accumulation and amount functions - ttuxiwang/3356/slide-ch-1.pdf · accumulation and amount...

Accumulation and Amount Functions

Definition

The accumulation function a(t) is a function which gives theaccumulated value at time t of an original investment of 1.

Definition

If the original investment is k , then the accumulated value attime t will be ka(t). The function

A(t) = ka(t)

is called the amount function.

Note that k = A(0) and a(t) = 1A(0)A(t).

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Interest

Definition

The amount of increasing of the investment during the nthperiod is called the interest eared during that period, i.e.

In = A(n)− A(n − 1).

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Effective Rate of Interest

Definition

The effective rate of interest during the nth period is the ratioof interest to the beginning amount of the period, i.e.

in =A(n)− A(n − 1)

A(n − 1)=

a(n)− a(n − 1)

a(n − 1).

We haveA(n) = A(n − 1)(1 + in)

where un = 1 + in is called the accumulation factor for the nthperiod.

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Example (Exercise 1.1)

Consider the amount function A(t) = t2 + 3t + 5

a) Find the corresponding accumulation function a(t).

b) Find In and in.

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a) A(0) = 5 and

a(t) =1

A(0)A(t) =

1

5(t2 + 3t + 5).

b)

In = A(n)−A(n−1) = (n2+3n+5)−((n−1)2+3(n−1)+5) = 2n+2.

in =1

A(n − 1)In =

2n + 2

(n − 1)2 + 3(n − 1) + 5=

2n + 2

n2 + n + 1

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Assume that A(t) = 350 + 3t.

a) Find i4.

b) Find i11.

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a)

i4 =A(4)− A(3)

A(3)=

3

359.

b)

i11 =A(11)− A(10)

A(10)=

3

380.

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Assume that A(t) = 250(1.09)t .

a) Find i4.

b) Find i11.

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a)

i4 =A(4)− A(3)

A(3)= 0.09.

b)

i11 =A(11)− A(10)

A(10)= 0.09.

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Assume that A(4) = 2500 and in = 0.05n. Find A(9).

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The accumulation factor for the nth period is

un = (1 + in) = 1 + 0.05n.

So

A(9) = A(8)u9 = A(7)u8u9 = · · · = A(4)u5u6u7u8u9,

i.e.

A(9) = 2500(1+0.05(5))(1+0.05(6)) · · · (1+0.05(9)) = 11133.28.

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It is known that a(t) is of the form kt2 + b. If $100 invested attime 0 accumulates to $172 at time 3, find the accumulated valueat time 10 of $100 invested at time 5.

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We need to find A(10) under the condition that A(5) = 100.From a(t) = kt2 + b we immediately know that b = 1 becausea(0) = 1. By the condition 100a(3) = 100(k32 + 1) = 172,

k =1

9

(172

100− 1

)=

2

25, and a(t) =

2

25t2 + 1.

We have

A(t) = A(0)a(t) = A(0)

(2

25t2 + 1

).

From 100 = A(5) = A(0)(

22552 + 1

)= 3A(0) we can solve

A(0) = 1003 and

A(10) =100

3a(10) =

100

3

(2

25102 + 1

)= $300.

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Simple Interest

Definition

Simple interest:a(t) = 1 + it.

Note that the effective rate of a simple interest investment is notconstant, and is decreasing:

in =a(n)− a(n − 1)

a(n − 1)=

(1 + in)− (1 + i(n − 1))

1 + i(n − 1)=

i

1 + i(n − 1).

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a) At what rate of simple interest will $500 accumulate to $620 in800 days?b) In how many years will $500 accumulate to $630 at 7.8% simpleinterest?

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a) 800 days = 800365 years. We have 620 = 500

(1 + i 800365

). Solve for

i :

i =365

800

(620

500− 1

)= 0.1095 = 10.95%.

b) Let 630 = 500 (1 + 0.078t) and solve for t:

t =1

0.078

(630

500− 1

)= 3.33 years.

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At a certain rate of simple interest $1,000 will accumulate to$1,110 after a certain period of time. Find the accumulated valueof $500 at a rate of simple interest 3/4 as great over twice as longa period of time.

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By the condition, 1000(1 + it) = 1110. So

it =1110

1000− 1 =

110

1000

and

500

(1 +

(3

4i

)(2t)

)= 500

(1 +

3

2it

)= $582.50.

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Simple interest of i = 4% is being credited to a fund. In whichperiod is this equivalent to an effective rate of 2.5%?

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The effective rate for the nth period for simple interest is given byi

1+(n−1)i Let 0.025 = i1+(n−1)i = 0.04

1+(n−1)0.04 and solve for n:

n =1

0.04

(0.04

0.025− 1

)+ 1 = 16.

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Compound Interest I

If the interest earned is periodically reinvested, then we have thecompound interest. Assume the initial investment 1, at the end ofthe 1st period, the amount becomes (1 + i). Then if we reinvestthe whole (1 + i) at the beginning of the 2nd period, then at theend of the 2nd period, the amount becomes(1 + i)(1 + i) = (1 + i)2, etc. So if we reinvest the whole amountat the end of every period, then at the end of the nth period, theamount is

(1 + i)n.

Compound interest is used almost exclusively for financialtransactions over one or multi-years. Simple interest is occasionallyused for short-term transactions over a fractional period. So unlessstated otherwise, we will use compound interest.

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Compound Interest II

For compound interest, the future value A(n) at the time n of aninitial investment A(0) is given by

A(n) = A(0)un

where u = 1 + i is the accumulation factor per period. Conversely,the present value A(0) of a future amount A(n) at the time n isgiven by

A(0) = A(n)vn

where v = 1u = 1

1+i is called the discount factor per period.

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Time Diagram I

Time diagram is a very useful tool to analyze the values ofinvestments. Almost all the problem need to be solved by drawinga time diagram first.Following are some simple examples of time diagrams:

a) If $100 is deposited into a bank account, what is the accountbalance 7 years from today?

0 1 2 3 4 5 6 7

100 FV

The direction (toward to or away from the number line) of thepayment arrows represents the direction of the cash flows.

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Time Diagram II

b) How much needs to be deposited into a bank account in orderto generate $1,000 ten years from to day?

0 1 2 3 4 5 6 7 8 9 10

PV 1000

Payment Moving Rule

For compound interests, if a payment P is moved n periods to theright (future), then its value becomes Pun, and if it is moved mperiods to the left (past), then its value becomes Pvm

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Time Diagram III

0 1 2 3 4 5 6 7 8 9 10

P Pu Pu2 Pu3 Pu4 Pu5 Pu6 Pu7 Pu8 Pu9 Pu10

0 1 2 3 4 5 6 7 8 9 10

PPv9 Pv8 Pv7 Pv6 Pv5 Pv4 Pv3 Pv2 Pv1Pv10

So for a): FV = 100u7, and for b): PV = 1000v10.

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What about in the middle of a period when t = n + k, 0 < k < 1?There are 2 ways to do it:

Definition

Compound interest throughout:

A(t) = A(0)ut = A(0)unuk .

Definition

Compound interest with simple interest during final fractionalperiod:

A(t) = A(0)un(1 + ki).

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Financial Calculator

The TVM (time-value of money) solver of a financial calculatorcan compute any one of the following values if the other values aregiven: present value (PV); periodic payments (PMT); future value(FV); nominal percentage interest rate (I); the number of totalperiods (n).Here is the time diagram of payments represented by the end modeof TVM solver. It is called the end mode because the kth periodicpayment is at the end of the kth period [k − 1, k]

0 1 2 3 4 5 6 n-2 n-1 n

PV

PMT PMT PMT PMT PMT PMT PMT PMT PMT

FV

return

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Financial Calculator: Beginning Mode

Here is the time diagram of payments represented by the beginningmode of the TVM solver. It is called the beginning mode becausethat the kth periodic payment is at the beginning of the kthperiod [k − 1, k].

0 1 2 3 4 5 6 n-2 n-1 n

PV

PMT PMT PMT PMT PMT PMT PMT PMTPMT

FV

Keep in mind that cash flow in different directions must haveopposite signs.

If the periodic payments (PMTs) are all 0, then the both modesare the same, and we can use either one of them.

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Example (Example 1.3, 1.4)

Find the accumulated value of $5,000 invested for 5 years with 6%per annum.a) invested in simple interest.b) invested in compound interest compounded yearly.

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a)

5000(1 + 0.06(5)) = $6500.00.

b)

0 1 2 3 4 5

5000 FV

So FV = 5000u5 = 5000(1.055).Financial Calculator:

N I PV PMT FV P/Y C/Y E/B?

5 6 -5000 0 1 1 either⇓

6691.13

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Example (Example 1.6)

Find the accumulated value of $8000 at the end of 6 years and 7months invested at a rate of 8.5% compound interest per annum

(1) Assuming compound interest throughout.

(2) Assuming simple interest during the final fractional period.

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(1)

8000u6+712

Financial Calculator:N I PV PMT FV P/Y C/Y E/B?

6 + 712 8.5 -8000 0 1 1 either

⇓13687.87

(2)

8000u6(

1 + 0.0857

12

)= $13698.89.

where u = 1 + 0.085 = 1.085

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Example

Find the accumulated value of $1000 invested for 5 years, if therate of compound interest is 7% per annum.

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0 1 2 3 4 5

1000 FV

So FV = 1000u5 = 1000(1.075).Financial Calculator:


5 7 -1000 0 1 1 either⇓

1402.55

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It is known that $600 invested in compound interest for two yearswill earn $264 interest. Find the accumulated value of $2000invested at the same rate of compound interest for three years.

0 1 2 3

600 600+264

0 1 2 3

2000 FV

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We have

600u2 = 600 + 264 = 864,

so

u =

√864

600and FV = 2000u3 = 2000

(864

600

) 32

.

Financial calculator: Interest rate:N I PV PMT FV P/Y C/Y E/B?

2 -600 0 864 1 1 either⇓i

Future value of initial $2000:N I PV PMT FV P/Y C/Y E/B?

3 i -2000 0 1 1 either⇓

3456.00

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An investor age 20 deposits $20, 000 in a fund earning 8%compound interest until retirement at age 65. Find the amount ofinterest earned between ages 20 and 30, between ages 30 and 40,between ages 40 and 50, between ages 50 and 60, and betweenages 60 and 65.

0 (age 20) 10 (age 30) 20 (age 40) 30 (age 50) 40 (age 60) 45 (age 65)

A(0)=20000 A(10) A(20) A(30) A(40) A(45)

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u = 1 + 0.08 = 1.08, and time = age−20. The interest earned inany time period equals the increasing amount of the beginningbalance. So the interests are:

I1 = A(10)− A(0) = 20000(u10 − 1)

I2 = A(20)− A(10) = 20000(u20 − u10)

I3 = A(30)− A(20) = 20000(u30 − u20)

I4 = A(40)− A(30) = 20000(u40 − u30)

I5 = A(45)− A(40) = 20000(u45 − u40)

Financial calculator:Between ages 20 and 30:


10 8 -20000 0 1 1 either⇓

A(10)I1 = A(10)− 20000 = $23178.50.

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Between ages 30 and 40:N I PV PMT FV P/Y C/Y E/B?

20 8 -20000 0 1 1 either⇓

A(20)I2 = A(20)− A(10) = $50040.64.


30 8 -20000 0 1 1 either⇓

A(30)i3 = A(30)− A(20) = $108033.99.

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40 8 -20000 0 1 1 either⇓

A(40)I4 = A(40)− A(30) = $233237.29.


45 8 -20000 0 1 1 either⇓

A(45)I5 = A(45)− A(40) = $203918.56.

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Example

How long will it take for an investment to double if it is invested incompound interest of r% percent per annum?

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Solve t from(1 + r

100

)t= 2:

t =ln 2

ln(1 + r

100

)

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Example

If an investment in a fixed rate compound interest account doublesevery 5 years, what is the interest rate?

0 1 2 3 4 5

1 2

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We need to solve u5 = (1 + i)5 = 2 for i and it is much easier touse a financial calculator.


5 -1 0 2 1 1 either⇓

14.87%

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Present Value I

It is often necessary to determine how much an initial invest mustbe so that the balance will be 1 at the end of the investment.Clearly it is the reciprocal of the accumulation function, i.e. 1

a(t) .Such a function is called the discount function.

Definition

The discount function b(t) is a function which gives theaccumulated value 1 at time t of an original investment of b(t).

So

Simple Interest: b(t) =1

1 + it,

Compound Interest: b(t) =1

(1 + i)t= v t ,

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Present Value II

where

v =1

1 + i=

1

u

is the discount factor per period.Note that the discount function for compound interest can bewritten as

b(t) =1

(1 + i)t= (1 + i)−t = a(−t)

i.e. for compound interest, the discount function is theaccumulation function with negative time.

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Present Value III

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Find the amount which must be invested at 8% per annum inorder to accumulate $2000 at the end of four years:

(1) Assuming simple interest.

(2) Assuming compound interest.

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(1)2000

1 + 0.08(4)= $1515.15.

(2)

0 1 2 3 4

PV 2000

PV = 2000v4 where v = 11.08 .

Financial Calculator:N I PV PMT FV P/Y C/Y E/B?

4 8 0 -2000 1 1 either⇓

1470.06

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Example

Bubba Jones has an offer to play pro football in the Arena League.Part of his contract calls for him to receive a signing bonus,payable as follows: $200,000 immediately, $500,000 in two years,and $700,000 in 6 years. What is the present value of the signingbonus, assuming annual 4% effective interest?

0 1 2 3 4 5 6

200000 500000 700000

500000v2

700000v6

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Answer = 200000 + 500000v2 + 700000v6, v =1

1.04.

Financial Calculator:


2 4 0 -500000 1 1 either⇓A


6 4 0 -700000 1 1 either⇓B

Answer =A+B+200000=$1,215,498.27

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Example

Grandparents for a newborn baby wish to invest enough moneyimmediately to pay $10,000 per year for four years toward collegecosts starting at age 18. If the effective rate of interest is 6% perannum, find the contribution of the grandparents.

0 1 2 3 17 18 19 20 21

10000 10000 10000 10000

10000v18

10000v19

10000v20

10000v21

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Answer = 10000v18 + 10000v19 + 10000v20 + 10000v21

= 10000(v18 + v19 + v20 + v21), v =1

1.06.

To use a financial calculator to compute the value, recall the ENDmode of a financial calculator.

Value at the time 17:N I PV PMT FV P/Y C/Y E/B?

4 6 -10000 0 1 1 END⇓A

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Value at the time 0:N I PV PMT FV P/Y C/Y E/B?

17 6 0 -A 1 1 END⇓

12868.17

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It is known that an investment of $500 will increase to $4,000 atthe end of 30 years. Find the sum of the present values of threepayments of $10,000 each which will occur at the end of 20, 40,and 60 years.

0 10 20 30 40 50 60

500 4000

0 10 20 30 40 50 60

10000 10000 10000

10000v20

10000v40

10000v60

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We have 4000v30 = 500. So v30 = 5004000 = 1

8 and v = 1

8130

Therefore

PV = 10000v20+10000v40+10000v60 = 10000

(1

82030

+1

84030

+1

86030

).

To use a financial calculator, we first compute the annual interestrate


30 -500 0 4000 1 1 either⇓i

then for n=20,40,60, computeN I PV PMT FV P/Y C/Y E/B?

n i 0 -10000 1 1 either⇓An

PV = A20 + A40 + A60 = 3281.25

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The Effective Rate of Discount I

To understand discounts, let’s consider 2 examples:

If a person borrows $100 for one year at an effective rate 7% ofinterest, he/she receives $100 and pay back $107 one year later.

If a person borrows $100 for one year at an effective rate 7% ofdiscount, he/she receives $93 and pay back $100 one year later.The effective rate of interest is actually

100− 93

93= 7.53%.

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The Effective Rate of Discount II

Definition

The effective rate of discount during the nth period is the ratioof the interest earned to the investment amount at the end ofthe period, i.e.

dn =A(n)− A(n − 1)

A(n)=

a(n)− a(n − 1)

a(n).

Since a(t) = 1b(t) , we also have

dn =

1b(n) −

1b(n−1)

1b(n)

=b(n − 1)− b(n)

b(n − 1)

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The Effective Rate of Discount III

For compound interest, a(n) = un anda(n − 1) = un−1 = unv = a(n)v . so

d =a(n)− a(n − 1)

a(n)=

a(n)− a(n)v

a(n)= 1− v

We have

i = u − 1 = u(1− v) = ud , and d =1

ui = vi

d iu→←v

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The Effective Rate of Discount IV

Accumulation factor:

u = 1 + i =1

1− d

Discount factor:

v = 1− d =1

1 + i

Relationship between u and v :

u =1

v, v =

1

u

Relationship between i and d :

i =d

v, d =

i

u.

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People also use the simple discount (it is not the discountfunction for simple interest!), where the discount function islinearization at 0 of the discount function of the compound interestb(t) = v t = (1− b)t

Simple discount function: b(t) = 1− dt for 0 ≤ t <1

d

(in order to keep the present value positive). Simple discount isonly used for short time transactions and as an approximation forcompound discount over fractional period. It is not as widely usedas simple interest.

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Find the amount you received if you borrow $2000 from a bank forfour years with a simple discount of 8% per annum.

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2000(1− 0.08(4)) = $1360

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Find the amount you received if you borrow $2000 from a bank forfour years with a compound discount of 8% per annum.

0 1 2 3 4

PV 2000

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PV = 2000v4 = 2000(1− 0.08)4.

We can also use a financial calculator to compute. Note that thepercentage interest

I% =D%

v.


4 8/(1-0.08) 0 -2000 1 1 either⇓

1432.79

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a) Find d4 if the rate of simple interest is 12%.

b) Find d4 if the rate of simple discount is 12%.

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a) For simple interest, a(t) = 1 + it

d4 =a(4)− a(3)

a(4)=

(1 + 0.12(4))− (1 + 0.12(3))

1 + 0.12(4)

= 0.08108108108 = 8.11%.

b) For simple discount, b(t) = 1− dt

d4 =b(3)− b(4)

b(3)=

(1− 0.12(3))− (1− 0.12(4))

1− 0.12(3)

= 0.1875000000 = 18.75%.

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Find the annual effective rate of discount at which a payment of$200 immediately and $300 one year from today will accumulatesto $600 two years from today.

0 1 2

200 300 600

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From the time diagram we have

600v2 = 300v + 200, or 6v2 − 3v − 2 = 0

So

v =3±

√9 + 4(6)2

12=

3 +√

57

12, (the other solution is negative)

and

d = 1− v = 1− 3 +√

57

12=

9−√

57

12= 0.1208471304 = 12.08%.

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The amount of interest earned on A is $336, while the equivalentdiscount on A is $300. Find A.

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By the given condition

Au = A + 336, Av = A− 300.

A + 336

A= u =

1

v=

A

A− 300,

(A− 300)(A + 336) = A2,

A2 + 36A− 300(336) = A2,

A =300(336)

36= $2800.

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Nominal Rates of Interest and Discount I

Definition

For compound interest or discount payable (or compound, orconvertible) m times per year, the given rate per year is calledthe nominal rate of interest, and is denoted by

i (m).

The 1m th of a year used to compute the interest is called

interest conversion period.

For compound interest or discount payable m times per year, theinterest used in each of the interest conversion period is

nominal rate of interest

m=

i (m)

m.

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Nominal Rates of Interest and Discount II

Definition

The accumulation function for nominal rate of interest i (m)

compound m times per year is given by

a(t) =

(1 +

i (m)

m

)mt

.

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Nominal Rates of Interest and Discount III

So the accumulation factor per year is given by

u =

(1 +

i (m)

m

)m

and the accumulation factor per conversion period is given by

u = 1 +i (m)

m

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Nominal Rates of Interest and Discount IV

Definition

The discount function for nominal rate of discount d (m)

convertible m times per year is given by

b(t) =

(1− d (m)

m

)mt

.

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Nominal Rates of Interest and Discount V

So the discount factor per year is given by

v =

(1− d (m)

m

)m

and the discount factor per conversion period is given by

v = 1− d (m)

m

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Nominal Rates of Interest and Discount VI

We can easily find the relationship between different nominal rateof interests and/or discounts through a common accumulation ordiscount factor.For examples, for an effective interest rate i per year,

1 + i = u =

(1 +

i (m)

m

)m

is the accumulation factor per year.For an effective discount rate d per year,

1− d = v =

(1− d (m)

m

)m

is the discount factor per year.

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Nominal Rates of Interest and Discount VII

For any m and k, (1 +

i (m)

m

)m

=1(

1− d (k)

k

)kbecause both of them are the accumulation factor per year.

Since i (m)

m and d (m)

m are the effective rates of interest and discount

per conversion period, and u = 1 + i (m)

m and v = 1− d (m)

m are theaccumulation and discount factors per conversion period,

respectively, we have i (m)

m =d(m)

mv , and d (m)

m =i(m)

mu , which means

that

i (m) =d (m)

v, and d (m) =

i (m)

u

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Nominal Rates of Interest and Discount VIII

i (m) =d (m)

discount factor per conversion period.

d (m) =i (m)

accumulation factor per conversion period

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Find the accumulated value of $100 at the end of two years:

a) If the nominal annual rate of interest is 6% convertiblequarterly.

b) If the nominal annual rate of discount is 6% convertible onceevery 4 years.

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For both of them, FV = 100u2, where u is the accumulation factorper year.

a)

u =

(1 +

0.06

4

)4

and FV = 100

(1 +

0.06

4

)8

Financial Calculator (N is the number of conversion periods):N I PV PMT FV P/Y C/Y E/B?

4*2 6 -100 0 4 4 either⇓

112.65

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b)

u =1(

1− 0.061/4

)1/4 and FV = 1001

(1− 0.24)1/2

A TVM solver does not allow a fractional number as C/Y (numberof compounding per year). One way to get around of this is toconsider a 4-year period. The rate of discount per 4-year period is4(6%) = 24%, and the rate of interest per 4-year period is

24

v% =

24

1− 0.24%,

and 2-year period is 0.5 of 4-year period.N I PV PMT FV P/Y C/Y E/B?

0.5 24/(1-0.24) -100 0 1 1 either⇓

114.71

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Find the present value of $2000 to be paid at the end of 5 years at8% per annum payable in advance and convertible quarterly.

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“payable in advance” means that the 8% is the nominal rate ofdiscount.

PV = 2000v5 = 2000

(1− 0.08

4

)4(5)

By using the formula

i (4) =d (4)

v=

d (4)

1− d (4)

4

we can also use a financial calculator to computeN I PV PMT FV P/Y C/Y E/B?

4*5 8/(1-0.08/4) 0 -2000 4 4 either⇓

1335.22

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Find the nominal rate of interest convertible monthly which isequivalent to a nominal rate of discount of 8% per annumconvertible quarterly.

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We have (1 +

i (12)

12

)12

=1(

1− 0.084

)4because both of them are the accumulation factor per year. So

i (12) = 12

(1(

1− 0.084

)4/12 − 1

)= 0.08108354 = 8.11%.

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Given that i (m) = 0.1844144 and d (m) = 0.1802608, find m.

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We have

d (m) =i (m)

u=

i (m)

1 + i (m)

m

i.e.

0.1802608

(1 +

0.1844144

m

)= 0.1844144

m =0.1844144

0.18441440.1802608 − 1

= 8

88 / 105

Forces of Interest and Discount I

Definition

The force of interest δ(t) is defined by

δ(t) =A′(t)

A(t)=

a′(t)

a(t).

δ(t) is the relative rates of change of the accumulation functiona(t).From the definition we can see that

δ(t) =d

dtln a(t).

So ∫ t

0δ(r) dr = ln a(t)− ln a(0) = ln a(t)

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Forces of Interest and Discount II

and

a(t) = e∫ t0 δ(r) dr .

δ(t) =d

dtln a(t) = − d

dtln b(t)

a(t) = e∫ t0 δ(r) dr

b(t) = e−∫ t0 δ(r) dr

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Forces of Interest and Discount III

The force of interest is a constant for the compound interestbecause

δ =d

dtln(1 + i)t =

d

dtt ln(1 + i) = ln(1 + i).

For compound interest or discount:

δ = ln(1 + i) = − ln(1− d).

u = 1 + i = eδ, a(t) = ut = eδt .

v =1

1 + i= e−δ, b(t) = v t = e−δt .

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Forces of Interest and Discount IV

Note that limn→∞(1 + i

n

)nt= e it . So δ is also called the nominal

rate of interest compound continuously.

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Forces of Interest and Discount V

For simple interest

δ(t) =d

dtln(1 + it) =

i

1 + it

and for simple discount

δ(t) = − d

dtln(1− dt) =

d

1− dt.

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Find the accumulated value of $500 invested for 15 years if theforce of interest is 6%.

94 / 105

a(t) = e∫ t0 δ(r)dr = e

∫ t0 0.06dr = e0.06t

FV = 500e0.06(15) = $1229.80

95 / 105


Find the accumulate function if the force of interest is δ(t) = 11+t2

96 / 105

a(t) = e∫ t0 δ(r)dr = e

∫ t0

11+r2

dr= etan

−1 t

97 / 105


Fund A accumulates at a simple interest rate of 10%. Fund Baccumulates at a simple discount rate of 5%. Find the point intime t at which the forces of interests on the two funds are equal.

98 / 105

For simple interest, a(t) = 1 + it and δ(t) = ddt ln a(t) = i

1+it .

For simple discount, b(t) = 1− dt and δ(t) = − ddt ln b(t) = d

1−dt .

0.1

1 + 0.1t=

0.05

1− 0.05t,

0.1− 0.1(0.05)t = 0.05 + 0.1(0.05)t,

t =0.1− 0.05

2(0.1)0.02= 5.

99 / 105

Varying Interest


Find the level effective rate of interest over a three-year periodwhich is equivalent an effective rate of discount 8% the first year,7% the second year, and 6% the third year.

100 / 105

(1

1− 0.08

)(1

1− 0.07

)(1

1− 0.06

)= (1 + i)3

So

i = 3

√(1

1− 0.08

)(1

1− 0.07

)(1

1− 0.06

)− 1

We can also use a financial calculator to compute. By the

condition, a(3) =(

11−0.08

)(1

1−0.07

)(1

1−0.06

)= 1

0.92(0.93)0.94


3 -1 0 1/(0.92*0.93*0.94) 1 1 either⇓

7.53%

101 / 105


An investor makes a deposit today and earns an average continuousreturn (force of interest) of 6% over the next 5 years. Whataverage continuous return must be earned over the subsequent 5years in order to double the investment at the end of 10 years?

102 / 105

e0.06(5)e5i = e0.3+5i = 2

i =1

5(ln(2)− 0.3) = 0.07862943606 = 7.86%.

103 / 105


In Fund X money accumulates at a force of interest

δt = 0.1t + 0.1

In fund Y , money accumulates at an annual effective interest ratei . An amount of 1 is invested in each fund, and 20 years later thevalues of fund X and fund Y are equal. Calculate the value offund Y at the end of 1.5 years.

104 / 105

The accumulation functions of funds X and Y are

aX (t) = e∫ t0 0.1r+0.1 dr = e0.05t

2+0.1t , aY (t) = ut ,

respectively. By the given condition

e0.05(20)2+0.1(20) = u20

So

u = e0.05(20)2+0.1(20)

20 = e0.05(20)+0.1 = e1.1

and

aY (1.5) = u1.5 = e1.5(1.1) = 5.21.

105 / 105

accumulation and amount functions - ttuxiwang/3356/slide-ch-1.pdf · accumulation and amount...

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