Interest Rate, Annuity
Goals:
• Study continuous compounding and force of interest • Discuss real interest rate
• Learn annuity-immediate, and its present value
• Study annuity-due, and compare with annuity-immediate • Learn continuous annuity and perpetuity.
Suggested Textbook Readings: Chapter 1: §1.6 - §1.7; Chapter 2: §2.1-2.2 Practice Problems:
All exercises in §1.6-§1.7 and $ 2.1, without an asterisk, and Section 2.2: 1-5, 7-10, 12-17
Continuous Compounding
At nominal rate i(m), the accumulated value of an initial investment of a dollar after a
year is a(m) = 1 + i (m) m m
With fixed i(m), the more often compounding taking place during the year, the larger
the accumulated value is.
Example 1: Suppose that the interest is compounded continuously. Find the accumu-lated value a(m) when m → ∞.
Force of Interest
If the interest is compounded continuously, the accumulated amount function A(t) is a continuous function of t. The nominal rate is called the force of interest and denoted by δt (sometimes i(∞)). The notation i(∞) makes sense, since we can think of the force
of interest as the limit as the number of times we credit the compound interest goes to infinity. that is, δt = i(∞)= lim
m→∞i (m).
Force of Interest
For an investment that grows according to accumulated amount function A(t), the force of interest at time t, is defined to be
δt=
A0(t) A(t)
Example 2: (Example 1.13, page 40) Derive an expression for δt if accumulation
is based on
1. simple interest at annual rate i, and 2. compound interest at annual rate i.
Force of Interest and Accumulation function
We can recover the accumulation function from the force of interest δt= A0(t)
A(t).
We may also calculate the interest earned from time t = 0 to time t = n from the force of interest δt.
Inflation, Real Rate of Interest
Inflation is the growth in prices from year to year. It is generally measured via the rate of change in the price of a ”basket of goods”. The rate of inflation is denoted by r. Real rate of interest With annual interest rate i and annual inflation rate r, the real rate if interest for the year is
ireal =
value of amount of real return (yr-end dollars) value of invested amount (yr-end dollars) =
i − r 1 + r Note that when r is small, i − r is a good estimate for the real rate of interest.
Annuities
An annuity is a finite sequence of payments made at fixed periods of time over a given interval. The fixed periods of time that we consider will always be of equal length. Example 3: Smith wants to save $100 each month for a vacation in the summer of 2013. At the end of each month he makes deposits into an account earning a nominal rate of i(12) = 9% starting from May 31, 2012. How much will Smith have saved after
his last deposit on April 30, 2013?
Geometric Sequence 1 + x + x2 + · · · + xn−1= x n− 1 x − 1 = 1 − xn 1 − x Annuity-Immediate
The series of payments in the example is referred to as an accumulated annuity-immediate. The notation sn|i is used to express the accumulated value at the time of (and including)
the final payment of a series of payment of 1 each made at equally spaced intervals of time, where the rate of interest per payment period is i.
sn|i = (1 + i)n−1+ (1 + i)n−2+ · · · + (1 + i) + 1 =
(1 + i)n− 1
i
The notation sn|i can be used to express the accumulated value of an annuity if the
following conditions are satisfied:
1. the payments are of equal amount;
2. the payments are made at equal intervals of time, with the same frequency as the interest rate is compounded;
Example 4: (Example 3 continued) If i(12) = 9%, how much will Smith have to
save each month to reach $2000 on April 30, 2013?
Present value of an annuity
Example 5: Suppose Smith has decided to borrow money to go on vacation now. If he has to pay back bank $110 each month starting one month from now for 12 months to pay back the loan. How much money did Smith borrow? Suppose i(12)= 9%.
Present value of an n-payment annuity-immediate of 1 per period
The symbol an|i is used to denote the present value of a series of equally spaced payments
of amount 1 each when the valuation point is one payment period before the payments begin.
an|i = v + v2+ · · · + vn =
1 − vn i
In order to use the above notation and formula, the following conditions have to be satisfied.
1. the n payments are of equal amount;
2. the payments are made at equal intervals of time, with the same frequency as the interest rate is compounded;
More Annuities
Example 6: Smith wants to take a vacation in May, 2013. Starting from May 31, 2011, he deposits $100 each month from his monthly paycheck into an account earning nominal rate i(12) = 9%. He was laid off on August 31, 2012 when he made his final deposit. If
the money he has deposited continues to accumulate until April 30, 2013, what is the balance in the account at that time?
Example 7: Smith bought a large TV on January 1, 2012. He worked out the financing so that he does not have to pay for the first 12 months, and then pays $50 at the end of each month for 36 months (his first deposit is on January 31, 2013). If i(12)= 9%, what
is the price of the TV?
Such an annuity is called a deferred annuity. The present value of a k-period deferred, n-payments annuity of 1 per period is
Annuity-Due
Another form of annuity is that of annuity-due. Total of n equal payments occur at time 0, 1, 2, · · · , n − 1. In the case of present value, the annuity-due refers to the valuation at the time of the first payment, and in the case of future value (accumulated value), the annuity-due refers to the valuation one period after the last payment.
Annuity-Due For n-payment annuities with payment of amount 1 each, the annuity-due present value is at the time of the first payment,
¨ an|i = 1 + v + v2 + · · · + vn−1= 1 − vn 1 − v = 1 − vn d and the accumulated value is one period after the final payment ,
¨
sn|i = (1 + i) + (1 + i)2+ · · · + (1 + i)n=
(1 + i)n− 1
d where d is the discount rate corresponding to i.
Annuity-Immediate v.s. Annuity-Due Present value Accumulate Value Annuity-Immediate an|i = 1 − vn i sn|i = (1 + i)n− 1 i Annuity-Due ¨an|i = 1 − vn d s¨n|i = (1 + i)n− 1 d
Differing interest and payment periods
It may happen that the quoted interest rate has a compounding period that doesn’t coincide with the annuity payment period. For the purpose of evaluation we can find the interest rate per payment period that is equivalent to the quoted interest rate, or find the equivalent payment in each quoted interest period.
Example 8: (Example 2.12 (a)) On the last day of March, June, September and De-cember, Smith makes a deposit of $1000 into a saving account that earns nominal rate i(12) = 9%. The first deposit is on Mar 31, 1995 and the last is December 31, 2010. What is the balance on January 1, 2011?
m-thly payable annuities
Example 9: (Example 2.12 (b)) In the above example, if the interest rate is quoted at an effective annual rate of 10%, what is the balance in Smith’s account on January 1, 2011?
mthly payable annuity-immediate
If the effective annual interest rate is i, and m payments of X are made each year, then the accumulated value over n years is
Ks(m)n|i = K(1 + i)
n− 1
i(m) = Ksn|i
i i(m)
where K = mX. The present value of this series of payments is Ka(m)n|i = Kan|i
i i(m)
Perpetuities
If an annuity has no end point, it is called a perpetuity. We cannot find the future value of a perpetuity, but we can always calculate the present value.
Annuity-immediate: a∞|i = lim n→∞an|i =
1 i Similarly, for annuity-due: ¨a∞|i =
1 d
and for mthly payable annuity-immediate: a(m)∞|i = 1 i(m)
Continuous Annuity
If payments are made more frequently, it is more convenient to approximate the calcu-lation by assuming the payments are made continuously.
The accumulated value of the continuous annuity, paid at 1 per period for n periods, denoted by sn|i, is sn|i = i δ · sn|i Similarly an|i = i
δ · an|i. If accumulation is based on force of interest δr, then an|δr = Z n 0 e−R0tδrdrdt and sn|δr = Z n eRtnδrdrdt. Also s n|δr = an|δr · e Rt 0δrdr.
