### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title Theory of Interest Code MATH167 Coordinator Professor DC Constantinescu-Loeffen Mathematical Sciences C.Constantinescu@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2023-24 Level 4 FHEQ Second Semester 15

### Aims

This module aims to provide students with an understanding of the fundamental concepts of Financial Mathematics, and how the concepts above are applied in calculating present and accumulated values for various streams of cash flows. Students will also be given an introduction to financial instruments, such as derivatives and the concept of no-arbitrage.

To teach students:

To understand and calculate all kinds of rates of interest, find the future value and present value of a cash flow and to write the equation of value given a set of cash flows and an interest rate.

To derive formulae for all kinds of annuities.

To understand an annuity with level payments, immediate (or due), payable monthly, (or payable continuously) and any three of present value, future value, interest rate, payment, and term of annuity as well as to calculate the remaining two items.

To calculate the outstanding balance at any point in time.

To calculate a schedule of repayme nts under a loan and identify the interest and capital components in a given payment.

To calculate a missing quantity, being given all but one quantities, in a sinking fund arrangement.

To calculate the present value of payments from a fixed interest security, bounds for the present value of a redeemable fixed interest security.

Given the price, to calculate the running yield and redemption yield from a fixed interest security.

To calculate the present value or real yield from an index-linked bond.

To calculate the price of, or yield from, a fixed interest security where the income tax and capital gains tax are implemented.

To calculate yield rate, the dollar-weighted and time weighted rate of return, the duration and convexity of a set of cash flows.

### Learning Outcomes

(LO1) Ability to understand, communicate, and solve straightforward problems and calculated quantities in the theory of interest.

(LO2) Ability to apply concepts and methods of theory of interest to well defined contexts, and interpret results.

### Syllabus

1. Time value of money

Simple interest, compound interest, force of interest, the effective and nominal rates of interest, accumulated and discount factors, m-thly convertible rate of interest, future value, present value/net present value, inflation and real rate of interest and the general cash flow.

2. Annuities

Present value and accumulated value of annuity-immediate, annuity-due, annuity-immediate/due payable m-thly, continuous annuity and respective deferred annuities, increasing/decreasing annuities-immediate/due, increasing continuous annuities-immediate/due, and respective deferred annuities as well as perpetuity.

3.Loans and the equation of value

Principal, interest, term of loan, outstanding balance, final payment (drop payment, balloon payment), amortization, sinking fund, the prospective and the retrospective methods as well as the equation of value for certain and uncertain payments and receipts, respectively.

4. Cash flow models &am p; Investment projects

Yield rate, current value, duration, cash flow techniques for investment projects, the time-weighted/money-weighted rate of return, spot rates, forward rates, yield curve, convexity, portfolio and investment year allocation methods.

5. Bonds, Fixed interest security and index-linked security

Including the following concepts:

-premium, redemption value, par value, term of bond,

-present value of payments from a fixed interest securities when the redemption is in one instalment and the coupon rate is constant,

-upper and lower bounds for the present value of a fixed interest security that is redeemable on a single a date within a given range,

-calculate price/running yield/redemption yield of a fixed interest security, index-linked bonds under a rate of inflation,

-arbitrage,

-price a forward contract in an arbitrage free environment,

-the value of a forward rate at any time during the term of the contra ct,

-hedging implied by a forward contract.

6.Term structure of interest rates & Stochastic interest rates models

Par yield, yield to maturity, immunisation, explaining the use of duration and convexity in the (Redington) immunisation of a portfolio of liabilities, the stochastic interest rate model, calculating the mean and variance of the accumulated amount of a single premium/an annual premium, the distribution functions for the accumulated amount of a single premium and for the present value of a sum due at a given specified future time provided (1+i) is log-normally distributed, calculating the probability that a simple sequence of payments will accumulate to a given amount at a specific future time.

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Final Assessment Open book and remote Assessment Schedule: Semester 2  120    70
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Class Test  60    30