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Time value of money chapte
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Questions
• Why is time value a big deal in M&B?
• What’s the PV and FV of an asset?
• What’s the PV and FV of a simple asset (one-shot cash flowin the future)?
• What’s the PV of a perpetuity?
• What’s the PV and FV of an annuity (ordinary- and -due) andthose of a mixed-stream asset?
1
• What’s frequent compounding and how does that affect assetvaluation?
• How much do we need to deposit periodically in order toaccumulate a fixed sum of money in the future?
• How are loans amortized – i.e., how do we determine equalperiodic payments to repay a loan principal plus a stipulatedinterest?
• How are interest or growth rates found? How do we find thenumber of periods it takes for an initial deposit to grow to acertain future amount, given the interest rate?
Time value of money
Assets generate benefits over their lifetime. Benefits (measuredas cash flows) happen at different points in time.
USD 1 on 10-5-2006 6= USD 1 on 10-5-2011
Time line diagrams.
Financial tables/calculators/spreadsheets. We’ll use Excel.
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One-shot cash flows
FV: Value of $100 today on 10/5/2006.
Principal: Amount on which interest is paid. Compound interest:Interest earned on previous interest that has increased the previousprincipal.
Say the banks pay an annual interest rate of 5%, which is thenadded to the principal.
Look at the whiteboard →
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FV and compounding
FVn = PV× (1 + i)n
If PV = $100, n = 5, and i = 5% = 0.05, then
FV10 = $100× (1 + 0.05)5 =?
(1 + i) is called the gross interest rate. (1 + i)n is called the futurevalue interest factor. What’s the meaning of the FVIF in plainterms?
In general, compound growth of any variable means that the valueof the variable increases each period by the factor (1 + g). Whenmoney is invested at the compound interest rate i, the growth rateis i.
Example in Excel.
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PV and discounting
PV =FVn
(1 + i)n
PV = FVn ×[ 1
(1 + i)n
]If FV10 = $162.89 and i = 5% = 0.05, then
PV = $162.89×[ 1
(1 + 0.05)10
]=?
Here, the interest rate i is called the discount rate and 1(1+i)n is
called the discount factor. What’s the meaning of the DF in plainterms?
Show example in Excel.
5
Finding i
Suppose a company needs to borrow. It issues bonds for $129each promising to pay its holder $1,000 at the end of 25 years.No coupons. One single payment at the end of 25 years. What(fixed) annual interest rate is the bond paying?
We know the PV ($129), the FV ($1,000). We don’t know theinterest rate i. Let’s figure it out:
PV = FVn ×[ 1
(1 + i)n
](1 + i)n =
FVn
PV
1 + i =(FVn
PV
)1/n
i =(FVn
PV
)1/n− 1
(1)
Excel to solve the problem.
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Frequent compounding
So far, we have implicitly assumed that the interest rate com-pounds annually (once a year). What if it compounds more often?
Suppose you have $100 and earn an annual interest rate of 6%to be compounded monthly. Each month the bank pays you 1/12of the annual rate, 0.06/12 = 0.005 or a half percent (50bp).∗
Since the interest is compounded monthly, your $100 earn [1 +(i/12)]12 − 1 = (1.005)12 − 1 = 1.0617 − 1 = 0.617. Your trueannual interest rate is not 6% but 6.17%! [Show in Excel or withcalculator.]
We need to modify the FV formula when dealing with more fre-quent compounding. Let m be the number of periodic paymentsper year (e.g. 2 semi-annually, 4 quarterly, 12 monthly,etc.).Then:
FVn = PV[1 +
i
m
]mn(2)
∗1%=0.1=100bp.
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WARNING!
Do NOT ever compare, add up, or subtract cash flows
that occur at different times without previously dis-
counting them (or compounding them) to a common
date. They are apples and oranges!
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FV and PV of multiple cash flows
If an asset generates multiple cash flows, how do we find its FV(or PV)? One way is to use computational brute force, i.e. wecalculate the FV (or PV) of each cash flow with the formulas weknow and then add up all those FVs (or PVs).
Example with Excel: You save $300 each year for 3 years to buya computer. You earn 8% annually on your savings balance. Howmuch money will you have in 3 years?
It’s better if we find a way to simplify the computation of assetthat generate multiple cash flows.
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PV of a perpetuity
An asset that promises to pay a fixed annual payment foreverstarting at the end of the present year (or beginning of the second).The principal is not repaid, you just receive those annual paymentsforever. Say the annual payment is C. What is the annual interestrate on this perpetuity? Assume the market decides that the rightvalue of this security is PV Ai,∞ or PV P .∗ Then obviously: i =
CPV Ai,∞
.
Usually, we know i of assets of similar risk and C. We can thenfind the present value of the perpetuity as: PV = C
i. This is the
PV of the perpetuity at zero or, to be more precise, PV P0.
∗How does the market do it?
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PV of a perpetuity
What is its PV at n some year in the future (PV Pn)? This perpe-tuity would begin generating a cash flow C starting in year n + 1.The value of the perpetuity at n will be the same as today’s!PV Pn = C
i.
Note that PV Pn is a FV (PV at n), not really a PV! An actualpresent value is PV P0. So, to find the PV of that perpetuitythat generates an eternal cash flow beginning in year n + 1, weneed to discount PV Pn to the present to find its PV P0. That is,
PV P0 = PV Pn
(1+i)n =(Ci
)(1
(1+i)n
).
See how this lego toy works?
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PV of a perpetuity
In the textbook, the formula appears as:
PV Ai,∞ =C
i(3)
We will prefer this notation:
PV P =C
i(4)
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PV of a perpetuity
Example: Find the PV P at the beginning of 2006 of a perpetuitythat yields $100 annually starting at the end of 2006. The marketinterest rate of equally safe investments is 5%. Do it in Excel:PV P0 = ?
Example: Find the PV P at the beginning of 2006 of a perpetuitythat yields $100 annually starting at the end of 2010. The marketinterest rate of equally safe investments is 5%. Do it in Excel:PV P4 = ?
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Annuities (FV)
What will be the balance of a savings account in 10 years if onedeposits $1,000 at the end of each year and i is 5%? What willthe balance be if the deposits are made at the beginning of eachyear? Do example in Excel.
These are annuities. The payments are the same each year. Theformer (payments at the end of each year) is known as an ordinaryannuity. The latter (payments at the beginning of each year) isknown as an annuity due.
An annuity is just a finite number of periodic constant flows. Ap-plying the FV formula we know, we construct a formula for theFV of annuities.
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FV of an ordinary annuity
Let C be the annual deposit (or payment), i the interest rate, andn the term of the annuity. The FVA will be the sum of the FV’sof each annual deposit (or payment).
FV An = C(1 + i)n−1 + C(1 + i)n−2 +. . . + C(1 + i)2 + C(1 + i) + C (5)
FV An = C
n−1∑t=0
(1 + i)t (6)
The interest factor is FV IFAi,n =∑n−1
t=0(1 + i)t.
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FV of an annuity due
What will be the balance of a savings account in 10 years if onedeposits $1,000 at the beginning of each year? Do example inExcel.
FV ADn = C(1 + i)n + C(1 + i)n−1 +
. . . + C(1 + i)2 + C(1 + i) (7)
FV ADn = C
n∑t=1
(1 + i)t (8)
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PV of an ordinary annuity
To find the PV of an ordinary annuity that produces a cash flowC from year 1 to year n we discount the FV to the present.
What is the PV of our retirement account earning 5% in annualinterest to be withdrawn in 20 years if we deposit $1,000 at theend of each year (ordinary annuity)? What if the deposits aremade at the beginning of each year (annuity due)?
Solve in Excel taking the PV of each cash flow separately and addthem all up.
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PV of an ordinary annuity
To find the PV of an ordinary annuity that produces a cash flowC from year 1 to year n we discount the FV to the present:
PV An = C[ 1
(1 + i)n
n∑t=1
(1 + i)t]
= C
n∑t=1
1
(1 + i)t(9)
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PV of an ordinary annuity
There’s an easier way: The PV of an ordinary annuity can beviewed as the PV of a perpetuity that begins to produce cashflows in year 1 minus the PV of a perpetuity that produces itsfirst cash flow in year (n + 1). Hence:
PV An = C[1
i−
1
i(1 + i)n
](10)
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PV of an annuity due
Try to derive the formula yourself:
PV ADn = C
[1 +
1
i−
1
i(1 + i)n−1
](11)
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Miscellaneous items
How do we figure out the FV and PV of mixed-stream assets(different cash flows each period)? Mix and match the previousformulas as may be required. If no fancy formula can be applies,use the basic formulas for FV and PV of single cash flows. Thatalways works!
Compounding and discounting in continuous time: FV = PV ein
and PV = FV e−in
Solve examples in pp. 151, 153, 155, 157, 159-163.
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And?
What we’ve learned about FV and PV, compounding and dis-counting, applies to all types of investments. Not only to savingsor bonds with a fixed interest rate (fixed return), but to variable-return investment as well (e.g. stocks, variable interest-rate in-struments, etc.)
With variable return instruments, the return is uncertain. But eventhe most certain (‘fixed’) return is in fact uncertain. The futureis essentially unknown and no human institution is eternal.
When we compare FV’s and PV’s of different investments, we canonly make a meaningful comparison when the degree of risk in-volved in the investments we compare are is similar. If the degreesof risk are different, then we are comparing apples and oranges.
We need to find an analytical way to translate risk into return.Then we will be able to convert uncertain cash flows into ‘certain’cash flows (i.e. adjusted for risk) and thus use what we’ve learnedto compare these different instruments on an apples-to-apples ba-sis.
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What did we learn?
• Why is time value a big deal in M&B?
• What’s the PV and FV of an asset?
• What’s the PV and FV of a simple asset (one-shot cash flowin the future)?
• What’s the PV of a perpetuity?
• What’s the PV and FV of an annuity (ordinary- and -due) andthose of a mixed-stream asset?
23
• What’s frequent compounding and how does that affect assetvaluation?
• How much do we need to deposit periodically in order toaccumulate a fixed sum of money in the future?
• How are loans amortized – i.e., how do we determine equalperiodic payments to repay a loan principal plus a stipulatedinterest?
• How are interest or growth rates found? How do we find thenumber of periods it takes for an initial deposit to grow to acertain future amount, given the interest rate?
