Intensive Actuarial Training for Bulgaria

January 2007

Lecture 1 – Theory of Interest and Applications

By Michael Sze, PhD, FSA, CFA

Objective

• What will you get out of this session? • You will understand how to calculate:

– Mortgage payment – Annuity value– Market value and duration of bonds– This serves as preparation for study of life

annuity

Agenda

• Concept of present value and future value• Annuities• Amortization and amortization schedule• Bonds• Interest sensitivity of each of the above

– Why are they affected by interest rate changes?– How are they affected and how much?

Present Value and Future Value• Assume that the interest rate is i• One dollar at beginning of period will grow to 1 + i at the

end of the period

1 1+ i

1+i is called the future value of 1.

1 is called the present value of (1 + i).

•For P at the beginning of the period, the value at the end of the period = P x (1 + i).

•Conversely, for P at the end of the period, the value at the beginning of the period = P / (1 + i) or P x [1/ (1 + i)].

•If we denote v = [1/ (1 + i)], then the value at the beginning of period = P x v.

Example on Compound Interest

• Starting with $1,000• Interest rate is 5%• Then in one year, it will grow to

$1,000 x (1 + 5%) = $1,050• $1,050 is called the future value of $1,000• $1,000 is called the present value of $1,050• Conversely, given $1,000 at the end of one year.• The present value of that is $1,000 / (1 + 5%) =

$952.38. We denote v = 1/(1+5%).

Compound Interest (Continued)• Starting with $1,000 with 5% interest • Future value (FV) at the end of two years equals

$1,000x(1+5%)x(1+5%)=$1,000x(1+5%)^2=$1,102.50• Future value at the end of three years equals

$1,000x(1+5%)^3=$1,157.63• Conversely, present value (PV) of $1,000 paid at the end of

one year = $1,000/(1+5%)= $1,000 x v = $952.38 • Present value of $1,000 paid at the end of two years =

$1,000/(1+5%)^2 = $1,000 x v^2 = $907.03• Present value of $1,000 paid at the end of three years =

$1,000/(1+5%)^3 = $1,000 x v^3 = $863.84

More Frequent Payments• Very often, input are provided on an annual

basis, however, payments are made more frequently, e.g. monthly payments

• Easiest way to deal with the problem is to– Calculate the monthly interest (/ 12)– Calculate the monthly payment (/ 12)– Calculate number of monthly payments (x 12)

• Then compute annuity values in the usual manner

More Frequent Compounding• Nominal interest rate is interest rate for a year,

assuming no compounding during the year• If there are m compounding each year, we

represent the nominal interest rate by i(m)

• The effective interest rate i is higher:• 1 + i = ( 1 + i(m)/m)m

• Hint: Always find equivalent interest, then the problem is the same as no special compounding

• When compounding happens every n years, just treat n years as one measurement period

Annuities• Annuities are just a stream of n payments,

of the same amount paid in each period

1 1 1 1 1 1 1

PV 1 v v2 v3 v4 vn-2 vn-1

Value of an annuity is just the sum of these present values.

Denote value of an annuity with payments at the beginning of each period by Ab. As far as the value of money is concerned, to have Ab upfront is the same as having a stream of n payments of 1 paid at the beginning of each period, and vice versa. In symbols,

Ab = 1 + v + v2 + v3 + v4 +…+ vn-2 + vn-1. If periodic payments are made at the end of each period, the value Ae is given by:

Ae = v + v2 + v3 + v4 +…+ vn-1 + vn.

Annuity Immediate, Annuity Due

• When there are n periods with one payment for each period

• Annuity due AD is sum of PV for the payments if the payments are made at beginning of each period

• AD = 1 + v + v2 +…+ v(n-1)

• Annuity immediate is sum of PV for payments at the end of each period

• AI = v + v2 + … + vn

Perpetuity

• Perpetuity is annuity with infinite payments

• Perpetuity immediate, PI, is given by• PI = 1 / i• Perpetuity due, PD, is given by• PD = 1/ d

Mortgage

• Buy a car, a house, etc., by installment: – Borrow a lump sum amount upfront Ae– Repay by a stream of n periodic payments – The values of the lump sum borrowed must

equal the value of the periodic payments• We shall do the detailed calculations

Mortgage (continued)

Determination of monthly mortgage payment• Monthly interest i = nominal interest / 12• Number of payment periods n = mortgage

period (in years) x 12• Calculate AI(i,n) by the usual method• Loan balance LB = purchase price x (1– dp)

– dp is down payment percentage, e.g. 20%• Monthly mortgage payment = LB / AI(i,n)

Amortization Schedule• A spread sheet showing the progress of the

payments of a loan each payment period• Input items

– Initial loan balance LB0

– Amount of payment P– Interest rate i– Number of payment periods n

• Items shown for each period t– Amount of payment P– Interest charge IC– Principal payment PP– Loan balance at the end of the period LBt

Amortization Schedule(continued)

Calculations of the items for period t• Start with the LB(t-1) at end of last year• Interest charge IC = LB(t-1) x interest rate i• Principal payment PP = total payment P- IC • Loan balance at the end of period t

LBt = LB(t - 1) – PP• Detailed schedule will is shown in spread sheet

Bonds• Debts:

– Government, or large corporation, (called issuer of bond) borrows from buyers (bond holders)

– Amount borrowed (face value of bond)– Fixed interest rate (coupon rate), semiannual – Time for repayment of principle (maturity date)– Note: the coupon rate is usually different from

the interest rate that the investment market pays

Price of Bonds

• Market value of bond = present value of coupon payments and the principal payment on maturity

• Market value of a bond is affected by market interest rate:– When market interest rate is higher than the coupon

rate, bond price must drop to attract buyers– When market interest rate is lower than the coupon

rate, bond price will increase

Factors Affecting Bond Price• MV increases with FV, and c• MV increases with n• MV decreases as i increases• For a specific bond, FV, c, and n are fixed• However, i fluctuates all the time• It is important to know how MV varies with i• This is the study of interest sensitivity of bonds• It is called the modified duration of bonds

Duration of Bonds

• Measure of interest rate sensitivity of a bond• The % change in bond price for 1% change in

market interest rate• Calculation:

– Change market interest rate a tiny bit (say .01%)– Find change % in the value of the bond as % of

original bond price– Interest sensitivity = % / .01%– Interest sensitivity is called modified duration of bond

Modified Duration of Bond• Modified duration D of bond = % change in

MV for each unit change in i• D = - [(MV / i) / MV] • Since the partial derivative is negative, D is

always expressed in positive• Macaulay duration = (xvx)/(vx)• Modified duration = Macaulay duration/(1+i)• We shall explain the concept through several

special cases

Convexity of Bond

• Convexity is rate of change of duration with respect to change of interest i

• Convexity = (MV+ + MV- - 2MVo ) /(2MVox (i)2)• It is a measure of the curvature of the yield

curve• It is positive for bond with no call option

Case 1 – Strip Bond with Only 1 Payment FV at End of n Periods

• MV = FV x (1 + i)-n

MV/i = - n FV (1 + i)-(n+1)

• (MV/i)/MV = - n /(1 + i)• The modified duration D = - n/(1 + i)• Observations

– The sign is negative: as i increases, MV decreases– For a strip bond, the duration is equal to the

maturity period n divided by (1 + i)

Case 2 – 2 bullet payments FV/2 each, paid at n, m

• MV = (FV/2) x [(1+i)-n + (1+i)-m] MV/i=(FV/2)x[-n(1+i)-n -m(1+i)-m]/(1+i)• (MV/I)/MV= -[n(1+i)-n+m(1+i)-m]

/[(1+i)-n+(1+i)-m]/(1+i)• Observations

– D is weighted average between n and m– Where the weights are related to the periods of the

payments– The shorter the payment period, the heavier the

weight

Case 3 – Mortgage Payments P at End of Each of n Periods

• MV = P x [(1+i)-1 +(1+i)-2 +…+ (1+i)-n]MV/i=-P x[(1+i)-1+2(1+i)-2+…+n(1+i)-n]

/(1+i)• (MV/i)/MV = - [(1+i)-1+2(1+i)-2 +…+

n(1+i)-n]/[(1+i)-1+(1+i)-2+…+(1+i)-n]/(1+i)• Observations

– D is weighted average of 1 through n– Weights are heavier for shorter periods

Case 4 – Bond with Very Small Coupons and Large FV

• D is dominated by FV at the end of period n

• D is close to – n/(1+i)

Case 5 – Bond with Large Coupons c and Not so Large FV

• D is more heavily affected by the coupons• D is more closely related to the duration

for that of a mortgage