1 | 2 | 3 | 4 | … |
| Cash flow | $0 | 62,000 | 62,000 × (1+g) | 62,000 × (1+g)2 | 62,000 × (1+g)3 | … |
| Present value | 850,000 |
The cash flows reflect a growing perpetuity
PV = C1 / (r – g), so g = r – (C1 / PV)
PV = $850,000 r = .0830 C1 = $62,000
g = r – (C1 / PV)
= .0830 – (62,000 / 850,000)
= .0830 – .0729
= .0101
= 1.01%
=============================================
Lecture Problem 6
What is the price of an investment that pays you $1,000 per month with the first payment in 1 month and the last payment in 6 months if the expected return for the investment is 1.0% per month?
From the timeline, we can see that the cash flows reflect a 6-period ordinary annuity
| Time | 0m | 1m | 2m | 3m | 4m | 5m | 6m |
| Pmt # | 1 | 2 | 3 | 4 | 5 | 6 | |
| CF | 0 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 |
Time reflects the number of periods from time 0
Pmt # is a count of the number of payments
END Mode
Enter 6 1.0 1,000 0
N I% PV PMT FV
Solve for -5,795.48
======================================
Lecture Problem 7-a
What is the price of a car if the loan from the bank to buy it involves annual payments of $7,300 to the bank for 5 years at an annual interest rate of 9.4 percent with the first annual payment made to the bank in 1 year and a special payment of $5,000 to the bank in 5 years?
The car price equals the opposite of the present value of the loan payments, including the special payment, discounted by the interest rate of the loan.
| Time | 0 | 1 | 2 | 3 | 4 | 5 |
| CF from regular payments | 0 | -7,300 | -7,300 | -7,300 | -7,300 | -7,300 |
| CF from special payment | -5,000 | |||||
| Present value | ? |
The present value of the loan payments equals:
1) The present value of a 5-period annuity with regular cash flows of -$7,300 and a discount rate of 9.4%
2) A cash flow of -$5,000 in 5 years with a discount rate of 9.4%
1) The PV of the annuity
END mode
Enter 5 9.4 -7,300 0
N I% PMT PV FV
Solve for 28,102
The present value of the cash flows associated with the regular payments is -$28,102
2) PV of -$5,000 in 5 years
Note that mode is not relevant
Enter 5 9.4 0 -5,000
N I% PMT PV FV
Solve for 3,191
The present value of the cash flows associated with the special payment is -$3,191
Combine the 2 pieces
The present value of the loan payments = -$28,102 + (-$3,191) = -$31,293
So the price of the car is $31,293
(Answers may differ slightly due to rounding)
Note: can be done in 1 step, since N = 5 for annuity and N = 5 for special payment
END mode
Enter 5 9.4 -7,300 -5,000
N I% PMT PV FV
Solve for 31,293
The present value of the cash flows associated with the payments is -$31,293
So the price of the car is $31,293
==================================
Lecture Problem 7-b
What is the price of a car if the loan from the bank to buy it involves annual payments of $6,200 to the bank for 5 years at an annual interest rate of 11.7 percent with the first annual payment made to the bank in 1 year and a special payment of $5,000 to the bank in 3 years?
The car price equals the opposite of the present value of the loan payments, including the special payment, discounted by the interest rate of the loan.
| Time | 0 | 1 | 2 | 3 | 4 | 5 |
| CF from regular payments | 0 | -6,200 | -6,200 | -6,200 | -6,200 | -6,200 |
| CF from special payment | -5,000 | |||||
| Present value | ? |
The present value of the loan payments equals:
1) The present value of a 5-period annuity with regular cash flows of -$6,200 and a discount rate of 11.7%
2) A cash flow of -$5,000 in 3 years with a discount rate of 11.7%
1) The PV of the annuity
END mode
Enter 5 11.7 -6,200 0
N I% PMT PV FV
Solve for 22,517
The present value of the cash flows associated with the regular payments is -$22,517
2) PV of -$5,000 in 3 years
Note that mode is not relevant
Enter 3 11.7 0 -5,000
N I% PMT PV FV
Solve for 3,588
The present value of the cash flows associated with the special payment is -$3,588
Combine the 2 pieces
The present value of the loan payments = -$22,517 + (-$3,588) = -$26,105
So the price of the car is $26,105
(Answers may differ slightly due to rounding)
Note: must be done in 2 steps, since N = 5 for annuity and N = 3 for special payment
=====================================
Lecture Problem 8-a
What is the value of an investment that pays you $1,000 a month with the first payment today and the last payment in 6 months if the discount rate is 1.0% per month?
From the timeline, we can see that the cash flows reflect a 7-period annuity due
| Time | 0m | 1m | 2m | 3m | 4m | 5m | 6m |
| Pmt # | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| CF | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 |
Time reflects the number of periods from time 0
Pmt # is a count of the number of payments
N = 7 (number of payments, not necessarily when last payment is expected to be made)
BEGIN Mode
Enter 7 1.0 1,000 0
N I% PV PMT FV
Solve for -6,795.48
==============================
Lecture Problem 8-b
What is the value of an investment that pays you $1,000 per month for 7 months if the first payment is today and the discount rate is 1.0% per month?
Monthly payments “for 7 months” tells us that there are 7 payments
From the timeline, we can see that the cash flows reflect a 7-period annuity due
| Time | 0m | 1m | 2m | 3m | 4m | 5m | 6m |
| Pmt # | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| CF | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 |
Time reflects the number of periods from time 0
Pmt # is a count of the number of payments
N = 7 (number of payments, not necessarily when last payment is expected to be made)
BEGIN Mode
Enter 7 1.0 1,000 0
N I% PV PMT FV
Solve for -6,795.48
====================================
Lecture Problem 8-c
What is the value of an investment that pays you $2,000 in 5 months plus $1,000 per month with the first $1,000 payment today and the last $1,000 payment in 5 months if the discount rate is 1.0% per month?
From the timeline, we can see that the cash flows reflect a 6-period annuity due plus a cash flow of $2,000 in 5 months
| Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| Pmt # | 1 | 2 | 3 | 4 | 5 | 6 | |
| CF | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 + 2000 | 0 |
Value of investment = PV of the6-period annuity due + PV of the cash flow of $2,000 in 5 months
PV of the 6-period annuity due
BEGIN Mode
Enter 6 1.0 1,000 0
N I% PV PMT FV
Solve for -5,853.43
PV of the cash flow of $2,000 in 5 months
PV = C5 / (1+r)5
r = .010
C5 = 2,000 (note: this CF reflects the cash flow not associated with the annuity due)
PV = 2,000 / (1.010)5 = 1,902.93
Mode is not relevant, since PMT = 0
Enter 5 1.0 0 2000
N I% PV PMT FV
Solve for -1,902.93
Aggregate the two sources of value
The value of the investment = 5,853.43 + 1,902.93 = $7,756.36
Important note
You can not do this problem in one calculator step: if you input in $2,000 as FV, the calculator will treat that cash flow as taking place in N periods, which would be in 6 periods and you would produce the following:
BEGIN Mode
Enter 6 1.0 1,000 2,000
N I% PV PMT FV
Solve for -7,737.52
Time line reflected by the preceding:
| Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| CF | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 2000 |
========================================
Lecture Problem 8-d
What is the value of an investment that pays you $2,000 in 6 months plus $1,000 a month with the first $1,000 payment today and the last $1,000 payment in 11 months if the discount rate is 1.0% per month?
From the timeline, we can see that the cash flows reflect a 12-period annuity due plus a cash flow of $2,000 in 6 months
| Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Pmt # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
| CF | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 | 1000 + 2000 | 1000 | 1000 | 1000 | 1000 | 1000 | 0 |
Time reflects the number of periods from time 0
Pmt # is a count of the number of payments
Value of investment = PV of the 12-period annuity due + PV of the cash flow of $2,000 in 6 months
PV of the 12-period annuity due
BEGIN Mode
Enter 12 1.0 1,000 0
N I% PV PMT FV
Solve for -11,367.63
PV of the cash flow of $2,000 in 6 months
PV = C6 / (1+r)6
r = .010
C6 = 2,000 (note: this CF reflects the cash flow not associated with the annuity due)
PV = 2,000 / (1.010)6 = 1,884.09
Mode is not relevant, since PMT = 0
Enter 6 1.0 0 2000
N I% PV PMT FV
Solve for -1,884.09
Aggregate the two sources of value
The value of the investment = 11,367.63 + 1,884.09 = $13,251.72
Note that the investment is worth more when the $2,000 comes sooner (in 6 months vs. 11 months in the previous problem).
===================================
Lecture Problem 9
If Arturo has $50,000 to invest, how much can he expect to receive each year as a fixed annual payment if he receives his first fixed annual payment today, he receives his last payment in 7 years, and he expects to earn 1.0% per year?
The cash flows that Arturo expects to receive reflect an annuity due with 8 annual payments. We want to find the size of the payments such that the annuity due would have a present value of $50,000.
A timeline shows that the number of payments is 8 (even though the last payment occurs in 7 years from now):
| Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Pmt # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| CF | X | X | X | X | X | X | X | X |
BEGIN Mode
Enter 8 1.0 -50,000 0
N I% PV PMT FV
Solve for 6,469.82
Arturo would receive annual payments of $6,469.82
=====================================
Lecture Problem 10
If Arturo has $50,000 to invest in an investment that makes fixed monthly payments of $1,000, how many monthly payments will he receive if he receives his first fixed monthly payment today and his return is 1.0% per month?
| Time | 0 | 1 | 2 | … | N – 2 | N – 1 | N |
| 0y0m | 0y1m | 0y2m | … | ||||
| Pmt # | 1 | 2 | 3 | … | N – 1 | N | |
| CF | 1000 | 1000 | 1000 | … | 1000 | 1000 | |
| Investment value | 50,000 |
The cash flows that Arturo expects to receive reflect an annuity due with $1,000 monthly payments. We want to find the number of annuity due payments that are necessary for the annuity due to have a present value of $50,000.
BEGIN Mode
Enter 1.0 -50,000 1,000 0
N I% PV PMT FV
Solve for 68.67
Arturo would receive 68.67 payments
==============================
Lecture Problem 11
What is the quarterly interest rate for Tony’s loan if he borrowed $5,400 today, he must make equal quarterly payments of $830, with the first quarterly payment due later today and the last quarterly payment due in 6 quarters?
Timeline tip for FNAN 303: the cash flows occur quarterly so the timeline period is a quarter
| Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Payment # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
| Regular payments | -$830 | -$830 | -$830 | -$830 | -$830 | -$830 | -$830 | 0 |
| Present value | -$5,400 |
The payments associated with the loan reflect an annuity due with 7 regular cash flows of -$830 and a present value of -$5,400. Therefore, the quarterly rate can be found as the rate such that a 7-period annuity due with regular cash flows of -$830 for 7 periods has a present value of -$5,400.
BEGIN mode
Enter 7 5,400 -830 0
N I% PV PMT FV
Solve for 2.51
The interest rate for the loan is 2.51%
============================
Lecture Problem 12
In 5 years, Arturo expects to have $40,000 to invest in a security that will make fixed payments. What annual expected return does his investment need in order for him to receive $9,100 per year with the first fixed annual payment received in 6 years from today and the last annual payment in 12 years from today?
Timeline
| Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Re-time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |||||
| Pmt # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ||||||
| Pmt amt | 9,100 | 9,100 | 9,100 | 9,100 | 9,100 | 9,100 | 9,100 | ||||||
| Inv val | 40,000 |
We want to find the annual discount rate such that the present value of a 7-period ordinary annuity of 9,100 per year is $40,000. This is the appropriate approach because the annual payments start in 1 year (after the “re-timed” time 0) and end in 7 years (after the “re-timed” time 0), so there are 7 payments.
END Mode
Enter 7 -40,000 9,100 0
N I% PV PMT FV
Solve for 13.20
The investment must have an expected annual return of 13.20%
=================================
Lecture Problem 13-a
I Scream Ice Cream Company is analyzing several of its upcoming flavors. What is the value of peanut chocolate crunch if the discount rate is 6.2 percent and the flavor is expected to generate annual fixed cash flows forever with the first cash flow of $100,000 in 4 years?
The value can be found in 2 steps.
1) Find the value of the flavor as of one year before the first payment is made
2) Find the present value (as of today) of the flavor
| Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | … |
| CF | 0 | 0 | 0 | 0 | 100k | 100k | 100k | … |
1) The first of an infinite series of fixed payments occurs in 4 years, so we can find the value of the flavor as of 3 years from now as a fixed perpetuity
PV3 = C4 / r
C4 = $100,000 r = .062
PV3 = 100,000 / .062 = 1,612,903
2) The value today of something worth 1,612,903 in 3 years can be found as
PV0 = PV3 / (1+r)3
PV3 = 1,612,903 r = .062
PV0 = 1,612,903 / (1.062)3 = $1,346,588
Mode is not relevant, since PMT = 0
Enter 3 6.2 0 1,612,903
N I% PV PMT FV
Solve for -1,346,588
The value of the flavor is $1,346,588
================================
Lecture Problem 13-b
What is the value of nougat swirl if the cost of capital is 7.2 percent and the flavor is expected to generate annual cash flows forever with the first cash flow of $90,000 in 5 years and all subsequent cash flows growing annually by 1.5 percent?
The value can be found in 2 steps.
1) Find the value of the flavor as of one year before the first payment is made
2) Find the present value (as of today) of the flavor
| Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | … |
| CF | 0 | 0 | 0 | 0 | 0 | 90k | 90k × 1.015 | 90k × 1.0152 | … |
| CF | 0 | 0 | 0 | 0 | 0 | 90,000 | 91,350 | 92,720 | … |
1) The first of an infinite series of constantly growing payments occurs in 5 years, so we can find the value of the flavor as of 4 years from now as a perpetuity with constant growth
PV4 = C5 / (r – g)
C5 = $90,000 r = .072 g = .015
PV4 = 90,000 / (.072 – .015) = 90,000 / (.057) = 1,578,947
2) The value today of something worth 1,578,947 in 4 years can be found as
PV0 = PV4 / (1+r)4
PV4 = 1,578,947 r = .072
PV0 = 1,578,947 / (1.072)4 = $1,195,607
Mode is not relevant, since PMT = 0
Enter 4 7.2 0 1,578,947
N I% PV PMT FV
Solve for - 1,195,607
The value of the flavor is $1,195,607
===============================
Lecture Problem 13-c
I Scream Ice Cream Company is analyzing several of its upcoming flavors. What is the value of coffee toffee if the cost of capital is 8.0 percent and the flavor is expected to generate fixed annual cash flows of $250,000 with the first cash flow in 6 years and the last cash flow in 14 years?
The value can be found in 2 steps.
1) Find the value of the flavor as of one year before the first payment is made
2) Find the present value (as of today) of the flavor
| Time | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
| Pmt # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||||||
| CF | 0 | 0 | 0 | 0 | 0 | 0 | 250k | 250k | 250k | 250k | 250k | 250k | 250k | 250k | 250k |
1) The first of a series of 9 fixed payments occurs in 6 years, so we can find the value of the flavor as of 5 years from now as an ordinary annuity with 9 payments of $250,000. The nine fixed payments occur in 6, 7, 8, 9, 10, 11, 12, 13, and 14 years from now.
END Mode
Enter 9 8 250,000 0
N I% PV PMT FV
Solve for -1,561,722
2) The value today of something worth 1,561,722 in 5 years can be found as
PV0 = PV5 / (1+r)5
PV5 = 1,561,722 r = .08
PV0 = 1,561,722 / (1.08)5 = $1,062,882
Mode is not relevant, since PMT = 0
Enter 5 8 0 1,561,722
N I% PV PMT FV
Solve for -1,062,882
The value of the flavor is $1,062,882