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Concept 9: Present Value
Is the value of a dollar received today the same as received a year from today? A dollar today is worth more than a dollar tomorrow because of inflation, opportunity cost, and risk Bringing the future value of money back to the present is called finding the Present Value (PV) of a future dollar
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Discount Rate
To find the present value of future dollars, one way is to see what amount of money, if invested today until the future date, will yield that sum of future money The interest rate used to find the present value = discount rate There are individual differences in discount rates Present orientation=high rate of time preference= high discount rate Future orientation = low rate of time preference = low discount rate Notation: r=discount rate The issue of compounding also applies to Present Value computations.
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Present Value Factor
n
r
PVF
To bring one dollar in the future back to present, one
uses the Present Value Factor (PVF):
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Present Value (PV) of Lump Sum
Money
n
r
PV P PVF P
= × = ×
For lump sum payments, Present Value (PV)
is the amount of money (denoted as P) times
PVF Factor (PVF)
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An Example Using Annual
Compounding
PV = P × PVF = ×
Suppose you are promised a payment of $100,
after 10 years from a legal settlement. If your
discount rate is 6%, what is the present value of
this settlement?
An Example Using Monthly
Compounding
100 , 000 * 0. 302995 $ 30 , 299. 50 ( 11 %)
1 100 , 000 120 = =
PV = P × PVF = ×
You are promised to be paid $100,000 in 10 years. If you
have a discount rate of 12%, using monthly
compounding, what is the present value of this
First compute monthly discount rate
Monthly r = 12%/12=1%, n=120 months
An Example Comparing Two
Options
Suppose you have won lottery. You are faced with two
options in terms of receiving the money you have won:
(1) $10,000 paid now; (2) $15,000 paid five years later.
Which one would you take? Use annual compounding
and a discount rate of 10% first and an discount rate of
5% next.
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Your answer will depend on your discount rate:
Discount rate r=10% annually, annual compounding Option (1): PV=10,000 (note there is no need to convert this number as it is already a present value you receive right now). Option (2): PV = 15,000 (1/ (1+10%)^5) = $9,313. Option (1) is better Discount rate r= 5% annually, annual compounding Option (1): PV=10, Option (2): PV = 15,000(1/ (1+5%)^5) = $11,752. Option (2) is better
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Present Value (PV) of Periodical Payments
For the lottery example, what if the options are (1)
$10,000 now; (2) $2,500 every year for 5 years, starting
from a year from now; (3) $2,380 every year for 5 years,
starting from now?
The answer to this question is quite a bit more complicated because it involves multiple payments for two of the three options. First, let’s again assume annual compounding with a 10% discount rate.
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Annual discount rate r= 10%, annual compounding Option (1): PV=10, Option (2): PV of money paid in 1 year = 2500[1/(1+10%)^1 ] = 2272. PV of money paid in 2 years = 2500[1/(1+10%)^2 ] = 2066. PV of money paid in 3 years = 2500[1/(1+10%)^3 ] = 1878. PV of money paid in 4 years = 2500[1/(1+10%)^4 ] = 1707. PV of money paid in 5 years = 2500[1/(1+10%)^5 ] = 1552. Total PV = Sum of the above 5 PVs = 9,476. Option (3): PV of money paid now (year 0) = 2380 (no discounting needed) PV of money paid in 1 year = 2380[1/(1+10%) 1 ] = 2163. PV of money paid in 2 years = 2380[1/(1+10%)^2 ] = 1966. PV of money paid in 3 years = 2380[1/(1+10%)^3 ] = 1788. PV of money paid in 4 years = 2380*[1/(1+10%)^4 ] = 1625. Total PV = Sum of the above 5 PVs = 9,924. Option (1) is the best, option (3) is the second, and option (2) is the worst.
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Are there simpler ways to compute present
value for periodical payments?
Just as in Future Value computations, if the periodic payments are equal value payments, then Present Value Factor Sum (PVFS) can be used.
Present Value (PV) is the periodical payment
times Present Value Factor Sum (PVFS). In the
formula below Pp denotes the periodical
payment:
PV=Pp*PVFS
Present Value Factor Sum (PVFS)
If the first payment is paid right now (so the first
payment does not need to be discounted), it is
called the Beginning of the month (BOM):
r
r
r r r
PVFS
n
n
1
0 1 1
−
−
30% down situation
(^780042). 580318183. 18
- 5 %
( 10. 5 %) 1 1
7800
( 0. 5 %, 48 , )
7800
48
= =
−
=
= =
PVFSr n EOM M
Option 2. The amount borrowed: 12,000*(1-30%)=8, Monthly r=3%/12=0.25%, n=48 months
(^840045). 178695185. 93
- 25 %
( 10. 25 %)
(^840011)
( 0. 25 %, 48 , )
8400
48
= =
−
=
= =
PVFSr n EOM M
Option 1. Amount borrowed is 12,000(1-30%) – 600 =7,* Monthly r=6%/12=0.5%, n=48 months
Option 1 is better because it has a lower monthly payment
19
5% down situation
10 , (^80042). 580318253. 64
- 5 %
( 10. 5 %) 1 1
10 , 800
( 0. 5 %, 48 , )
10 , 800
48
= =
−+
=
= =
PVFSr n EOM M
Option 2. The amount borrowed: 12,000*(1-5%)=11, Monthly r=3%/12=0.25%, n=48 months
11 , (^40045). 178695252. 33
- 25 %
( 10. 25 %) 1 1
11 , 400
( 0. 25 %, 48 , )
11 , 400
48
= =
−
=
= =
PVFSr n EOM M
Option 1. Amount borrowed is 12,000(1-5%) – 600 =10,* Monthly r=6%/12=0.5%, n=48 months
Option 2 is better now because it has a lower monthly payment
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Application of Present Value:
Annuity
Annuity is defined as equal periodic payments which a
sum of money will produce for a specific number of
years, when invested at a given interest rate.
Example: You have built up a nest egg of $100,
which you plan to spend over 10 years. How much can
you spend each year assuming you buy an annuity at
7% annual interest rate, compounded annually?
21
10
= × = =
PVFSr n EOM
M
M PVFSr n EOM
Annuity calculation is an application PVFS
because the present value of all future annuity
payments should equal to the nestegg one has
built up.
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If you know how much money you want to have
every year, given the interest rate and the initial
amount of money, you can compute how long the
annuity will last. Say you have $10,000 now, you
want to get $2,000 a year. The annual interest rate is
7% with annual compounding (EOM)
Approximate solution: Step 1: $10,000/$2,000 = 5 Step 2: Find a PVFS that is the closest possible to 5 PVFS(r=7%, n=5, EOM) = 4. PVFS(r=7%, n=6, EOM) = 4.76654 close to 5 PVFS(r=7%, n=7, EOM) = 5.389289 close to 5 Because 5 is in-between PVFS(n=6) and PVFS(n=7), this annuity is going to last between 6 and 7 years Exact solution: $10,000/$2,000= 5=PVFS (r=7%, n=?, EOM) => 5=[1- 1/(1+7%)^n]/7% 0.35=1-1/(1.07)^n 0.65=1/(1.07)^n 1/0.65=(1.07)^n Log(1/0.65)=n log(1.07) n=log(1/0.65)/log(1.07)=6.37 years Note: Homework, Quiz and Exam questions will ask for approximate solution, not the exact solution, although for those who understand the exact solution the computation can be easier.
Appendix: A Step-by-Step Example for PVFS
Computation
(^555)
PVFSn = r = EOM =
