Present value on a parametric yield curve¶

Course exercise 53 · Part I

Problem¶

The yield curve is given by a quadratic function of maturity. Use it to price a five-year bond paying an annual 6% coupon.

Method¶

The yield to maturity at date \(s\) and the corresponding annual rate are:

with \(\alpha = 0.0024\), \(\beta = 0.0097\), \(\gamma = 0.0033\). The price is \(\sum_k F_k\,(1 + i(s_k))^{-s_k}\).

Code¶

clear all, clc

% Point 1
    % Parameters
    F = [6, 6, 6, 6, 106];
    s = [1, 2, 3, 4, 5];
    alpha = 0.0024;
    beta = 0.0097;
    gamma = 0.0033;

    % Present Value Calculation
    Present_value = F_es53(s, F, alpha, beta, gamma)
    % xlswrite('CashFlows2.xls', Present_Value_F);


function [valore_attuale_flussi] = F_es53(s, F, alpha, beta, gamma)
% inputs:   s := schedule
%           F := the flows
% output:   present_value
% s and F must have the same dimension

h_sk = alpha + beta * s + gamma * s.^2;                 % yield to maturity
i_sk = exp(h_sk) - 1;                                   % rate of interest

valore_attuale_flussi = F * ((1 + i_sk).^(-s))';        % present value of the cash flow (flows times discount factors)
end

import numpy as np


# %% function F_es53

def F_es53(s, F, alpha, beta, gamma):
    # inputs:   s := schedule
    #           F := the flows (same length as s)
    # output:   present value of the flows

    h_sk = alpha + beta * s + gamma * s**2      # yield to maturity
    i_sk = np.exp(h_sk) - 1                     # rate of interest

    valore_attuale_flussi = F @ ((1 + i_sk) ** (-s))   # flows times discount factors
    return valore_attuale_flussi


# %% Point 1
F = np.array([6, 6, 6, 6, 106])
s = np.array([1, 2, 3, 4, 5])
alpha = 0.0024
beta = 0.0097
gamma = 0.0033

Present_value = F_es53(s, F, alpha, beta, gamma)
print("Present_value =", Present_value)

Results¶

Present value: 75.02.

The curve rises steeply: the annual rate is 1.55% at 1 year and 14.27% at 5 years, so the final payment of 106 is discounted heavily.

Takeaways¶

• A parametric curve gives a rate for any maturity with three numbers; the quadratic term dominates at long maturities.
• A 6% bond priced at 75 is consistent with the curve: at 5 years the rate is far above the coupon.