1. Chapter The Growth of Money
ACTSC231 Mathematics of Finance
Department of Statistics and Actuarial Science University of Waterloo Fall 2010
Instructor: Chengguo Weng
C. Weng ([e-mail])
p. 1/3
Interest
(p10) Monday has time value investment opportunities theory. time preference theory.
(p10) Interest is the payment by a borrower to a lender (investor) in return for the use of the capital. Interest may be paid annually, monthly, weekly, or daily...... = continuously. The original sum invested (or borrowed) is the capital or principal. The total repaid is the principal plus interest.
C. Weng ([e-mail])
p. 2/3
John lends Tom $10,000 for 5 years. Tom offers the following three options of repayments:
Example 1.1.
Opt1: Pay $10,500 at the end of 5 years. Opt2: Pay $100 at the end of each year, and repay the principal after 5 years. Opt3: Make payment of principal and interest of $2,100 at the end of each year. Which option is most favorable for John and which one is best for Tom?
C. Weng ([e-mail])
p. 3/3
Accumulation Function
(p11) Consider a single investment of $1 at time 0 without any withdrawal up to time t > 0 At time t, the accumulated value (or accumulation) of the investment is the total of the principal plus interest added.
Notation: a(t)
is used to denote the accumulation at time t of $1 invested at time 0.
a(t) as a function of t is called accumulation function. a(0) = 1 if interest is positive, a(t) is an increasing function of t
C. Weng ([e-mail])
p. 4/3
Amount Function
(p11) We use AK (t) to denote the accumulated value at time t of $K invested at time 0. Is it true: AK (t) = K a(t)? It may not be true e.g. A bank account earns 3% if K 5000, and 2.5% otherwise. Unless stated otherwise, we generally assume AK (t) = K a(t).
C. Weng ([e-mail])
p. 5/3
Example 1 of accumulation functions (p12-13)
e.g.1. a(t) = 1 + 0.05t (simple interest). 5% of original capital is added each year, paid continuously. Graph of a(t):
C. Weng ([e-mail])
p. 6/3
Example 2 of accumulation functions (p12-13)
e.g.2. a(t) = 1 + 0.05t, where t denotes the oor of t, i.e. the integer part of t. interest is added only at the end of year Graph of a(t):
C. Weng ([e-mail])
p. 7/3
Example 3 of accumulation functions (p12-13)
e.g.3. a(t) = 1.05t (compound interest). 5% of the accumulation (capital+interest earned) is added continually. Graph of a(t):
C. Weng ([e-mail])
p. 8/3
Simple Interest (p15)
An investment accumulates with simple at rate i per time unit (e.g. one year) paid continuously if and only if a(t) = 1 + it, t 0. Example 1.2. Consider an investment of $500 for 5 years and suppose it earns simple interest at a rate of 10% per year payable continuously. What is its accumulation at time 4.25? What if the interest is added only at the end of year?
C. Weng ([e-mail])
p. 9/3
Simple Interest (p15)
Simple interest is inconsistent: Compare the following two investments by analyzing their accumulation at the end of 3 years: A: Invest $100 for 3 years to earn simple interest at a rate of 8% per year; B: Invest $100 for 1 year to earn simple interest at a rate of 8% per year; then withdraw the proceeds and reinvest it for 2 more years.
Example 1.3.
C. Weng ([e-mail])
p. 10/3
Compound interest (p19)
An investment accumulates with compound interest at rate i per time unit (e.g. one year) if and only if
a(t) = (1 + i)t , t 0.
Compound interest is consistency: a(t + s) = a(t) a(s)
C. Weng ([e-mail])
p. 11/3
Simple interest Versus compound interest
Suppose you put $1,000 in a bank account which earn interest at a rate of 4% per annum.
Example 1.4
How much is the account balance at the end of one year? How much is the balance at the end of 2 years?
C. Weng ([e-mail])
p. 12/3
The Effective Interest Rate (p14)
The effective interest rate on period [t1 , t2 ] is dened as
i[t1 ,t2 ] a(t2 ) a(t1 ) interest earned on [t1 , t2 ] = = a(t1 ) capital at the beginning of the period
) in = a(na(na(n1) , for integer n. 1) in is the nth time unit. AK (t2 ) AK (t1 ) i[t1 ,t2 ] = AK (t1 ) Yes only when AK (t) = K a(t) (We usually assume this)
?
Generally, it is still called interest rate, but maybe not effective interest rate
C. Weng ([e-mail])
p. 13/3
Examples of effective interest rates
a(t) = 1 + it (simple interest) in a(n) a(n 1) [1 + in] [1 + i(n 1)] = = a(n 1) 1 + i(n 1) i = , decreasing in n. 1 + i(n 1)
a(t) = (1 + i)t (compound interest) in a(n) a(n 1) (1 + i)n (1 + i)(n1) = = = (1 + i) 1 (n1) a(n 1) (1 + i) = i, a constant for all n.
C. Weng ([e-mail])
p. 14/3
Examples
Example 1.5.
Compare the proceeds of 20-year investment of $1,000 earning simple interest and compound interest. Assume the rate is 8% per year for both the simple interest and compound interest.
You are given that $600 invested for 2 years accumulates to $864. What is the effective annual rate of interest?
Example 1.6.
C. Weng ([e-mail])
p. 15/3
Examples
Example 1.7.
An investor lends $1,000 at 10% of effective rate of interest and receives $2,000 after t years. Whats t?
Example 1.8.
An investor wants to buy a car which costs $12,000 in 1.5 years. How much should he/she invest now if he/she earns interest at 8% per year effective?
C. Weng ([e-mail])
p. 16/3
Examples
Suppose the investor in the above example can only invest $900 at time 0, and another $X at time 1. Find X .
Example 1.9.
An investor can get interest 4% per half year or 8.1% per year. Which is better for the investor?
Example 1.10.
C. Weng ([e-mail])
p. 17/3
Example: varying interest
Deposit $1,000 at an account which earns interest at annual rate of 10% in rst year and 15% in the second year. Whats the balance at the end of 2 years?
Example 1.11.
C. Weng ([e-mail])
p. 18/3
Example:Present Value
Suppose that Alice owes $1,000 due in 1 year. How much should she invest now (at time 0) to accumulate to exactly enough to pay her debt at time 1? Assume the effective annual rate of interest is i.
Example 1.12.
C. Weng ([e-mail])
p. 19/3
Present Value (p30)
The amount invested at time 0 so as to accumulate to $C at time t is called the Present Value (PV) of $C at time t, denoted PV($C at t)
1 PVa(t) = C = PV= C a(t) Let v (t) = 1/a(t), then v (t) is called the discount function, so that the PV of $C due at time t is simply $C v (t)
C. Weng ([e-mail])
p. 20/3
Discount Function (p28)
In compound interest world:
a(t) = (1 + i)t = v (t) = 1 1+i
t
1 v := is called discount factor per time unit, 1+i e.g., annual discount factor, i.e. PV of $1 in 1 year.
e.g., PV of $1,000 due in ve years is $1, 000 v 5 . e.g. PV of $P due in t years is $P v t .
In simple interest world:
1 a(t) = (1 + ti) = v (t) = , not constant. 1 + it
C. Weng ([e-mail])
p. 21/3
Geometric Series (progression)
a + ar + ar2 + (n terms in total), where arst term, rcommon ratio and nnumber terms of a(1 rn ) a(rn 1) . Formula for sum: sum = = 1r r1 e.g. Find the sum of 1 + 2 + 4 + 8 + 16 + 32
e.g. Find the sum of 1 + x + x2 + + xn
e.g. If |r| < 1, then limn rn = 0 and the sum a + ar + ar2 + (innitely many items) is a rst term = 1r 1 common ratio
C. Weng ([e-mail])
p. 22/3
Example for PV
Example 1.13.
Paul invests $100 at the start of each month for 10 years. The interest is earned at 1% per month effective. 1) Calculate the accumulated value at time 10. 2) Calculate the present value at time 0.
C. Weng ([e-mail])
p. 23/3
Example for PV
Example 1.14.
Paul invests $100 at the start of each month for 10 years. The interest is earned at 1% per month effective. 1) Calculate the accumulated value at time 10. 2) Calculate the present value at time 0.
C. Weng ([e-mail])
p. 24/3
Example for PV
Assume the interest is earned at 8% per year effective. Which of the following two cash ows has the higher value: A: 50 at time 0.5 and 50 at time 1.5. B: 100 at time 1.
Example 1.15.
C. Weng ([e-mail])
p. 25/3
Effective Rate of Discount
If the borrower agrees to repay $X in one year after receiving $(1 d)X today, then d is called the annual effective rate of discount. In perspective of the lender: invest $(1 d)X today and receive $X after one year.
D = dX is the discount amount, interest amount paid in advance instead of in arrear.
e.g. A bank might loan you $1,000 at an annual discount rate of 8% by giving you $920 now and asking you to pay back $1,000 after one year.
C. Weng ([e-mail])
p. 26/3
General Effective Rate of Discount (p27)
Discount amount on interval [t1 , t2 ] is AK (t2 ) Ak (t1 ) Effective discount rate on [t1 , t2 ] is Discount Amount AK (t2 ) AK (t1 ) = Balance at the end AK (t2 ) a(t2 ) a(t1 ) , if AK (t) = K a(t). = a(t2 ) The EDR in the nth period is d[t2 t1 ] = dn = = AK (n) AK (n 1) AK (n) a(n) a(n 1) , if AK (t) = K a(t). a(n 1)
C. Weng ([e-mail])
p. 27/3
Simple and Compound Discount Rates
Simple discount rate Lender makes use of $(1 td)X and repay $X , where d is called (e.g., annual) simple discount rate 1 equivalently, a(t) = . 1 td Compound discount rate Lender makes use of $(1 d)t X and repay $X , where d is called (e.g., annual) compound discount rate equivalently, a(t) =
X (1d)t .
Equivalence:
Rates of interest and discount are equivalent if any given amount of principle invested for any period accumulates to the same sum.
C. Weng ([e-mail])
p. 28/3
Example
Example 1.16.
What an annual rate of compound interest is equivalent to a rate of compound discount of 6% per year for, respectively, (a) t = 2 and (b) t = 4?
C. Weng ([e-mail])
p. 29/3
Some Facts
Facts under equivalence principle: (1 + i)(1 d) = 1
d+v =1 d=iv
C. Weng ([e-mail])
p. 30/3
Example
Example 1.17.
What an annual rate of compound interest is equivalent to a rate of simple interest at 6% per year effective for, respectively, (a) t = 2 and (b) t = 4?
C. Weng ([e-mail])
p. 31/3
Nominal rate of interest (p43)
In name only, not the real interest rate Interest of 4% per 0.5 year is expressed as an annual rate of 8%. But it equivalent to
(1 + 4%)(1 + 4%) 1 = 8.16%
In this case, we say: 8% is the annual nominal rate of interest convertible (or payable, or compounded) semi-annually notation:
i
(2)
i(2) 1 = 0.08 per year effective 2 2 1+ i(2) 2
2
1 per year effective
C. Weng ([e-mail])
p. 32/3
Example
Example 1.18.
If i = 10%, whats i(2) ?
C. Weng ([e-mail])
p. 33/3
Nominal Rate of Interest (p46-47)
(payable, or compounded) m times per year i(m) 1 (m) is denoted by i and is equivalent to an effective rate per years. m m If i is the equivalent annual effective rate of interest, then
Nominal rate of interest convertible
1+i= 1+ e.g., m = 4:
i
(m) m
m
C. Weng ([e-mail])
p. 34/3
Nominal Rate of Interest(p46-47)
i= 1+
i(m) m
m
1 & i(m) = m (1 + i)1/m 1
Find the accumulated value of $5,000 invested for 5 years at a rate of i(4) = 5%.
Example 1.19.
C. Weng ([e-mail])
p. 35/3
Nominal Rate of Discount(p46-47)
Nominal rate of discount convertible
(payable, or compounded) m times per year d(m) 1 (m) is denoted by d and is equivalent to an effective rate per years. m m If d is the equivalent annual effective rate of discount, then d(m) 1 m which is the PV of $ in 1 year. e.g., m = 4:
m
= 1 d = v,
C. Weng ([e-mail])
p. 36/3
Nominal Rate of Interest
i(m) d=1 1 m
Example 1.20.
m
& d(m) = m 1 (1 d)1/m
Suppose interest is credited at an annual discount rate of 8% compounded quarterly. Find (1) Find AV of $5,000 invested for 5 years. (2) Find the PV of $1,000 due in 3 years.
C. Weng ([e-mail])
p. 37/3
Nominal Rate of Interest
Example 1.21.
Given i(4) . Find the equivalent i, d and d(4) .
Given the value of i. How to nd d(4) ?
Relation i(m) between d(m)
?
C. Weng ([e-mail])
p. 38/3
Force of Interest (p54)
Force of Interest corresponding
to a general accumulation function a(t) is
dened as d 1 dat a (t) t = ln a(t) = = . dt a(t) dt a(t) Since da(t) = a(t + dt) a(t), the above display implies da(t) a(t + dt) a(t) = , t dt = a(t) a(t) which is the effective rate of interest on the innitesimal [t, t + dt].
Tow formulae a(t) v (t) = exp
t 0
r dr
t 0
= exp
r dr
C. Weng ([e-mail])
p. 39/3
Proof
Proof of
a(t) = exp
t 0
r dr
C. Weng ([e-mail])
p. 40/3
Constant Force of Interest (p54)
In the compound interest world, a(t) = (1 + i)t . So ln a(t) = t ln(1 + i) and hence da(t) d t = = [t ln(1 + i)] = ln(1 + i), a constant. dt dt A fact: if i > 0 and m > 1, then i > i(m) > > d(m) > d; see p53. (p52) In fact, we have
m
lim i(m) = = lim d(m) .
m t t2
Example 1.22.
Given a(t) = a a nd t .
C. Weng ([e-mail])
p. 41/3
Examples
Example 1.23.
given
Calculate AV at time t=10 of $1,000 invested at time 0. You are 0.05, 0 < t 4; t = 0.06, 4 < t 10.
C. Weng ([e-mail])
p. 42/3
Ination (58)
Ination
may be dened as the loss of purchasing power per dollar.
Many measures. The most common one is Consumer Price Index (CPI) Suppose the index is Q(t) at time t. The rate of ination on period [t1 , t2 ] is dened by Q(t2 ) Q(t1 ) r[t1 ,t2 ] = Q(t1 ) e.g. Ination index Date Index Jan01,2006 121.5 June01,2006 125.3 Dec31,2006 142.7
r[0,1/2] r[1/2,1]
= =
125.3 121.5 = 3.13% 121.5 142.7 125.3 = 13.89% 125.3
C. Weng ([e-mail])
p. 43/3
Example with Ination
Let i be the effective annual rate of interest and r the annual ination rate. Then, the ination adjusted interest rate (the real interest rate) is dened as the value of j satisfying 1+i ir 1+j = , i.e., j = . 1+r 1+r
Invest $1,000 for one year at an annual interest rate of 25% on Jan01,2006. The ination index is as in the previous slide. What is the real rate of interest over the 2006 year.
Example 1.24.
C. Weng ([e-mail])
p. 44/3