Nominal rates of interest and discount

Nominal rates of interest and discount

• i(m): The nominal rate of interest payable m times per period, where m is a positive integer > 1. By a nominal rate of interest i(m), we mean a rate payable mthly, i.e. the rate of interest is i(m)/m for each mth of a period and not i(m). Thus,

1 + i =

1 + i

(m)

m

and

i =

1 + i

(m)

m

m

− 1 i(m) = m[(1 + i)m1 − 1]

• d(m): The nominal rate of discount payable m times per period. Thus,

1 − d =

1 − d

(m)

m

and

d = 1 −

1 − d

(m)

m

m

d(m) = m[1 − (1 − d)m1 ] = m[1 − vm1 ]

Generally,

1 + i

(m)

m

m

=

1 − d

(p)

p

If m = p,

1 + i

(m)

m

=

1 − d

(m)

m

−1

=⇒ i

(m)

m −

d(m)

m =

i(m) m ·

d(m) m .

Force of interest and discount

δ_t = a

0_(t)

a(t) =

A0(t) A(t)

= d

dt ln A(t) = d

dt ln a(t) Thus,

Z t

0

δ_rdr =

Z t

0

d

dr ln A(r)dr = ln A(r)|

t

0 = ln

A(t) A(0)

Hence,

eR0t δrdr = A(t)

A(0) =

a(t)

a(0) = a(t) Note that

Z n

0

A(t)δ_t =

Z n

0

A0(t)dt = A(n) − A(0).

If δ_t = δ, 0 ≤ t ≤ n, then

so that

eδ = 1 + i =⇒ i = eδ − 1 =⇒ δ = ln(1 + i)

• δ_t0: the force of discount at time t, is defined by

δ_t0 = −

d dta

−1_(t)

a−1(t) .

δ_t0 = −

d dta

−1_(t)

= a

−2_(t) d

dta(t)

a−1(t)

= a

−2_(t)a(t)δ

t

a−1(t) = δ_t

Some properties of force of interest

• For simple interest

δ_t =

d

dta(t)

a(t)

=

d

dt(1 + it)

1 + it

= i

• For simple discount

δ_t = δ_t0 = −

d dta

−1_(t)

a−1(t)

= −

d

dt(1 − dt)

1 − dt

= d

1 − dt for 0 ≤ t < 1 d.

• i > δ

i = eδ − 1 = δ + δ

2

2! + δ3

3! + δ4

• lim

m→∞ i

(m)

= δ

1 + i

(m)

m

m

= eδ

i(m) = m

emδ − 1

= m δ m + 1 2! δ m 2 + 1 3! δ m 3 + · · ·

= δ + δ

2

2!m +

δ3

• lim

m→∞ d

(m)

Varying interest

Let i_k denote the rate interest applicable for period k. We consider first the present value of an n-period

annuity-immediate.

a_n| = (1 + i₁)−1 + (1 + i₁)−1(1 + i₂)−1 + · · · + (1 + i₁)−1(1 + i₂)−1 · · · (1 + i_n)−1

=

n

X

t=1

t

Π

s=1

(1 + i_s)−1

The second pattern would be to compute present values using rate i_k for the payment made at time k over all k

periods. In this case the present value becomes

a_n| = (1 + i₁)−1 + (1 + i₂)−2 + · · · + (1 + i_n)−n

=

n

X

t=1

(1 + i_t)−t

Similarily, ¨

s_n| = (1 + i_n) + (1 + i_n)(1 + i_n−1) + · · ·

+ (1 + i_n)(1 + i_n−1) · · · (1 + i₁) =

n

X

t=1

t

Π

s=1

Alternately, if the payment made at time k earns at rate i_k over the rest of the accumulation period, we have

¨

s_n| = (1 + i_n) + (1 + i_n−1)2 + · · · + (1 + i₁)n

=

n

X

t=1

Linear interpolation of

(1 +

i)

, v

(1 + i)n+k = (1. − k)(1 + i)n + k(1 + i)n+1 = (1 + i)n[(1 − k) + k(1 + i)] = (1 + i)n(1 + ki)

Similarily,

vn+k = (1 − d)n+k = (1. − k)(1 − d)n + k(1 − d)n+1 = (1 − d)n[(1 − k) + k(1 − d)] = vn(1 − kd)

Method of equated time

Let amounts s₁, s₂, · · · , s_n be paid at times t₁, t₂, · · · , t_n respectively. The problem is to find t, such that

s₁ + s₂ + · · · + s_n paid at time t is equivalent to the payments. of s₁, s₂, · · · , s_n made separately.

(s₁ + s₂ + · · · + s_n)vt = s₁vt1 _{+ s}

2vt2 + · · · + snvtn

As a first approximation, t is calculated as a weighted average of the various times of payment, where the

weights are the various amounts paid, i.e.

¯

t = s1t1 + s2t2 + · · · + sntn s₁ + s₂ + · · · + s_n =

n

X

k=1

s_kt_k

n

X

k=1

s_k

This approximation is denoted by t¯ and is often called using the method of equated time.

Consider s₁ quantities each equal to vt1, s

2 quantities

each equal to vt2, and so forth until there are s

quantities each equal to vtn_{. The arithmetic mean of}

these quantities is

s₁vt1 + s

2vt2 + · · · + snvtn

s₁ + s₂ + · · · + s_n

The geometric mean of these quantities is v

s₁t₁₊s₂t₂₊···+_sntn

s₁₊s₂₊···+_{sn = v}¯t

And,

s₁vt1 + s

2vt2 + · · · + snvtn

s₁ + s₂ + · · · + s_n > v

¯

=⇒ s₁vt1 _{+ s}

2vt2 + · · · + snvtn > (s1 + s2 + · · · + sn)v

¯

t

This means that the value of t¯ is always greater than the true value of t.

Basic annuities

a_n| = v + v2 + · · · + vn−1 + vn = v1 − v

n

1 − v = v1 − v

n

iv = 1 − v

i

= (1 + i)

n _{− 1}

(1 + i) − 1 = (1 + i)

n _{− 1}

i ? 1 = ia_n| + vn

? s_n| = a_n|(1 + i)n

? 1

a_n| = 1

s_n| + i

1

s_n| + i =

i

= i + i(1 + i)

n _{− i}

(1 + i)n − 1

= i

1 − vn

= 1

a_n|

• annuity-due ¨

a_n| = 1 + v + v2 + · · · + vn−1 = 1 − v

n

= 1 − v

n

iv = 1 − v

n

d

¨

s_n| = (1 + i) + (1 + i)2 + · · · + (1 + i)n

= (1 + i)(1 + i)

n _{− 1}

(1 + i) − 1 = (1 + i)

n _{− 1}

= (1 + i)

n _{− 1}

d

? s¨_n| = ¨a_n|(1 + i)n

? 1

¨

a_n| = 1 ¨

s_n| + d ? ¨a_n| = a_n|(1 + i) ? s¨_n| = s_n|(1 + i) ? ¨a_n| = 1 + a_n−1| ? s¨_n| = s_n+1| − 1

• Perpetuities

a_∞| = v + v2 + v3 + · · ·

= v

1 − v

= v

iv

= 1

i Alternatively, we have

a_∞| = lim

n→∞an| = limn→∞

1 − vn

i =

1 i .

For a perpetuity-due, we have ¨

a_∞| = 1 d.

Nonstandard terms

a

,

0

< k <

1

a_n+k| = 1 − v

n+k

i = 1 − v

n _{+ v}n _{− v}n+k

i = a_n| + vn+k

(1 + i)k − 1 i

Yiele rate

Consider a situation in which an investor makes deposits or contributions into an investment of C₀, C₁, · · · , C_n at times 0, 1, 2, · · · , n. Thus, we can denote the returns as R₀, R₁, · · · , R_n at times 0, 1, · · · , n. Then we have

R_t = −C_t for t = 0, 1, · · · , n.

Assume that the rate of interest per period is i. Then the net present value at rate i of investment returns by the discounted cash flow technique is denoted be P (i) and is

given by

P (i) =

n

X

t=0

vtR_t.

An important special case of this formula is the one in which P (i) = 0,

P (i) =

n

X

t=0

vtR_t = 0.

The rate of interest i which satisfies P (i) = 0 is called the yield rate on the investment. Stated in words:

The yield rate is that rate of interest at which the

present value of returns from the investment is equal to the present value of contributions into the investment. It is often called the internal rate of return.

Dollar-weighted rate of interest

Consider finding the effective rate of interest earned by a fund over one measurement period. We make the

following definitions:

• A = the amount in the fund at the beginning of the period

• B = the amount in the fund at the end of the period

• C_t = the net amount of principal contributed at time t (positive or negative), where 0 ≤ t ≤ 1

• C = the total net amount of principal contributed during the period

C = X

t

C_t

• _ai_b = the amount of interest earned by 1 invested at time b over the following period of length a, where a ≥ 0, b ≥ 0, and a + b ≤ 1

Note that

B = A + C + I

I = iA + X

t

C_t · _1−ti_t.

Assuming compound interest throughout the period, we have

1−tit = (1 + i)1−t − 1

.

Hence,

i =. I

A + X

t

C_t(1 − t) .

Assume that the net principal contributions occur at time t = 1

2, we have i =. I

A + .5C =

I

A + .5(B − A − I) =

2I

A + B − I If it is known that net principal contributions occur at

time k on the average, then

i =. I

Time-weighted rate of interest

Let the amount of the net contribution to the

fund(Positive or negative) at time t_k be denoted by C_k0 for k = 1, 2, · · · , m − 1.

Let the fund values immediately before each contribution to the fund be denoted by B_k0 for k = 1, 2, · · · , m − 1.

Also the fund value at the beginning of the year is

denoted by B₀0 = B₀, while the fund value at the end of year is denoted by B_m0 = B₁.

time-weighted method are given by

1 + j_k = B

0 k

B_k−0 ₁ + C_k−0 ₁

The overall yield rate for the entire year is then given by 1 + i = (1 + j₁)(1 + j₂) · · · (1 + j_m)

Finding the outstanding loan balance

• L = B₀: The original loan balance.

• B_t: The outstanding loan balance at time t.

• The methods of finding the outstanding loan balance Consider a loan of L = a_n| at interest rate i per period being repaid with payments of 1 a the end of each period for n period.

1. Prospective method

B_tp = a_n−t|

2. Retrospective method

Amortization schedules

L = B₀ = a_n|

I₁ = ia_n| = 1 − vn

P₁ = 1 − (1 − vn) = vn B₁ = a_n| − vn = a_n−1|

I₂ = ia_n−1| = 1 − vn−1

P₂ = 1 − (1 − vn−1) = vn−1 B₂ = a_n−1| − vn−1 = a_n−2|

...

I_t = iB_t−1 = ia_n−t+1| = 1 − vn−t+1 P_t = 1 − (1 − vn−t+1) = vn−t+1

Payment Interest Principal Outstanding

Period amount paid repaid loan balance

0 a_n|

1 1 1 − vn vn a_n−1|

2 1 1 − vn−1 vn−1 a_n−2|

... ... ... ... ...

t 1 1 − vn−t+1 vn−t+1 a_n−t|

... ... ... ... ...

n − 1 1 1 − v2 v2 a_1|

n 1 1 − v v a_1| − v = 0