Time Value of Money Formulae
Notation
Let i or r represent rates of return (e.g., interest rate). Let n represent the number of periods. Let t
represent a length of time.
Formulae
To save time and space we use a functional notation for describing the value of money at various times
for given rates of return. FVIF stands for “future value interest factor.” PVIF stands for “present value
interest factor.” An annuity is a series of constant payments with the same frequency (e.g., $1 per week for
30 weeks). An ordinary annuity is an annuity with the first payment beginning in one period. FVIFA stands
for “future value interest factor of an annuity.” PVIFA stands for “present value interest factor of an
annuity.”
FVIF (i , n)
(1 i ) n
PVIF (i , n)
(1 i )
n
1
(1 i ) n
1
FVIF (i , n)
n 1
FVIFA(i , n)
(1 i )
n 1
(1 i )
n 2
 1
(1 i )
j 0
PVIFA(i , n)
1 PVIF (i , n)
i
1 (1 i )
i
1
j
(1 i ) n 1
i
n
FVIF (i , n) 1
i
n
(1 i )
n
(1 i )
( n 1)
 (1 i )
1
(1 i )
j
j 1
1
FVIF (i , n)
i
Perpetuities
An annuity which goes on forever is known as a “perpetuity.” One can compute the value of a perpetuity by
inspection. For example, if I can earn 10% per year (rate held constant over the infinite future), a $10
investment today should yield $1 per year forever. In other words, if one draws all the interest from an
investment, the investment’s value will never change in a static world (no changes in interest rates). Hence,
if PV i Payment , PV Payment / i . Note, this fails for non-positive i. More formally, the present value
of a perpetuity is just the limiting case of the present value of an annuity.
lim PVIFA(i , n)
n
1
i
The Present Value of an Annuity Beginning at the End of m+1 Periods.
Suppose one has an annuity for n periods at rate i where the first payment begins at the end of m+1
periods. To solve this problem, first use the annuity factor to convert a stream of payments into an
equivalent single sum. Under the assumptions, an individual would be indifferent between the single sum
and the stream of payments since given one the individual could replicate the other. Hence, given the
present value of the annuity in the future, one can treat it exactly the same as a single sum. Recall when one
applies the present value of the annuity factor, one obtains the present value one period before the first
payment (by definition since an ordinary annuity begins 1 period in the future). Therefore, if the first
payment begins in m+1 periods, the present value factor gives an equivalent single-sum in m periods in the
future. Using the present value of the single-sum factor for m periods on the equivalent single sum yields
the present value of the annuity which begins in the future.
PV
PVIF(m, i) PVIFA(n, i)
Note, the last payment of this annuity beginning in the future should be m+n periods from the present.
Example: Suppose we have an annuity of $1 per period for 2 years where the first payment is 2 years
from today and the rate is 10%.
time=0
PV?
1
2
$1
3
$1
Doing this term by term we obtain PV=PVIF(10%,2)+PVIF(10%,3)=$1.5777. Using the method
discussed above the result is PV=PVIF(10%,1)PVIFA(10%,2) = (1/1.1)(1.7353) = $1.5777.
Deriving the Present Value of an Annuity Formula
Since we know the present value of a perpetuity and the present value of an annuity commencing in the
future, we can derive the formula for the present value of an annuity.
PVIFA(n, i)
Perpetuity PV of a Perpetuity with first payment in n 1 periods
After substitution of the formulae and a very slight simplification, we obtain the formula for the present
value of an ordinary annuity.
PVIFA(n, i )
1
1
PVIF (n, i )
i
i
1 PVIF (n, i )
i
Deriving the present value of annuity formula does not require advanced mathematics, it just requires
knowledge of very basic time value of money concepts.
Deriving the Future Value of an Annuity Formula
Since we know the amount we need to invest at rate of return i to replicate the ordinary annuity
extending over n periods (i.e., the present value of the annuity), the future value of the this equivalent single
sum invested over n periods at rate i will yield the future value of the annuity.
FVIFA(n, i)
FVIF (n, i) PVIFA(n, i)
Substituting the formula for the present value of the annuity and the future value of a single sum yields the
future value of annuity formula.
1
FVIFA(n, i )
FVIF (n, i )
1 PVIF (n, i )
i
FVIF (n, i ) 1
i
Of course, the future value of the present value of a $1 sum is $1 ( FVIF(n, i) PVIF(n, i) 1). Again, deriving
the future value of annuity formula does not require advanced mathematics, it just requires knowledge of
more basic time value of money concepts.
Different Periods and their Effect on Compounding
There is a convention or custom in finance of quoting rates expressed as an annual rate with the
frequency of associated payment of interest. For example, we might read about 10% per year compounded
quarterly, or 12% per year compounded monthly. Suppose we compounded for two periods each year
(semiannual compounding). We would obtain,
FG i IJ
H 4K
2
(1 i / 2)(1 i / 2) 1 i
This exceeds the payoff from i rate-of-return paid annually (1+i). Similarly, compounding quarterly yields,
FG 6i
H 16
2
(1 i / 4)(1 i / 4)(1 i / 4)(1 i / 4) 1 i
i3
16
i4
64
IJ
K
which also exceeds the payoff from i rate-of-return paid annually (1+i). In both cases, the terms in
parenthesis on the right hand side of the equation represent interest being paid upon interest and this always
is positive. As the number of compounding periods increases per year this increases, but does have a limit.
Continuous Compounding
As the number of possible compounding periods per period (k) increases indefinitely, we reach the
limiting case of continuous compounding.
lim FVIF (r / k , k )
k
lim(1
r
k
k
)k
er
Time Value of Money in Continuous Time
Continuous time has similar factors to the ones we derived for discrete time.
FVIF (r , t )
e rt
PVIF (r , t )
e
FVIFA(r , t )
PVIFA(r , t )
rt
e rt 1
r
1 e rt
r
Time Value of Money Applied to a Sums Growing at a Constant Rate g.
2
Suppose some sum of money is growing at a constant rate g. Hence,
St
S0e gt .
S0 FVIF ( g , t )
Discounting this back at rate r gives,
PV of St
PVIF (r , t ) St
e rt e gt S0
e( r
g)t
S0 .
Essentially, one can replace r in the continuous compounding factors by r-g. The same holds approximately
true in the discrete factors. In particular, the present value of $1 growing at rate g forever and discounted
1
back at rate r is
for r>g.
r g
Derivation of the Infamous Rule-of-72
Let us examine all possible rates and time periods which produce a doubling of money.
FVIF (r , t )
ert
2
Taking logarithms of both sides says that when the rate times the number of periods equals 0.69, money
will double in that time.
rt
ln(2)
0.6931
To make this more useful, multiply each side by 100.
(r %)t
69.31
However, 69.31 is not easily divisible by other round numbers. The number 69 is only divisible by 3 and 23
while 72 is divisible by 2, 3, 4,6, approximately by 5,7,8,9, approximately by 10, and so forth. Hence,
modify the formula to use 72 instead of 69.31.
(r%)t
72
Between the rule-of-72, the time value of money factors in terms of the FVIF(r,t), and perpetuities, one can
usefully approximate many time value of money problems.
Equivalent Rates
We can translate from discrete to continuous rates and vice versa.
(1 rdiscrete ) n
rcontinuous
rdiscrete
e rcontinuousn
ln(1 rdiscrete )
e rcontinuous 1
We can do the same among rates with different compounding periods.
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(1
r1 k1
)
k1
(1
r1
k1 (1
r2 k2 k1
)
1
k2
FG
H
r2 k2
)
k2
IJ
K
These are specific examples of internal rate of return computations.
NPV
We define NPV as,
NPV=PV of the Benefits from some investment – PV of the associated costs
where the present value computations are done at the rate of return one could obtain on comparable,
alternative investments.
IRR
We define IRR as the rate which if obtained on alternative investments would make the NPV=0. In
other words, at the IRR the present value of the benefits and the present value of the costs are equal.
Average Return
Arithmetic average return is,
rarithemetic
br
1
g
r2  rn / n
while the geometric return is,
rgeometric
b(1
r1 )(1 r2 )(1 rn )
g
1
n
FG 1 IJ bln(1 r )
eH n K
1
ln(1 r2 )  ln(1 rn )
g
and the latter is a specific type of IRR computations.
1. Given the following, what were the real and nominal geometric average rates of return over 1929-32 on
the S&P 500 and T-Bills? What were the arithmetic returns?
Year
1929.0
1930.0
1931.0
1932.0
Inflation T-Bill Long Gov. Long Corp. S&P 500
0.2
4.7
3.4
3.3
-8.4
-6.0
2.4
4.7
8.0
-24.9
-9.5
1.1
-5.3
-1.9
-43.3
-10.3
1.0
16.8
10.8
-8.2
The price level fell from an arbitrary norm of $1 at the beginning of 1929 to the equivalent of $0.7616 at
the end of 1932.
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$1 FVIF (0.2%,1) FVIF ( 6%,1) FVIF ( 9.5%,1) FVIF ( 10.3%,1)
$1 (1 0.002)(1 0.06)(1 0.095)(1 0103
. ) (0.998)(0.94)(0.905)(.897) $0.7646
On an annualized basis, the fall in the price level was –6.49%.
$1=PVIF(IRR,4)$0.7646
IRR=-6.49%
Similarly, the $1 invested in the S&P 500 at the beginning of 1929 would produce $0.358 at the end of
1932.
(1 0.084)(1 0.249)(1 0.433)(1 0.082)
(0.916)(0.751)(0.567)(0.918)
0.358
This equates to a annualized rate of return (IRR) of –22.64% per year.
$1=PVIF(4,IRR)$0.358
IRR=-22.64% per year
In real (after inflation) terms, on an annualized basis $1 invested at the beginning of each year would yield
FVIF(-22.64,1)/FVIF(-6.58%,1) in constant purchasing power terms. Hence, the annualized real return was
0.7736/0.9342=0.828, or a loss of 17.2% per year. Over the entire period investors obtained 0.358/0.762 or
0.47 on each dollar invested. Hence, in real terms investors lost about 53% of their money if the invested in
the entire S&P 500 during the Great Depression.
Loan Conventions
When a lender writes into a contract a rate of % payable times per year, they really mean
they are charging a rate of % per period. For example, 12% per year with 12 monthly payments means
the contract rate of interest is 1% per month. Due to compounding, lenders receive a higher IRR than this
(given all payments are made on schedule).
Fundamental Loan Principles
All loans based upon the time value of money are priced so that the amount owed equals the
present value of the remaining payments discounted at the contract rate of interest each period for
the number of remaining periods on the loan.
The payments on fixed rate loans are a function only of the contract rate of interest per period, the
pattern of the payments, and the number of payments on the loan. Indexed loans (variable rates are just the
same as a series of fixed rate loans).
The following holds true for loans based on the time value of money,
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Interest t ct Balancet 1
Balance Reduction t Payment t
Balancet
Balancet
1
Interest t
Balance Reduction t
where ct refers to the loan contract rate at time t.
Naturally, interest over some period equals the sum of the payments less the change in the balances
over the period. Hence,
p
Interest k
Paymentst
p
( Balance p
Balancek )
t k
Example: Suppose you have a loan for $100,000 at 12% per year with 30 year term and monthly payments.
How much will you pay in interest over the life of the loan? Total payments equal
(360)(1028.61)=370,299.6 and hence the total interest equals 270,299.6 (370,299.6-100,000).
Loans with Payments or Balances Not Computed Using Time Value of Money Calculations
The following loans use idiosyncratic ways of computing balances or interest. These may be used in
isolation or as part of some more standard loan. For example, many real estate loans are really combinations
of fully amortized loans and discount loans.
Rule of 78 loans
Rule of 78 loans have the payments computed by some method and the balance is computed as,
Interest Rebate t
LoanPayofft
F
GG
GG
H
t
n
j
0
n
j
1
I
JJ (total interest)
JJ
K
Loan Balance t
(cumulative interest paid to t )
Interest Rebate t
The rebate is to the lender not the borrower. The term 78 refers to the sum of the month’s digits in one year
(1+2+…+12=12(12+1)/2=78). Illegal in some states. The rule-of-78 effectively exacts a heavy prepayment
penalty, with the penalty having the maximum dollar value approximately half-way through the loan.
Discount Loans
Discount loans are ancient. Essentially, the borrower receives less than the principal at the time of the
origination, but must repay the principal at maturity. Common examples of pure discount loans include Tbills. Historically, the discount loan has been used to escape prohibitions against interest.
Prepaid Interest
Loans with prepaid interest act much like discount loans. The interest is computed over the some period
in the loan and it must be paid at origination. Alternatively, the interest over the course of a year is prepaid
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at the beginning of the year. Obviously, this raises the lender’s true rate of return relative to a loan without
prepaid interest.
Prepayment Penalties
Prepayment penalties increase the amount needed to payoff the loan in the years prior to maturity.
Prepayment penalties can vary over the life of the loan.
Loans in General
As discussed earlier, the balance at time t equals the present value of the future payments using the
contract rate per period (i.e., Balancet PV ( payments, c) ).
Example: Suppose you have a loan for $1,000 with two annual payments where the 2nd payment is 10
times the first payment. The contract rate per year is 10%. Hence,
1000=PVIF(c,1)payment+PVIF(c,2)10payment. For c=10%, the payment equals
1000/(PVIF(1,10%)+PVIF(2,10%)10)=$109.09.
Special Types of Loans
A huge number of specific types of loans have been used at various times to solve some economic
problem.
Non-Indexed Loans
Non-indexed loans have fixed rates, but may have payments which change according to some
predetermined schedule.
Fully Amortized Loans
Fully amortized loans have constant payments each period. Hence, Balancet
PVIFA(n t , c) Payment .
Example: Suppose you borrow $100,000 for 360 months at 1% per month. Your payment equals
100000/PVIFA(360,1%)=1028.61.
Partially Amortized Loans
Partially amortized loans have constant payments, but also have some ending balance (a.k.a., balloon
payment, bullet payment). Hence, Balancet PVIFA(n t , c) Payment PVIF (n t , c) Balancen .
Example: Suppose you borrow $100,000 for 360 months at 1% per month. You pay $500 per month.
Your terminal balance equals (100000-PVIFA(360,1%)500)/PVIF(360,1%)= 3,591,469.17.
Interest Only Loans
Interest only loans are partially amortized loans where the ending balance equals the beginning balance.
Hence, no principal reduction happens over the life of the loan and the payments are all interest.
Example: Suppose you borrow $100,000 at 1% per month for 30 years. You pay $1,000 each month for
360 months and $100,000 at the end of 360 months.
Constant Amortization Loans
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Constant amortization loans have a constant amount of principal reduction each period. Hence,
Paymentt cBalancet 1 Balance0 / n.
Example: Suppose you borrow $10,000 for 2 years at 10% per year. The first payment is $6,000 and the
second payment is $5,500.
Constant Rate Increasing Payment Loans
A loan with the payments growing at a rate g and discounted at rate r acts like a loan with constant
payments discounted at rate r-g.
Example: Suppose you borrow $100,000 for 360 months at 1% per month. However, the payments
grow at 0.5% per month. Hence, the initial payment is 100000/PVIF(0.5%,360)=$599.55. The final
payment is 599.55FVIF(359,0.5%)= 3,592.87.
“Bought-Down” Loans
A buydown refers to taking some loan and reducing the payments over some period during the loan
through initial payments.
Example: Suppose the market interest rates are at 6% and you wish to “buydown” the loan in the first
year to 5%. You have a loan for $100K at 6%. How much do you need to pay for the buydown? The
buydown equals the present value of the difference in payments. In this case, the difference between a
payment of $599.55 versus $536.82 or $62.73. The present value of $62.73 for 12 months at 0.5% per
month equals PVIFA(12,0.5%)62.73=$728.86.
Wraparound Loans
A wraparound loan is where the borrower pays the lender with the second mortgage a single payment
which combines the payments for the first and second mortgages.
Example: Suppose you have a wraparound for $110,000 with payments of $1228.61 per month for 360
months. What is the IRR/mo on this loan? 110000=PVIFA(360,IRR/mo)1228.61 implies
IRR/mo=1.09475%. If you could obtain a loan for $100K with payments of $1028.61 as a first
(IRR/mo=1%), what rate are you paying on the implicit second mortgage?
(10000=PVIFA(IRR/mo,360)200, IRR/mo=1.998%).
Indexed Loans
Indexing is usually tied to the interest rate, price level, or revenues. In non-real estate lending, bonds
have even been issued with the payments tied to oil prices or to lift ticket prices at ski-resorts. In real estate
lending, loans have been indexed to interest rates, the price level, or de facto to real estate asset prices. Key
variables for indexed loans include (1) the index; (2) the frequency of adjustment; (3) the range of
adjustment per period; and (4) the maximum lifetime adjustment. Indexed loans can be thought of as a
sequence of shorter range loans, each refinancing the outstanding balance based upon the new value of the
index.
Variable Payment Loans
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These loans adjust the payments while holding the maturity constant.
Example: Suppose one has an adjustable rate mortgage (ARM) with beginning balance of $100K at
12% for 30 years. It adjusts every 12 months. In the first year the payments are $1028.61. At the end of 12
month the balance is $99,636.87. Suppose the new rate is 8%. One should think of this as refinancing the
old loan of $99,636.87 at the new rate of 8% over the remaining term of 348 months (360-12). Hence, the
new payment is 99636.87/PVIFA(348,0.666%)=$ 737.26.
Variable Balance
These loans hold the payment constant and adjust the balance as the rate changes.
Example: Suppose one has an adjustable rate mortgage with beginning balance of $100K at 12% for 30
years. The payments will be $1028.61 per month over the term of the loan. It adjusts every 12 months. At
the end of 12 month the balance is $99,636.87. Suppose the rate changes to 8%. One should think of this as
refinancing the old loan of $99,636.87 with a partially amortized loan with payments of $1028.61 per
month at the new rate of 8% over the remaining term of 348 months (360-12). After 24 months,
99636.87=PVIFA(12,0.666%)1028.61+PVIF(12,0.666%) MB24 . Hence, MB24 = $95,100.56.
Variable Maturity Loans
These loans adjust the term while holding the maturity constant as rates change.
Example: Suppose one has a loan at 12% for 360 months. Your payments are $1028.61. At the end of
one year you owe $99,636.87. Suppose rates go to 12.388%. What is your new term?
1028.61PVIFA(?,12.388%/12)=99636.87. The new term is 1031 months or 85.92 years. What would
happen if rates had climbed to 12.4%?
Equity Participation Loans
These loans have the payments also tied to some variable such as sales.
Reverse Annuity Mortgage
These loans allow one to refinance the home to purchase an annuity.
Shared Appreciation Loans
These loans split the appreciation between two borrowers, an investor and the party who lives in the
home. The financial institution can also be the borrower/investor.
Construction Loans
These loans are typically adjustable rate, with disbursements in the near future and repayment in the
near term.
Hybrid of Discount Loans and Other Types of Loans
One can combine discount loans with other types of loans. The discount reduces the amount actually
disbursed to the borrower, but the borrower makes payments and payoffs based upon the full principal of
the loan. Early payoffs can greatly affect the cost of the loan. As a definition, one point equals one percent
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of the principal. Also, APR denotes annual percentage rate which equals the IRR/period multiplied by the
number of periods per year when computed over the full maturity of the loan.
Example: Suppose one obtained a loan with a principal of $100K at 12% for 30 years with 5 points.
Hence, the payments are 100K/PVIFA(1%,360)=1028.61, but the amount disbursed is $100K(15/100)=$95K. The IRR/mo on this loan over the full maturity is determined by the following equation:
95K=PVIFA(360,IRR/mo)1028.61. Hence, the IRR/mo equals 1.05828%. The IRR/year is
(FVIF(1.05828,12)-1)100=13.465%.
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